The cost of the mission is approximately US$800 million  not including the Atlas V launch vehicle, which is about US$183.5 million.  It is the third planetary science mission selected in the New Frontiers program, after Juno and New Horizons . The principal investigator is Dante Lauretta from the University of Arizona.
Juno is a NASA space probe orbiting the planet Jupiter. It was built by Lockheed Martin and is operated by NASA's Jet Propulsion Laboratory. The spacecraft was launched from Cape Canaveral Air Force Station on August 5, 2011 (UTC), as part of the New Frontiers program, and entered a polar orbit of Jupiter on July 5, 2016, to begin a scientific investigation of the planet. After completing its mission, Juno will be intentionally deorbited into Jupiter's atmosphere.
A time capsule is a historic cache of goods or information, usually intended as a method of communication with future people and to help future archaeologists, anthropologists, or historians. Time capsules are sometimes created and buried during celebrations such as a world's fair, a cornerstone laying for a building, or at other events.
Rehearsals will be performed before the sampling event, during which the solar arrays will be raised into a Y-shaped configuration to minimize the chance of dust accumulation during contact and provide more ground clearance in case the spacecraft tips over (up to 45°) during contact.  The descent will be very slow to minimize thruster firings prior to contact in order to reduce the likelihood of asteroid surface contamination by unreacted hydrazine propellant. Contact with the surface of Bennu will be detected using accelerometers, and the impact force will be dissipated by a spring in the TAGSAM arm.
The OSIRIS-REx Laser Altimeter (OLA) is a scanning and LIDAR instrument that will provide high resolution topographical information throughout the mission.  The information received by OLA creates global topographic maps of Bennu, local maps of candidate sample sites, ranging in support of other instruments, and support navigation and gravity analyses.
The OSIRIS-REx Visible and IR Spectrometer (OVIRS) is a spectrometer, which measures light to provide mineral and organic spectral maps and local spectral information of candidate sample sites.  It also provides full-disc asteroid spectral data, global spectral maps (20 m resolution), and spectra of the sample site (blue to near-infrared, 400–4,300 nm, with a spectral resolution of 7.5–22 nm).  These data will be used in concert with OTES spectra to guide sample-site selection. These spectral ranges and resolving powers are sufficient to provide surface maps of mineralogical and molecular components including carbonates, silicates, sulfates, oxides, adsorbed water and a wide range of organic compounds. It provides at least two spectral samples per resolution element taking full advantage of the spectral resolution.
The Regolith X-ray Imaging Spectrometer (REXIS) will provide an X-ray spectroscopy map of Bennu to constrain the element abundances. It complements core OSIRIS-REx mission science.  REXIS is a collaborative development by four groups within Massachusetts Institute of Technology (MIT) and Harvard University, with the potential to involve more than 100 students throughout the process. REXIS is based on flight heritage hardware, thereby minimizing elements of technical risk, schedule risk, and cost risk.
The sample-return system, called Touch-And-Go Sample Acquisition Mechanism (TAGSAM), consists of a sampler head with an articulated 3.35-meter (11 ft) arm.   An on-board nitrogen source will support up to three separate sampling attempts for a minimum total amount of 60 grams (2.1 oz) of sample. The surface contact pads will also collect fine-grained material.
Hayabusa2 is an asteroid sample-return mission operated by the Japanese space agency, JAXA. It follows on from Hayabusa mission which returned asteroid samples in 2010. Hayabusa2 was launched on 3 December 2014 and rendezvoused with near-Earth asteroid 162173 Ryugu on 27 June 2018. It is in the process of surveying the asteroid for a year and a half, departing in December 2019, and returning to Earth in December 2020.
Michael Julian Drake, Regent's Professor, was the Director of the University of Arizona's Lunar and Planetary Laboratory and Head of the Department of Planetary Sciences. He was the principal investigator of the Origins Spectral Interpretation Resource Identification Security Regolith Explorer (OSIRIS-REx) mission of NASA's New Frontiers Program. The OSIRIS-REx mission, launched on September 8, 2016 and scheduled to arrive at Asteroid Bennu in December 2018, is the most ambitious University of Arizona planetary science project to date and will retrieve a sample of the asteroid and return it to Earth. He also made significant contributions to the study of HED meteorites and studied the origin of water in terrestrial planets.
The Planetary Missions Program Office is a division of NASA headquartered at the Marshall Space Flight Center, formed by the agency's Science Mission Directorate (SMD). Succeeding the Discovery and New Frontiers Program Office, it was established in 2014 to manage the Discovery and New Frontiers programs of low and medium-cost missions by third-party institutions, and the Solar System Exploration program of NASA-led missions that focus on prioritized planetary science objectives. The Discovery and New Frontiers programs were established in 1992 and 2001 respectively, and have launched fourteen primary missions together, along with two missions launched under the administration of the Planetary Missions Program Office. The Solar System Exploration Program was established alongside the office, with three missions planned for launch under the new program.
1 2 3 4 Brown, Dwayne C. (25 May 2011). "NASA To Launch New Science Mission To Asteroid In 2016". NASA. Retrieved 18 September 2016.
1 2 3 4 5 6 7 8 9 10 11 12 13 "OSIRIS-REx: Asteroid Sample Return Mission" (PDF) (Press Kit). NASA. August 2016. Retrieved 18 September 2016.
1 2 3 Graham, William (8 September 2016). "Atlas V begins OSIRIS-REx's round trip to the asteroid Bennu". NASA Spaceflight. Retrieved 18 September 2016.
1 2 3 Ray, Justin (9 September 2016). "OSIRIS-REx probe launched to asteroid in compelling search for the origins of life". Astronomy Now. Retrieved 18 September 2016.
↑ "NASA'S OSIRIS-REx Spacecraft Arrives at Asteroid Bennu". NASA. 3 December 2018. Retrieved 6 December 2018.
Brown, Dwayne; Neal-Jones, Nancy (31 March 2015). "RELEASE 15-056 - NASA's OSIRIS-REx Mission Passes Critical Milestone". NASA . Retrieved 4 April 2015.
Chang, Kenneth (5 September 2016). "NASA Aims at an Asteroid Holding Clues to the Solar System's Roots". The New York Times . Retrieved 6 September 2016.
Corum, Jonathan (8 September 2016). "NASA Launches the Osiris-Rex Spacecraft to Asteroid Bennu". The New York Times . Retrieved 9 September 2016.
Chang, Kenneth (8 September 2016). "The Osiris-Rex Spacecraft Begins Chasing an Asteroid". The New York Times . Retrieved 9 September 2016.
↑ "Canadian Space Milestones". www.asc-csa.gc.ca. 2007-12-04. Retrieved 2018-12-19.
↑ "About OSIRIS-REx". www.asc-csa.gc.ca. 2018-08-06. Retrieved 2018-12-19.
1 2 Chang, Kenneth (December 3, 2018). "NASA's Osiris-Rex Arrives at Asteroid Bennu After a Two-Year Journey". The New York Times . Retrieved December 3, 2018.
↑ "NASA Aims to Grab Asteroid Dust in 2020". Science Magazine. 26 May 2011. Archived from the original on 29 May 2011. Retrieved 26 May 2011.
↑ Buck, Joshua; Diller, George (5 August 2013). "NASA Selects Launch Services Contract for OSIRIS-REx Mission". NASA . Retrieved 8 September 2013.
1 2 3 4 Kramer, Herbert J. "OSIRIS-REx". Earth Observation Portal Directory. Retrieved 20 April 2015.
1 2 Hille, Karl (9 January 2018). "NASA Selects Participating Scientists for Mission to Asteroid Bennu". NASA. Retrieved 2 February 2018.
↑ Graham, William (8 September 2016). "Atlas V begins OSIRIS-REx's round trip to the asteroid Bennu". NASASpaceFlight.com .
1 2 Wall, Mike. "'Exactly Perfect'! NASA Hails Asteroid Sample-Return Mission's Launch". Space.com. Retrieved 10 September 2016.
↑ "OSIRIS-REx Mission & Trajectory Design". Spaceflight101.com. Sep 2016.
1 2 3 Neal-Jones, Nancy (17 January 2017). "Successful Deep Space Maneuver for NASA's OSIRIS-REx Spacecraft". NASA. Retrieved 7 March 2017.
1 2 Clark, Stephen (1 February 2017). "NASA's OSIRIS-REx probe moonlights as asteroid sleuth". Spaceflight Now. Retrieved 9 March 2017.
↑ "OSIRIS-REx Asteroid Search Tests Instruments". NASA . Retrieved 2018-12-20.
↑ Morton, Erin; Neal-Jones, Nancy (9 February 2017). "NASA's OSIRIS-REx Begins Earth-Trojan Asteroid Search". NASA. Retrieved 9 March 2017.
↑ "NASA's OSIRIS-REx Takes Closer Image of Jupiter". NASA. 15 February 2017. Retrieved 9 March 2017.
↑ Morten, Eric (31 December 2018). "NASA's OSIRIS-REx Spacecraft Enters Close Orbit Around Bennu, Breaking Record". NASA . Retrieved 1 January 2019.
↑ NASA's OSIRIS-REx Spacecraft Enters Close Orbit Around Bennu, Breaking Record. Lonnie Shekhtman, OSIRIS-Rex Mission home page. 31 December 2018.
↑ [https://www.asteroidmission.org/asteroid-operations/#detailedsurveybaseballdiamond Orbital B Phase. Osiris-Rex. Accessed on 22 March 2018.
↑ Lauretta, Dante (2014-02-05). "How Do We know When We Have Collected a Sample of Bennu?". Dslauretta.com. Retrieved 23 August 2016.
↑ "Sample Return Capsule". Spaceflight101.com. Retrieved 25 October 2017.
1 2 Davis, Jason (5 July 2018). "What's the benefit of sample return?". The Planetary Society. Retrieved 2 September 2018.
↑ "OSIRIS-REx Project". JAXA/Astromaterial Science Research Group. Retrieved 2 September 2018.
↑ "OSIRIS-REx". NASA/Astromaterials Research & Exploration Science Directorate. Retrieved 2 September 2018.
1 2 Wolchover, Natalie (27 May 2011). "NASAcronyms: How OSIRIS-REx Got Its Name". LiveScience. Retrieved 12 May 2015.
↑ Moskowitz, Clara (27 May 2011). "Why NASA Chose Potentially Threatening Asteroid for New Mission". Space.com. Retrieved 14 May 2017.
↑ Müller, T. G.; O'Rourke, L.; Barucci, A. M.; Pál, A.; Kiss, C.; Zeidler, P.; Altieri, B.; González-García, B. M.; Küppers, M. (December 2012). "Physical properties of OSIRIS-REx target asteroid (101955) 1999 RQ36. Derived from Herschel, VLT/ VISIR, and Spitzer observations". Astronomy & Astrophysics. 548. A36. arXiv: 1210.5370 . Bibcode:2012A&A...548A..36M. doi:10.1051/0004-6361/201220066.
↑ "Earth Impact Risk Summary for 101955 Bennu". Near Earth Object Program. NASA's JPL. 5 August 2010. Retrieved 29 April 2013.
↑ "OSIRIS-REx - The Mission". Asteroidmission.org.
↑ Lauretta, Dante (16 December 2014). "Integration of the OSIRIS-REx Main Propellant Tank". Dslauretta.com. Retrieved 20 April 2015.
1 2 3 4 5 6 7 "Instruments: Science Payload". University of Arizona. Retrieved 18 September 2016.
↑ Lauretta, Dante (11 January 2014). "OCAMS – The Eyes of OSIRIS-REx". Dslauretta.com. Retrieved 10 September 2016.
↑ "OLA, Canada's Contribution to OSIRIS-REx". Canadian Space Agency. 4 March 2013. Retrieved 15 October 2014.
↑ Simon-Miller, A. A.; Reuter, D. C. (2013). OSIRIS-REx OVIRS: A Scalable Visible to Near-IR Spectrometer for Planetary Study (PDF). 44th Lunar and Planetary Science Conference. 18–22 March 2013. The Woodlands, Texas. Bibcode:2013LPI....44.1100S.
↑ Lauretta, Dante (27 November 2013). "How To Get To Bennu and Back". Dslauretta.com. Retrieved 10 September 2016.
↑ Elifritz, T. L. (2012). OSIRIS-REx II to Mars - Mars Sample Return from Phobos and Deimos (PDF). Concepts and Approaches for Mars Exploration. 12–14 June 2012. Houston, Texas. Bibcode:2012LPICo1679.4017E.
↑ Templeton, Graham (31 May 2016). "OSIRIS-REx is about to go collect (and return) samples from an asteroid". ExtremeTech. Retrieved 24 November 2016.
↑ Dunn, Marcia (8 January 2019). "Asteroid-circling spacecraft grabs cool snapshot of home width2=200". AP News . Retrieved 8 January 2019.
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H&R Block Announces Fiscal 2013 Results
Releasedf87550e-dfb4-4b85-aa4f-9d6f5b0367e8_1829364.pdf 69.2 KB
KANSAS CITY, MO -- (Marketwired) -- 06/12/13 -- H&R Block, Inc. (NYSE: HRB)
Earnings per share(1) from continuing operations of $1.69, up 46% from prior year(2)
Total revenues increase 0.4% to $2.9 billion
Successful cost reduction initiatives contribute to $126 million, or 22%, increase in pretax earnings from continuing operations
EBITDA increased 15% to $874 million, or 30% of revenues(3)
H&R Block, Inc. (NYSE: HRB), the world's largest consumer tax services provider, today announced its financial results for the fiscal year ended April 30, 2013. Earnings per share from continuing operations increased 46 percent to $1.69. On an adjusted non-GAAP basis, earnings per share from continuing operations increased 25 percent to $1.59, largely driven by the successful execution of the company's previously announced cost reduction initiatives. Total revenues increased 0.4 percent to $2.9 billion, while total expenses fell 5 percent to $2.2 billion.
The 2013 U.S. tax season experienced unprecedented challenges, including significant tax legislation changes that occurred shortly before the traditional opening of the tax season, delays related to the opening of the Internal Revenue Service's (IRS) e-file system, and increased fraud controls at the IRS affecting several forms, among other matters. Based on data through April 30, the company estimates total filings at the IRS during tax season 2013 decreased approximately 0.6 percent to 133 million returns, and that the company maintained its share of total U.S. tax returns. International returns increased 1.3 percent to 3.3 million. Total tax returns prepared worldwide by and through H&R Block were 25.4 million in fiscal 2013.
CEO Perspective
"Considering the challenges the industry faced this tax season, we're pleased to have executed well and delivered improved profits," said Bill Cobb, H&R Block's president and chief executive officer. "While there is opportunity for improvement, we remain committed to our long-term strategy of balancing client acquisition with earnings growth. Consistent with this strategy, we made a number of decisions this year to optimize our promotional offerings and distribution channels in both the assisted and digital do-it-yourself categories. Though some of these actions negatively impact total client volume, we improved overall profitability, while maintaining our overall share of the U.S. market. We also gained share for the third consecutive year in the important digital online category," added Cobb.
Fiscal 2013 Results From Continuing Operations
Actual Adjusted*
in millions, except EPS Fiscal
2013 Fiscal
Revenue $2,906 $2,894 $2,906 $2,894
EBITDA* $874 $757 $883 $808
Pretax Income $702 $576 $710 $626
Net Income $465 $346 $437 $380
Weighted Avg. Shares - Diluted 274.4 298.6 274.4 298.6
EPS $1.69 $1.16 $1.59 $1.27
*Adjusted amounts and EBITDA (earnings before interest, taxes, depreciation and amortization) are non-GAAP financial measures. See "About Non-GAAP Financial Measures" below for more information regarding financial measures not prepared in accordance with generally accepted accounting principles (GAAP).
CFO Perspective
"While we would have preferred to see stronger revenue growth, I'm pleased that we remained disciplined and exceeded our previously stated goal of reducing costs $85 to $100 million," said Greg Macfarlane, H&R Block's chief financial officer. "Despite the industry challenges we faced, we were able to deliver modest revenue growth along with significant expansion in our EBITDA margin, which increased four full points to 30%. We remain committed to providing shareholder return, evidenced by share repurchases and dividends totaling $532 million this fiscal year, and appreciation in our stock price that outpaced the S&P 500 benchmark during the last 12 months."
Business Segment Results and Highlights
Revenues increased 0.5 percent to $2.9 billion, primarily due to changes in promotional offerings such as the Free Refund Anticipation Check (RAC) and an increase in digital online filings
U.S. assisted tax preparation fees and royalties declined 1 percent to $2.0 billion due to a 2.7 percent decrease in assisted returns prepared, partially offset by a 1.7 percent increase in price
Revenues related to core financial services increased 7 percent to $317 million, primarily due to the discontinuation of the Free RAC promotion in fiscal 2013, partially offset by lower Emerald Card fees compared to the prior year
International revenue increased 7 percent to $249 million, with strong performance in both Canada and Australia
Total expenses declined 4.7 percent to $2.1 billion, driven by previously announced cost reduction initiatives that included lower compensation and occupancy costs.
Adjusted, non-GAAP pretax income improved 9 percent to $823 million, primarily due to successful cost reduction initiatives mentioned above
Pretax loss improved by $9 million to $119 million, due to lower interest expense resulting from the refinancing of our medium term notes during this fiscal year. Additionally, the provision for loan loss decreased reflecting lower delinquencies in the mature and declining legacy mortgage loan portfolio noted below.
Net balance of mortgage loans held for investment declined $67 million to $339 million, while provision for loan losses declined 45 percent to $13.3 million
Effective tax rate from continuing operations improved approximately 6.2 points to 33.7%, driven by a $43 million tax benefit related to the settlement of substantially all outstanding issues in our 1999 through 2007 tax returns with the IRS
Net loss of $31 million compared to $80 million net loss in prior year as fiscal 2012 results included a loss on the sale of RSM McGladrey
Sand Canyon Corporation (SCC), a separate legal entity from H&R Block, Inc., received new claims during the quarter for alleged breaches of representations and warranties in the principal amount of $23 million
SCC reviewed claims in the principal amount of $26 million during the quarter, all of which were deemed invalid
During the latter half of fiscal 2013, SCC entered into tolling agreements with certain counterparties from which SCC had received a significant majority of its asserted claims. During the fourth quarter SCC engaged in settlement discussions with these counterparties related to previously denied and future claims. Based on these actions and other considerations, SCC recorded a provision of $40 million during the fourth quarter, increasing its accrual for contingent losses related to representations and warranties to $159 million at April 30.
As of April 30, 2013 the company had unrestricted cash of $1.7 billion and total outstanding debt of $0.9 billion
Shareholder equity at April 30 was $1.3 billion
Share Repurchases and Dividends
During fiscal 2013, the company repurchased and retired 21.3 million shares at an aggregate price of $315.0 million, or $14.82 per share. The purchase represented an approximate 8 percent reduction in shares outstanding. As of April 30, 2013, 272.6 million shares remained outstanding.
A previously announced quarterly cash dividend of 20 cents per share is payable on July 1, 2013 to shareholders of record as of June 17, 2013. The July 1 dividend payment will mark H&R Block's 203rd consecutive quarterly dividend since the company went public in 1962.
At 4:30 p.m. Eastern on June 12, 2013, the company will host a conference call for analysts, institutional investors and shareholders. To access the call, please dial the number below approximately 5 to 10 minutes prior to the scheduled starting time:
U.S./Canada (877) 809-6980 or International (706) 758-0071
The call will also be webcast in a listen-only format for the media and public. The link to the webcast can be accessed directly at http://investors.hrblock.com.
A replay of the call will be available beginning at 6:30 p.m. Eastern on June 12, 2013, and continuing until July 12, 2013, by dialing (855) 859-2056 (U.S./Canada) or (404) 537-3406 (International). The conference ID is 57796144. The webcast will be available for replay June 13, 2013 at http://investors.hrblock.com.
H&R Block, Inc. (NYSE: HRB) is the world's largest consumer tax services provider. More than 600 million tax returns have been prepared worldwide by and through H&R Block since 1955. In fiscal 2013, H&R Block had annual revenues of $2.9 billion with 25.4 million tax returns prepared worldwide. Tax return preparation services are provided in company-owned and franchise retail tax offices by approximately 90,000 professional tax preparers, and through H&R Block At Home™ digital products. H&R Block Bank provides affordable banking products and services. For more information, visit the H&R Block Online Press Center.
About Non-GAAP Financial Measures
This press release and the accompanying tables include non-GAAP financial measures. For a description of these non-GAAP financial measures, including the reasons management uses each measure, and reconciliations of these non-GAAP financial measures to the most directly comparable financial measures prepared in accordance with generally accepted accounting principles, please see the section of the accompanying tables titled "About Non-GAAP Financial Measures."
This press release may contain forward-looking statements within the meaning of the securities laws. Forward-looking statements can be identified by the fact that they do not relate strictly to historical or current facts. They often include words or variation of words such as "expects," "anticipates," "intends," "plans," "believes," "seeks," "estimates," "projects," "forecasts," "targets," "would," "will," "should," "could" or "may" or other similar expressions. Forward-looking statements provide management's current expectations or predictions of future conditions, events or results. All statements that address operating performance, events or developments that we expect or anticipate will occur in the future are forward-looking statements. They may include estimates of revenues, income, earnings per share, capital expenditures, dividends, liquidity, capital structure or other financial items, descriptions of management's plans or objectives for future operations, products or services, or descriptions of assumptions underlying any of the above. All forward-looking statements speak only as of the date they are made and reflect the company's good faith beliefs, assumptions and expectations, but they are not guarantees of future performance or events. Furthermore, the company disclaims any obligation to publicly update or revise any forward-looking statement to reflect changes in underlying assumptions, factors, or expectations, new information, data or methods, future events or other changes, except as required by law. By their nature, forward-looking statements are subject to risks and uncertainties that could cause actual results to differ materially from those suggested by the forward-looking statements. Factors that might cause such differences include, but are not limited to, a variety of economic, competitive and regulatory factors, many of which are beyond the company's control and which are described in our Annual Report on Form 10-K for the fiscal year ended April 30, 2012 in the section entitled "Risk Factors," as well as additional factors we may describe from time to time in other filings with the Securities and Exchange Commission. You should understand that it is not possible to predict or identify all such factors and, consequently, you should not consider any such list to be a complete set of all potential risks or uncertainties.
(1) All per share amounts are based on fully diluted shares.
(2) Unless otherwise noted, all comparisons, including those made to the "prior year," refer to the current period compared to the prior year period.
(3) EBITDA (earnings before interest, taxes, depreciation and amortization) is non-GAAP financial measures. See "About Non-GAAP Financial Measures" below for more information regarding financial measures not prepared in accordance with generally accepted accounting principles (GAAP).
KEY OPERATING RESULTS
Unaudited, amounts in thousands, except per share data
Three months ended April 30,
Revenues Income (loss)
Tax Services $ 2,193,261 $ 1,994,234 $ 1,156,346 $ 1,015,735
Corporate and Eliminations 6,951 6,440 (26,510 ) (34,109 )
$ 2,200,212 $ 2,000,674 1,129,836 981,626
Income taxes 440,914 389,923
Net income from continuing operations 688,922 591,703
Net loss from discontinued operations (24,582 ) (5,600 )
Net income $ 664,340 $ 586,103
Basic earnings per share:
Continuing operations $ 2.53 $ 2.02
Discontinued operations (0.09 ) (0.02 )
Consolidated $ 2.44 $ 2.00
Basic shares 272,384 293,103
Diluted earnings per share:
Diluted shares 274,715 293,985
Twelve months ended April 30,
Tax Services $ 2,877,967 $ 2,862,378 $ 821,143 $ 704,002
Corporate and Eliminations 27,976 31,393 (119,132 ) (127,932 )
$ 2,905,943 $ 2,893,771 702,011 576,070
Net loss from discontinued operations (31,210 ) (80,036 )
NOTES TO CONSOLIDATED FINANCIAL STATEMENTS
Basic earnings per share is computed using the two-class method and is based on the weighted average number of shares outstanding. The dilutive effect of potential common shares is included in diluted earnings per share, except in those periods with a loss from continuing operations.
On October 25, 2012, we issued $500.0 million aggregate principal amount of our 5.50% Senior Notes due 2022 for aggregate proceeds of $497.2 million. The notes bear interest at 5.50% per annum, subject to adjustment based upon our credit ratings. Interest is payable on May 1 and November 1 of each year beginning on May 1, 2013 until the stated maturity date of November 1, 2022. The notes were issued by our wholly-owned subsidiary, Block Financial LLC (Block Financial), and were fully and unconditionally guaranteed by H&R Block, Inc.
On October 25, 2012, we provided notice to the trustee of our intention to redeem the entire $600.0 million aggregate principal amount of our 7.785% Senior Notes that were due to mature in January 2013. The redemption settled on November 26, 2012 for an aggregate price of $623.0 million, which included full payment of principal, a make-whole premium of $5.8 million and interest accrued up to the redemption date of $17.2 million. Proceeds of the issuance of our 5.50% Senior Notes, together with cash balances on hand, were used to redeem the 7.875% Senior Notes. We recognized a loss on the extinguishment of this debt of $5.8 million in fiscal year 2013, which primarily represents the interest that would have been paid on these notes if they had not been redeemed prior to maturity. This loss is included in other income, net on our consolidated statements of income.
CONSOLIDATED BALANCE SHEETS
April 30, April 30,
Cash and cash equivalents $ 1,747,584 $ 1,944,334
Cash and cash equivalents - restricted 117,837 48,100
Receivables, net 206,835 193,858
Prepaid expenses and other current assets 390,087 314,702
Total current assets 2,462,343 2,500,994
Mortgage loans held for investment, net 338,789 406,201
Investments in available-for-sale securities 486,876 371,315
Property and equipment, net 267,880 240,772
Intangible assets, net 284,439 276,664
Goodwill 434,782 427,566
Other assets 262,670 426,055
Total assets $ 4,537,779 $ 4,649,567
Customer banking deposits $ 936,464 $ 827,549
Accounts payable, accrued expenses and other current liabilities 523,921 567,079
Accrued salaries, wages and payroll taxes 134,970 163,992
Accrued income taxes 416,128 336,374
Current portion of long-term debt 722 631,434
Total current liabilities 2,012,205 2,526,428
Long-term debt 905,958 409,115
Other noncurrent liabilities 356,069 388,132
Total liabilities 3,274,232 3,323,675
Common stock, no par, stated value $.01 per share 3,166 3,979
Additional paid-in capital 752,483 796,784
Accumulated other comprehensive income 10,550 12,145
Retained earnings 1,333,445 2,523,997
Less treasury shares, at cost (836,097 ) (2,011,013 )
Total stockholders' equity 1,263,547 1,325,892
Total liabilities and stockholders' equity $ 4,537,779 $ 4,649,567
CONSOLIDATED STATEMENTS OF INCOME
Three months ended April 30, Twelve months ended April 30,
Service revenues $ 1,884,472 $ 1,717,064 $ 2,443,000 $ 2,434,307
Product and other revenues 274,943 243,547 364,114 359,664
Interest income 40,797 40,063 98,829 99,800
2,200,212 2,000,674 2,905,943 2,893,771
Cost of revenues:
Compensation and benefits 514,731 512,634 769,161 828,773
Occupancy and equipment 107,553 118,122 354,612 381,200
Provision for bad debt and loan losses 39,287 23,734 90,685 92,157
Interest 15,062 22,737 79,957 92,089
Depreciation and amortization of property and equipment 19,081 16,470 68,192 61,390
Other 126,021 112,561 242,181 246,086
821,735 806,258 1,604,788 1,701,695
Impairment of goodwill - 3,152 - 7,409
Selling, general and administrative expenses 251,667 210,231 604,469 618,375
Operating income 1,126,810 981,033 696,686 566,292
Other income, net 3,026 593 5,325 9,778
Income from continuing operations before taxes 1,129,836 981,626 702,011 576,070
Income taxes 440,914 389,923 236,853 230,102
Net income from continuing operations 688,922 591,703 465,158 345,968
Net loss from discontinued operations (24,582 ) (5,600 ) (31,210 ) (80,036 )
Net income $ 664,340 $ 586,103 $ 433,948 $ 265,932
Discontinued operations (0.09 ) (0.02 ) (0.11 ) (0.27 )
Consolidated $ 2.44 $ 2.00 $ 1.59 $ 0.89
Basic shares 272,384 293,103 273,057 297,863
Diluted shares 274,715 293,985 274,359 298,601
Unaudited, amounts in thousands
Net cash provided by operating activities $ 497,108 $ 362,049
Purchases of available-for-sale securities (227,177 ) (256,173 )
Sales, maturities and payments received on available-for-sale securities 118,411 66,382
Principal repayments on mortgage loans held for investment, net 44,031 49,142
Purchases of property and equipment, net (113,239 ) (82,457 )
Payments made for acquisitions of businesses and intangibles, net (20,742 ) (15,258 )
Proceeds from sale of businesses, net 3,785 560,499
Franchise loans:
Loans funded (70,807 ) (46,246 )
Payments received 83,445 56,591
Surrender of company-owned life insurance policies 81,125 -
Other, net (9,769 ) 19,387
Net cash provided by (used in) investing activities (110,937 ) 351,867
Repayments of commercial paper (1,214,238 ) (664,167 )
Proceeds from commercial paper 1,214,238 664,167
Repayments of long-term debt (636,621 ) -
Proceeds from issuance of long-term debt 497,185 -
Repayments of FHLB borrowings - (25,000 )
Customer banking deposits, net 103,608 (26,091 )
Dividends paid (217,201 ) (208,801 )
Repurchase of common stock, including shares surrendered (340,413 ) (180,592 )
Proceeds from exercise of stock options, net 25,139 12,275
Other, net (16,238 ) (16,853 )
Net cash used in financing activities (584,541 ) (445,062 )
Effects of exchange rates on cash 1,620 (2,364 )
Net increase (decrease) in cash and cash equivalents (196,750 ) 266,490
Cash and cash equivalents at beginning of the year 1,944,334 1,677,844
Cash and cash equivalents at end of the year $ 1,747,584 $ 1,944,334
Supplementary cash flow data:
Income taxes paid, net $ 155,617 $ 218,444
Interest paid on borrowings 73,559 69,681
Interest paid on deposits 5,665 6,843
Transfers of foreclosed loans to other assets 10,357 10,308
Accrued additions to property and equipment 4,261 801
Accrued purchase of common stock - 22,484
Tax Services Income Statement
Unaudited, dollars in thousands
Tax preparation fees:
U.S. $ 1,712,319 $ 1,749,032
International 220,870 205,466
1,933,189 1,954,498
Royalties 318,386 308,561
Fees from RACs 158,176 132,361
Fees from Emerald Card 98,896 104,143
Fees from POM guarantees 71,355 75,603
Interest and fee income on EAs 59,657 59,660
Other 238,308 227,552
Total revenues 2,877,967 2,862,378
Field wages 654,794 691,680
Other wages 150,306 150,908
Benefits and other compensation 148,492 183,037
Occupancy and equipment 354,430 381,572
Marketing and advertising 270,240 278,231
Depreciation and amortization 92,004 88,836
Bad debt 77,402 68,082
Supplies 40,131 44,236
Impairment of goodwill and intangible assets 3,581 11,389
Total expenses 2,056,824 2,158,376
Pretax income $ 821,143 $ 704,002
Pretax margin 28.5 % 24.6 %
U.S. Tax Operating Data
Fiscal Year to Date through 4/30/13 Fiscal Year to Date through 4/30/12 Percent change
Total returns prepared: (1)
H&R Block Company-Owned Operations 8,907 9,203 -3.2 %
H&R Block Franchise Operations 5,598 5,697 -1.7 %
Total H&R Block Assisted Returns 14,505 14,900 -2.7 %
H&R Block At Home Desktop 2,004 2,158 -7.1 %
H&R Block At Home Online 4,892 4,419 10.7 %
Sub-total 6,896 6,577 4.9 %
H&R Block Free File Alliance 774 861 -10.1 %
Total H&R Block at Home 7,670 7,438 3.1 %
Total H&R Block U.S. Returns 22,175 22,338 -0.7 %
(1) Prior year numbers have been reclassified between company-owned and franchise operations for offices that were refranchised during either year.
Unaudited, amounts in thousands, except per share amounts
April 30, Twelve months ended
EBITDA and Adjusted EBITDA (1) 2013 2012 2013 2012
Net income from continuing operations - as reported $ 688,922 $ 591,703 $ 465,158 $ 345,968
Add back :
Interest expense 15,062 22,737 79,957 92,089
Depreciation and amortization 25,165 23,030 92,407 89,157
481,141 435,690 409,217 411,348
EBITDA from continuing operations 1,170,063 1,027,393 874,375 757,316
Loss contingencies - litigation 364 (4,567 ) (4,579) 22,961
Impairment of goodwill and intangible assets 2,160 3,152 3,581 11,389
Severance 4,310 30,554 4,785 32,474
Loss on extinguishment of debt - - 5,790 -
Gains on sales of tax offices (396 ) (17,742 ) (1,272) (16,601)
6,438 11,397 8,305 50,223
Adjusted EBITDA from continuing operations $ 1,176,501 $ 1,038,790 $ 882,680 $ 807,539
Non-GAAP Pretax Results
Pretax income from continuing operations - as reported $ 1,129,836 $ 981,626 $ 702,011 $ 576,070
Pretax income from continuing operations - as adjusted $ 1,136,274 $ 993,023 $ 710,316 $ 626,293
Non-GAAP After-Tax Results
Add back (net of tax) :
Impairment of goodwill and intangible assets 1,331 1,895 2,203 6,912
Loss on extinguishment of debt 10 - 3,562 -
Gains on sales of tax offices (245 ) (10,770 ) (782) (10,075)
Discrete tax items 5,377 4,932 (33,302) 3,643
9,341 11,764 (28,192) 34,123
Net income from continuing operations - as adjusted $ 698,263 $ 603,467 $ 436,966 $ 380,091
(1) Earnings before interest, taxes, depreciation and amortization.
Non-GAAP EPS 2013 2012 2013 2012
EPS from continuing operations - as reported $ 2.51 $ 2.01 $ 1.69 $ 1.16
Loss contingencies - litigation - (0.01 ) (0.01) 0.04
Impairment of goodwill and intangible assets - 0.01 0.01 0.02
Severance 0.01 0.06 0.01 0.07
Gains on sales of tax offices - (0.04 ) - (0.03)
Loss on extinguishment of debt - - 0.01 -
Discrete tax items 0.02 0.02 (0.12) 0.01
0.03 0.04 (0.10) 0.11
EPS from continuing operations - as adjusted $ 2.54 $ 2.05 $ 1.59 $ 1.27
Non-GAAP Pretax Results - Tax Services segment
Pretax income - as reported $ 1,156,346 $ 1,015,735 $ 821,143 $ 704,002
Pretax income - as adjusted $ 1,162,255 $ 1,026,120 $ 822,884 $ 753,052
Stock-based compensation expense:
Pretax $ 3,879 $ 3,166 $ 15,293 $ 14,213
After-tax 2,407 1,897 9,408 8,626
Amortization of intangible assets:
After-tax 3,775 3,935 14,896 16,852
The accompanying press release contains non-GAAP financial measures. Non-GAAP financial measures should not be considered as a substitute for, or superior to, measures of financial performance prepared in accordance with GAAP. Because these measures are not measures of financial performance under GAAP and are susceptible to varying calculations, they may not be comparable to similarly titled measures in other companies.
We consider non-GAAP financial measures to be a useful metric for management and investors to evaluate and compare the ongoing operating performance of our business on a consistent basis across reporting periods, as it eliminates the effect of items that are not indicative of our core operating performance.
The following are descriptions of adjustments we make for our non-GAAP financial measures:
We exclude from our non-GAAP financial measures litigation charges we incur and favorable reserve adjustments. This does not include legal defense costs.
We exclude from our non-GAAP financial measures non-cash charges to adjust the carrying values of goodwill, intangible assets, other long-lived assets and investments to their estimated fair values.
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We may consider whether other significant items that arise in the future should also be excluded from our non-GAAP financial measures.
We measure the performance of our business using a variety of metrics, including EBITDA, adjusted EBITDA, adjusted pretax and net income of continuing operations, adjusted EPS and adjusted pretax results of our Tax Services segment. We also use EBITDA and pretax income of continuing operations as factors in incentive compensation calculations for our employees. These adjusted results eliminate the impact of items that we do not consider indicative of our core operating performance and, we believe, provide meaningful information to assist in understanding our financial results, analyzing trends in our underlying business, and assessing our prospects for future performance.
Colby Brown
Media Relations:
Gene King
Source: H & R Block

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

The first-degree based graph entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. In the remaining of the paper, it is just referred to as entropy.
It has attracted significant attention in organic chemistry, as measures of uniformity of a graph's structural aspect of interest \cite{mowshowitz1968entropy,bonchev1983chemical,dehmer2009entropy}. The interpretation of the entropy of a particular graph depends on knowing the extremal values.
In this paper, we continue our work~\cite{CM22+1, CM22+2}
on determining the extremal graphs (and thus extremal values) for the entropy among all graphs satisfying some natural restrictions.
Minimizing (resp. maximizing) the entropy corresponds to maximizing (resp. minimizing) the function $h(G)=\sum_i d_i \log(d_i)$, where $(d_i)_{1 \le i \le n}$ is the degree sequence of the graph $G$,
where $G$ ranges over all members of a class of graphs. This is since the entropy $I(G)$ equals $\log(2m)-\frac 1{2m} h(G),$ where $m$ is the size (number of edges) of $G$.
Here we determine the graphs with extremal entropy among all bipartite graphs with given size $m$, or given size $m$ and (upper bound on the) order $n$.
The maximum value for the entropy is obtained by the graphs for which the degree sequence is as balanced as possible. This is a corollary of Karamata's inequality.
Note that the random graphs $G_{n,m}$ and $G_{n,p}$ are close to balanced, i.e. the maximum degree $\deg_{max}$ and the minimum degree $\deg_{min}$ are nearly the same for $m=\Theta(n^2)$, and thus have an entropy which is almost maximal. This is in line with the intuition that entropy is a measure for randomness.

In graph theory, bipartite graphs are one of the main classes to investigate, because it finds several applications in pure and applied Mathematics.
Hall's Matchings Theorem is a famous theorem on bipartite graphs with several applications in scheduling- and matching problems,
e.g. the kidney matching process.
When edges represent bonds between positive and negative charges, the resulting graph is bipartite.
Furthermore, results on bipartite graphs often lead to results on more general classes of graphs. A famous example is given by the Kahn-Zhao theorem, where Kahn~\cite{Kahn01} considered a problem on the maximum number of independent sets for bipartite graphs and Zhao~\cite{Zhao10} reduced the general case to the bipartite case.

We will show that, given the size $m$, the bipartite graphs attaining the minimum entropy are precisely the complete bipartite graphs of size $m$.
So if $m$ has $\sigma(m)$ divisors, there are $\left\lceil \frac{\sigma(m)}{2} \right\rceil$ non-isomorphic extremal graphs, all which are of the form $K_{q,y}$ with $yq=m.$

\begin{thr}\label{thr:main}
    If $G=(U \cup V,E)$ is a bipartite graph of size $m$, then $h(G) \le m \log m$. Equality occurs if and only if $G$ is a complete bipartite graph.
\end{thr}

When there is an upper bound $n$ on the order (one can extend with isolated vertices since $0\log(0)=0$), such that $m$ cannot be written as a product $yq$ with $y+q \le n$, the problem is harder.
We will prove that the extremal bipartite graphs given the size $m$ and order no more than $n$, are of the form $B(n,m,y)$ for some $1 \le y \le \sqrt{m}$.
Here $B(n,m,y)$ is the graph formed by connecting an additional vertex with $m-yq$ vertices in $V$ of $K_{q,y}=(U \cup V,E),$ where $q=\left\lfloor \frac{m}{y} \right\rfloor.$

We will also consider a different representation of bipartite graphs by means of Young tableaux, as it simplifies the description of the graph operations needed in the proofs.
For a bipartite graph $G=(U \cup V,E)$, let us write $U=\{u_1 , u_2, \ldots, u_x\}$ and $V=\{v_1, v_2, \ldots, v_y\}$ such that $\deg u_i \ge \deg u_j$ and $\deg v_i \ge \deg v_j$ whenever $i \le j.$
Let $y_i=\deg(u_i)$ and $x_j=\deg(v_j)$ for every $1\le i \le x$ and $1\le j \le y.$
We associate the tableau $T$ which contains a cell $(i,j)$ if and only if $u_iv_j \in E.$
Note that this gives a one-to-one correspondence between tableaux and bipartite graphs.

An example of an extremal bipartite graph with $n=10$ and $m=22$ has been represented in these two ways in Figure~\ref{fig:(10,22)}.
Remark that the number of cells in column $i$ corresponds to the degree of $u_i$ and the number of cells in row $j$ equals the degree of $v_j.$

\begin{figure}[h]

\begin{minipage}[b]{.49\linewidth}
\begin{center}
    \begin{tikzpicture}
    {

	\foreach \x in {0,1,2,3,4}{\foreach \y in {1,2,3,4}{\draw[thick] (\x,2) -- (\y,0);}}

	\draw[thick] (5,2) -- (3,0);
	\draw[thick] (5,2) -- (4,0);

		\foreach \x in {0,1,2,3,4,5}{\draw[fill] (\x,2) circle (0.1);
		\foreach \x in {1,2,3,4,5,6}{\node at (\x-1,2.4) {$u_\x$};}
		\foreach \x in {1,2,3,4}{\draw[fill] (\x,0) circle (0.1);
	    \foreach \x in {4,3,2,1}{\node at (5-\x,-0.4) {$v_\x$};}

	}
	\end{tikzpicture}\\
\subcaption{Graph representation}
\label{fig:Graph(10,22)}
\end{center}
\end{minipage}\quad\begin{minipage}[b]{.49\linewidth}
\begin{center}
\ytableausetup{centertableaux}
\begin{ytableau}
  \none[5] &  &  & &  &  & \none \\
  \none[5] &  &  & &  &  & \none \\
  \none[6] &  &  & &  &  & \\
  \none[6] &  &  & &  &  &  \\
  \none & \none[4] & \none[4] & \none[4] & \none[4] & \none[4] & \none[2]
\end{ytableau}\\
\subcaption{Associated Young tableau}
\label{fig:Young(10,22)}
\end{center}
\end{minipage}
\caption{Two representations of the extremal bipartite $(10,22)$-graph $B(10,22,4)$}\label{fig:(10,22)}
\end{figure}

In Section~\ref{sec:bip_case}, we prove that the bipartite graphs minimizing the entropy given the size are exactly the complete bipartite graphs. We give two different approaches to prove this. One of them is a proof by induction, and the other one consists in proving that the associated tableau has to be a rectangle.
In Section~\ref{sec:denseY} we study the possible extremal bipartite graphs given order $n$ and size $m$, by proving that under certain restrictions they are of the form $B(n,m,y)$ and we estimate the value of $h$ for these graphs.
Next, in Section~\ref{sec:proofYdense}, we prove that the extremal graphs are indeed of the form $B(n,m,y).$
For this, we use the equivalent representation with the Young tableaux and give local operations that decrease the entropy (equivalently, such that $h$ increases).
Furthermore, in Section~\ref{sec:chemicalindex} we remark that some of the ideas can be applied to other chemical indices, also called degree-based topological indices, such as e.g. the Second Zagreb index and Reciprocal Randic index.
Just as has been done in~\cite{Cam21}, we notice that certain results hold more generally and we give the essence of the proofs of some known results.
Here, we give short proofs for the main results in~\cite{HLP22} and~\cite{Dong21} about the graphs and bipartite graphs maximizing the entropy in the more general context of a class of chemical indices.
Finally, in Section~\ref{sec:conc}, we give some conclusions on the precise extremal graphs $B(n,m,y)$. We do this based on computational results and the estimates in Section~\ref{sec:denseY}, as well as some number theory.

\subsection{Preliminary definitions and results}\label{subsec:prelimanaryresults}

In this subsection, we define the notions and help functions we will frequently use, as well as give some basic results.

\begin{defi}
    We define the function $f(x)=x\cdot \log (x)$
    for $x\geq 0$. Here $\log$ denotes the natural logarithm and $f(0):=0.$

    For a graph $G$ with degree sequence $(d_i)_{1 \le i \le n}$, we define $h(G)=\sum_i f(d_i).$
    The entropy $I(G)$ of a graph of size $m$ equals $\log(2m)-\frac{1}{2m}h(G).$
\end{defi}

In the following definition of majorization and the inequality of Karamata, we only consider non-increasing sequences.

\begin{defi}
    A sequence $(x_i)_{1 \le i \le n}$ majorizes the sequence $(y_i)_{1 \le i \le n}$ if and only if  $\sum_{1 \le i \le j } x_i \ge \sum_{1 \le i \le j } y_i$ for every $1 \le j \le n$ and equality does hold for $j=n$.
\end{defi}

\begin{thr}[Karamata's inequality, \cite{Kar32}]\label{thr:Kar}
    Let $(x_i)_{1 \le i \le n}$ be a sequence majorizing the sequence $(y_i)_{1 \le i \le n}$.
    For every convex function $g$ we have $\sum_{1 \le i \le n } g(x_i) \ge \sum_{1 \le i \le n } g(y_i).$ Furthermore this inequality is strict if the sequences are not equal and $g$ is a strictly convex function. For concave functions, the same holds with the opposite sign.
\end{thr}

A corollary of Karamata's inequality is the following

\begin{cor}\label{cor:majdeg_kar}
    Let $G$ and $G'$ be two graphs such that the degree sequence of $G'$ majorizes the degree sequence of $G$.
    Then $h(G') \ge h(G).$ Furthermore, equality occurs if and only if the two degree sequences are equal.
\end{cor}

This was also observed by Ghalavand et al.~\cite{GEA19}.

We will also make use of difference graphs, which were introduced by Hammer et al.~\cite{Hammer90}.
The following equivalent characterization is due to Mahadev and Peled~\cite{MP95}.

\begin{thr}[\cite{MP95}]\label{thr:MP95}
    Let $G=(U\cup V, E)$ be a bipartite graph.
    The graph $G$ is a difference graph if and only if one of the following equivalent conditions is true.
\begin{itemize}
      \item[(a)] there are no $u_1,u_2 \in U$ and $v_1,v_2\in V$ such that $u_1v_1,u_2v_2 \in E(G)$ and $u_1v_2,u_2v_1 \notin E(G)$;
      \item[(b)] every induced subgraph without isolated vertices has on each side of the bipartition a domination vertex, that is, a vertex which is adjacent to all the vertices on the other side of the bipartition.
\end{itemize}
\end{thr}

\begin{lem}\label{lem:extr_is_diff_graph}
If $G$ is a graph maximizing $h(G)$ among all bipartite graphs of size $m$, then $G$ is a difference graph.
\end{lem}

\begin{proof}
    Suppose that $G$ is not a difference graph. By Theorem~\ref{thr:MP95}, there are four vertices
    $u_1, u_2\in U$ and $v_1, v_2 \in V$ such that $u_1v_1, u_2v_2 \in E(G)$ and $u_1v_2, u_2v_1 \not \in E(G).$
    Assume without loss of generality that $\deg(u_1) \ge \deg(u_2)$. In that case the degree sequence of the graph $G'=\left(G\backslash \{u_2v_2\}\right) \cup \{u_1v_2\}$ will majorize the degree sequence of $G$ (strictly) and so we get the desired contradiction by Corollary~\ref{cor:majdeg_kar}.
\end{proof}

\begin{lem}\label{lem:max_with_balanced}
    Let $t,\ell>0$ be fixed positive integers.
    Under the condition that $\sum_{j=1}^x z_i =t$ and all $z_i \ge 0$ are integers, $ \sum_{j=1}^x f(z_j+\ell) -\sum_{j=1}^x f(z_j)$ is maximized when $z_1=z_2=\ldots=z_x=\left\lfloor \frac tx \right\rceil.$
\end{lem}

\begin{proof}
    Note that the function $\Delta^{\ell}(z)=f(z+\ell) - f(z)$ is a strictly concave function for every $\ell>0$ and every sequence of $x$ integers with sum $t$ do majorize the sequence with $z_1=z_2=\ldots=z_x=\left\lfloor \frac tx \right\rceil.$
    Now the result follows immediately by Karamata's inequality.
\end{proof}

For the function $\Delta^1$, we write $\Delta$ for ease of notation.
We define this one separately, as it will be used frequently.

\begin{defi}
    The function $\Delta$ is defined by $\Delta(x)=f(x)-f(x-1)=1+\int_{x-1}^{x} \log t \, \mathrm{d}t$.
    This is a strictly increasing and concave function.
\end{defi}

We also use Landau notation, such as $o()$ and $\omega()$.
A function $q(x,y)$ or expression is $o_y(p(x,y))$ or $\omega_y(p(x,y))$ if
$\frac{q(x,y)}{p(x,y)} \to 0$ respectively $\frac{\lvert q(x,y) \rvert }{\lvert p(x,y) \rvert } \to \infty$  when $y \to \infty$.
Sometimes the dependency on $y$ is removed to keep the notation light.

We also use the notation $[k]=\{1,2,\ldots, k\}$ and $[k..\ell]=\{k,k+1,\ldots, \ell-1,\ell\}.$

\section{Minimum entropy of bipartite graphs of fixed size}\label{sec:bip_case}

In this section, we prove Theorem~\ref{thr:main}.
In both approaches, we assume that our bipartite graph is $G=(U \cup V,E)$, $U=\{u_1 , u_2, \ldots, u_x\}$ and $V=\{v_1, v_2, \ldots, v_y\}$ are the vertices with degree at least $1.$
Here we assume these are ordered, i.e. $\deg u_i \ge \deg u_j$ and $\deg v_i \ge \deg v_j$ whenever $i \le j.$
Let $y_i=\deg(u_i)$ and $x_j=\deg(v_j)$ for every $1\le i \le x$ and $1\le j \le y.$
So with this notation, we have $y=y_1$ and $x=x_1.$

\subsection{Approach 1}

\begin{proof}[Proof of Theorem~\ref{thr:main}]
    We prove the statement by induction on $m.$
    The base case $m=1$ is trivial since we only have one edge.
    So assume the statement is true when the size is at most $m-1$ and let $G$ be a graph maximizing $h(G)$ among all graphs of size $m$.
    By Lemma~\ref{lem:extr_is_diff_graph} we can assume that $G$ is a difference graph.
    This implies that $u_1$ is a dominating vertex (dominates $V$). Let $G'=G \backslash u_1$.
    By the induction hypothesis applied to $G'$ and Lemma~\ref{lem:max_with_balanced} applied with $\ell=1$ and $z_i=x_i-1$, we know
    \begin{align*}\label{eq:upperboundh_cnG}
        h(G)&=h(G')+f(y)+\sum_{j=1}^y f(x_j) -\sum_{j=1}^y f(x_j-1)\nonumber \\
        &\le h(K_{1,m-y}) +f(y) + y\left(  f\left( \frac{m}{y} \right) - f\left( \frac{m}{y}-1 \right) \right).
    \end{align*}
    Equality here is only attained if $x_j=\frac{m}{y}$ for every $1\le j \le y.$
    Now the conclusion is direct as
     \begin{align*}
            h(G)\le (m-y)\log(m-y)+y\log(y)+m\left(\log(m)-\log(y)\right)-(m-y)\left(\log(m-y)-\log(y)\right)= m\log m.
    \end{align*}
    By induction, we see that equality is attained precisely for the complete bipartite graphs on $m$ edges.
\end{proof}

\subsection{Approach 2}\label{subsec:approach2}

In this subsection, we give an alternative proof for the fact that $K_{1,m}$ minimizes the entropy among all bipartite graphs of size $m$, which does not use any prerequisites.
Note that the proof can be formulated without the notion of a tableau and that we give a short proof of Lemma~\ref{lem:extr_is_diff_graph} as a claim in this notation.

In every cell $(i,j)$ of the associated tableau $T$ of a bipartite graph $G$, we put $\log(y_ix_j )=\log(x_j)+\log(y_i)$.
The sum over all cells, $h(T)=\sum_{(i,j) \in T} \log(x_j y_i)$ is now exactly equal to $\sum_j f(x_j)+\sum_i f(y_i)=h(G).$

\begin{proof}[Proof of Theorem~\ref{thr:main}]
    We first prove that the associated tableau $T$ is a Young tableau, i.e. if $(i,j) \in T$ and $0<i'\le i$ and $0<j'\le j,$ then $(i',j') \in T.$

    \begin{claim}
        If $G=(U \cup V,E)$ is a bipartite graph maximizing $h(G)$ among all bipartite graphs of size $m$, then its associated tableau is a Young tableau.
    \end{claim}
    \begin{claimproof}
        Assume $(i,j) \in T$ and $(i',j') \not\in T$.
        If $i'<i$ and $j'<j$, then $G'=G \backslash u_iv_j \cup u_{i'}v_{j'}$ satisfies
        $h(G')=h(G)- \Delta(y_i)-\Delta(x_j)+\Delta(y_{i'}+1)+\Delta(x_{j'}+1)>h(G).$ The latter due to $\Delta$ being strictly increasing and $y_i \le y_{i'}$ and $x_j\le x_{j'}.$
        If $i'=i$ or $j'=j$, it is analogous as $h(G')-h(G)$ equals $\Delta(y_{i'}+1)-\Delta(y_i)>0$ or $\Delta(x_{j'}+1)-\Delta(x_j)>0.$
        This implies that $G$ was not extremal and so we conclude by contradiction.
    \end{claimproof}

    Next, we note that
    $$  \sum_{(i,j) \in T} x_j y_i \le \sum_{i,j} x_j y_i = \left(\sum_j x_j \right)\left(\sum_i y_i\right)=m^2.$$
    So by Jensen's inequality, we have
    $$\sum_{(i,j) \in T} \log(x_j y_i) \le m \log\left(\frac{m^2}{m}\right)=m \log(m)$$ with equality if and only if $x_jy_i=m$ for every choice of $(i,j)$, i.e. $G$ is complete, bipartite.
\end{proof}

\section{Minimum entropy among dense Bipartite graphs}\label{sec:denseY}

We state two propositions in terms of the Young tableaux. Note that by taking $x=q+1$, we conclude that $B(n,m,y)$ is extremal in both cases.

\begin{prop}\label{prop:denseTxy}
    Fix integers $x>y>r>0$ and $m=xy-r$.
    Among all Young tableaux with $m$ cells in $[x]\times [y],$
    $h(T)$ is maximized by $$T'=\left([x]\times [y]\right) \backslash \left( x \times [y-r+1..y] \right).$$
    If $x>y=\omega(r^2)$, then the latter tableau has $h(T') \sim m \log m -r+o(1).$
\end{prop}

\begin{proof}
	    Let $T$ be a Young tableau which equals $[x]\times [y],$ except from the $r$ cells which have coordinates $(i_1,j_1),(i_2,j_2), \ldots, (i_r,j_r)$, where these are ordered in reverse lexicographic order. In particular $(i_1,j_1)=(x,y)$ and for every $1 \le k \le r$ the pair $(i_k,j_k)$ satisfies $i_k+j_k \ge x+y+1-k$, $j_k \le y$ and $i_k \le x.$
    	Since $\Delta$ is increasing and concave, by the inequality of Karamata, we have
    	$$ \Delta(i_k) + \Delta(j_k) \ge \Delta(x)+\Delta(y+1-k).$$
    	Now
    	\begin{align*}
    	    h(T)&=xy \log(xy) - \sum_{k=1}^{r} \left( \Delta(i_k) + \Delta(j_k) \right)\\
    	    &\le xy \log(xy) - \sum_{k=1}^{r} \left( \Delta(x)+\Delta(y+1-k) \right)\\
    	    &= h(T'),
    	\end{align*}
    	and furthermore equality occurs if and only if $(i_k,j_k)=(x,y+1-k)$ for every $k$, i.e. $T=T'$.
    	Now since $1+\log(x)-\frac 1{x-1} \le \Delta(x) < 1+\log(x)$ and $1+\log(y) - \frac k{y-k})\le \Delta(y+1-k) < 1+\log(y)$, we note that when $y=\omega(r^2),$ we have
    	\begin{align*}
    	    h(T')&=xy \log(xy) - r \Delta(x)  -\sum_{k=1}^{r} \Delta(y+1-k)\\
    	    &= m\log(m+r)  -2r -o(1)\\
    	    &= m\log(m)-r-o(1),
    	\end{align*}
    	where the $o(1)$ term tends to zero as $x,y \to \infty$ for fixed $r$.
\end{proof}

\begin{prop}\label{prop:denseTbq}
	Fix integers $q\ge y>r>0$ and $m=qy+r$.
	Among all Young tableaux containing $[q]\times [y]$ with $m$ cells,
	$h(T)$ is maximized by $$T'=\left([q]\times [y]\right) \cup \left( (q+1) \times [r] \right).$$
	If $q=\omega(r)$, the latter tableau satisfies $h(T') \sim m\log (m)-r \log\left( \frac{y}{r} \right).$
\end{prop}

\begin{proof}
	Let $T$ be a Young tableau which equals $[q]\times [y],$ with $r$ additional cells which have coordinates $(i_1,j_1),(i_2,j_2), \ldots, (i_r,j_r)$, where these are ordered in lexicographic order. We note that for every $1 \le k \le r$ the pair $(i_k,j_k)$ satisfies $i_k+j_k \le q+k+1$ and $\min\{i_k,j_k\} \le k.$
    Since $\Delta$ is increasing and concave, by the inequality of Karamata, we have
    $$ \Delta(i_k) + \Delta(j_k) \le \Delta(\min\{i_k,j_k\})+\Delta(q+1+k-\min\{i_k,j_k\}) \le \Delta(q+1)+\Delta(k).$$
    	Now     	\begin{align*}
    	    h(T)&=yq \log(yq) + \sum_{k=1}^{r} \left( \Delta(i_k) + \Delta(j_k) \right)\\
    	    &\le yq \log(yq) + \sum_{k=1}^{r} \left( \Delta(q+1)+\Delta(k) \right)\\
    	    &= h(T'),
    	\end{align*}
    	and furthermore equality occurs if and only if $(i_k,j_k)=(q+1,k)$ for every $k$, i.e. $T=T'$.

    	Now since $1+\log(q)\le \Delta(q+1) < 1+\log(q)+\frac 1q$ and $\sum_{k=1}^r \Delta(k)=f(r)=r \log(r)$, we note that when $q=\omega(r),$ we have
    	\begin{align*}
    	    h(T')&=yq \log(yq) + r \Delta(q+1)  +f(r)\\
    	    &=qy \log(q)+ qy\log(y)+r \log(q) +r+f(r) +o(1) \\
    	    &= m\log(m-r)-r \log(y)  +r+r \log(r) +o(1)\\
    	    &= m\log(m)-r\log\left( \frac yr \right)+o(1),
    	\end{align*}
    	where the $o(1)$ term tends to zero when $q \to \infty$ for fixed $r.$
\end{proof}

\section{Minimum is attained by dense Young tableaux}\label{sec:proofYdense}

In this subsection, we list and prove a number of lemmas that show that the extremal Young tableaux are dense, in the sense that they are of the form presented in Section~\ref{sec:denseY}.

We start with proving that the extremal Young tableau cannot have the following specific form.

\begin{figure}[h]
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\begin{tikzpicture}
\draw [fill = medium-gray](0,0) -- (7.5,0) -- (7.5,4) --(0,4) -- cycle;
\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=0pt] (2,4.05) -- (7.5,4.05) node [black,midway,yshift=0.35cm]{$r$};
\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=0pt] (7.55,4.) -- (7.55,2) node [black,midway,xshift=0.35cm]{$s$};
\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=0pt] (7.45,0.05) -- (7.45,1.95) node [black,midway,xshift=-0.35cm]{$t$};
\draw (3.5,-0.35) node { $x$};
\draw (-.35,2) node {$y$};
\draw [fill = white] (2,3.5) -- (7,3.5) -- (7,2) -- (7.5,2)--(7.5,4)-- (2,4) -- cycle;
\end{tikzpicture}
\subcaption{Young tableau $T$}
\label{fig:T}
\end{center}
\end{minipage}\quad
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\begin{tikzpicture}
\draw [fill = medium-gray](0,0) -- (7.5,0) -- (7.5,4) --(0,4) -- cycle;

\draw (3.5,-0.35) node { $x$};
\draw (-.35,2) node {$y$};
\draw [fill = white] (6.5,0) -- (7.5,0) -- (7.5,4) -- (6,4)--(6,3)--(6.5,3) -- cycle;
\draw [decorate,decoration={brace,amplitude=4pt},xshift=3pt,yshift=0pt] (6.,3.95) -- (6.,3.05) node [black,midway,xshift=0.35cm]{$r'$};
\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=0pt] (6.45,0.05) -- (6.45,2.95) node [black,midway,xshift=-0.35cm]{$t'$};
\draw [decorate,decoration={brace,amplitude=0.1cm},xshift=0.cm,yshift=0.pt] (6.55,0.05) -- (7.45,0.05) node [black,midway,yshift=0.35cm]{$w$};
\end{tikzpicture}
\subcaption{Young tableau $T'$}
\label{fig:T'}
\end{center}
\end{minipage}
\caption{The Young tableaux from
Lemma~\ref{lem:crucialcomparison}}\label{fig:Ywithdeletedhook}
\end{figure}

\begin{lem}\label{lem:crucialcomparison}
    Let $m=xy-r-s+1$ with $1\le r\le x-y$ and $1\le s<y$, such that $r+s-1=wy+r'$ for some integers $0<w$ and $0\le r' <y.$
    If
    \begin{align*}
        T&=\left([x]\times [y]\right) \backslash \left( [x-r+1..x] \times y \right) \backslash \left( x \times [y-s+1..y-1] \right) \mbox{ and}\\
        T'&=\left([x-w]\times [y]\right) \backslash \left( (x-w) \times [y-r'+1..y] \right)
        ,
    \end{align*}
    then
    $h(T) < h(T').$ See Figure~\ref{fig:Ywithdeletedhook}.
\end{lem}

\begin{proof}
    We prove this in a number of steps. First we prove that it is sufficient to prove when $s=1.$
    \begin{claim}
        If Lemma~\ref{lem:crucialcomparison} does hold whenever $s=1$, then so it does for all other cases.
    \end{claim}
    \begin{claimproof}
        Assume there are choices of $x,y,r,s$ (and thus Young tableaux $T$ and $T'$) for which Lemma~\ref{lem:crucialcomparison} is false.
        Let $t=y-s$ and $t'=y-r'$, and assume $0<t<y-1.$
        We now consider two cases.\\
        \textbf{Case 1: $t' \le t$.}
        In this case, we note that $\Delta(t+1)+\Delta(x)>\Delta(t'+1)+\Delta(x-w)$ and thus we also have a counterexample with $s-1$ instead of $s$.
        We can repeat this until $s=1.$\\
        \textbf{Case 2: $t' > t$.}
        Note that $y\cdot (x-w)>m>x\cdot (y-1)$ implying $\frac{x-w}{x}>\frac{y-1}{y} > \frac{t}{t+1} \ge \frac{t}{t'}$ and hence $(x-w)t'>xt.$
        Now we have $\Delta(t)+\Delta(x)<\Delta(t')+\Delta(x-w)$.
        When $t+x \le (x-w)+t'$, this is by Karamata's inequality and the fact that $\Delta$ is increasing and concave.
        When $t+x> (x-w)+t'$, we note that
        $(x-u)(t-u)=xt-u(x+t)+u^2< (x-w)t'-u((x-w)+t')+u^2=  (x-w-u)(t'-u)$ for every $u \ge 0$ and thus
        $$\int_{u=0}^{1} \log\left( (x-u)(t-u) \right) \d u < \int_{u=0}^{1} \log\left( (x-w-u)(t'-u) \right) \d u.$$
        The latter being equivalent to $\Delta(t)+\Delta(x)<\Delta(t')+\Delta(x-w)$.
        This implies that we can construct a counterexample with $s+1$ instead of $s$ as well and we can repeat this until $s=y,$
        which corresponds to an example with $s=1.$
    \end{claimproof}
    Next, we show that we can assume that $x-r=y.$

    \begin{claim}\label{clm:rmax}
        If Lemma~\ref{lem:crucialcomparison} does hold whenever $s=1$ and $r=x-y$, then so it does for all other cases.
    \end{claim}
    \begin{claimproof}
        Assume there are Young tableaux $T$ and $T'$ for which Lemma~\ref{lem:crucialcomparison} is false, where $s=1$ and $y+r<x.$
        Now deleting the first column of both tableaux, implies that
        $h(T)$ decreases by $\Delta(x-r)+(y-1)\Delta(x)+f(y)$ and $h(T')$ has been decreased by $(y-r')\Delta(x-w)+r'\Delta(x-w-1)+f(y).$
        Since $\{\underbrace{x,x,\ldots, x}_{y-1},x-r\}$ majorizes $\{\underbrace{x-w,x-w,\ldots,x-w}_{y-r'},\underbrace{x-w-1,\ldots x-w-1}_{r'}\}$, as $\Delta$ is strictly concave, the value $h(T')$ has decreased by a larger amount than $h(T).$
        So Lemma~\ref{lem:crucialcomparison} is false for a construction with parameters $(x-1,y,r).$
        We can repeat this, until $x=y+r.$
    \end{claimproof}

    \begin{comment}
    \begin{figure}[h]
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\begin{tikzpicture}
\draw [fill = medium-gray](0,0) -- (7.5,0) -- (7.5,4) --(0,4) -- cycle;
\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=0pt] (2,4.05) -- (7.5,4.05) node [black,midway,yshift=0.35cm]{$r$};
\draw (3.5,-0.35) node { $x$};
\draw (-.35,2) node {$y$};
\draw [fill = white] (2,3.5) -- (7.5,3.5) --(7.5,4)-- (2,4) -- cycle;
\draw [dashed] (0.5,0)--(0.5,4);
\end{tikzpicture}
\subcaption{Young tableau $T$}
\label{fig:T}
\end{center}
\end{minipage}\quad
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\begin{tikzpicture}
\draw [fill = medium-gray](0,0) -- (7.5,0) -- (7.5,4) --(0,4) -- cycle;
\draw [dashed] (0.5,0)--(0.5,4);
\draw (3.5,-0.35) node { $x$};
\draw (-.35,2) node {$y$};
\draw [fill = white] (6.5,0) -- (7.5,0) -- (7.5,4) -- (6,4)--(6,3)--(6.5,3) -- cycle;
\draw [decorate,decoration={brace,amplitude=4pt},xshift=3pt,yshift=0pt] (6.,3.95) -- (6.,3.05) node [black,midway,xshift=0.35cm]{$r'$};
\draw [decorate,decoration={brace,amplitude=0.1cm},xshift=0.cm,yshift=0.pt] (6.55,0.05) -- (7.45,0.05) node [black,midway,yshift=0.35cm]{$w$};
\end{tikzpicture}
\subcaption{Young tableau $T'$}
\label{fig:T'}
\end{center}
\end{minipage}
\caption{The two Young tableaux from Claim~\ref{clm:rmax} }\label{fig:Ywithshortremovedrectangle}
\end{figure}
\end{comment}

Now we finish the proof by proving that Lemma~\ref{lem:crucialcomparison} is true when $s=1$ and $r=x-y.$
In the latter case, we can write $x=ay+b$ where $a>0$ is an integer and $0 \le b <y$, in which case we know the precise shapes of $T$ and $T'$. These have been represented in Figure~\ref{fig:Yremainingcase}.

\begin{figure}[h]
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\begin{tikzpicture}
\draw [fill = medium-gray](0,0) -- (7.5,0) -- (7.5,2.5) --(3,2.5) --(3,3)--(0,3)-- cycle;
\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=0pt] (0.02,3.05) -- (2.98,3.05) node [black,midway,yshift=0.35cm]{$y$};
\draw (3.75,-0.35) node { $ay+b$};
\draw (-.35,2) node {$y$};
\end{tikzpicture}
\subcaption{Young tableau $T$}
\end{center}
\end{minipage}
\quad
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\begin{tikzpicture}
\draw [fill = medium-gray](0,0) -- (7,0) -- (7,1.5) --(6.5,1.5) --(6.5,3)--(0,3)-- cycle;
\draw (3.5,-0.35) node { $a(y-1)+b+1$};
\draw (-.35,2) node {$y$};
\draw [decorate,decoration={brace,amplitude=4pt},xshift=0pt,yshift=0pt] (6.55,2.95)--(6.55,1.55)  node [black,midway,xshift=0.35cm]{$b$};
\end{tikzpicture}
\subcaption{Young tableau $T'$}
\end{center}
\end{minipage}
\caption{The two Young tableaux in the remaining case }\label{fig:Yremainingcase}
\end{figure}

When $a=1$, we have $h(T)\le h(T')$ by Proposition~\ref{prop:denseTxy} and the inequality is even strict when $b>1.$
We now finish the proof by induction on $a$. So assume it is proven for the values $a,b,y.$
Going from $a$ to $a+1$, the value $h(T)$ increases by
\begin{equation}\label{eq:T_a_a+1}
    I_1=y \cdot f(y-1) + (y-1) \left( f((a+1)y+b)-f(ay+b) \right)
\end{equation}
while $h(T')$ increases by at least
\begin{equation}\label{eq:T'_a_a+1}
    I_2= (y-1) \cdot f(y)+ y \left( f\left(\left(a+1+\frac{b}{y}\right)(y-1)\right)-f\left(\left(a+\frac{b}{y}\right)(y-1)\right) \right).\\
\end{equation}
Now one can note that $I_1=I_2$ by implementing the following two equalities in Equation~\eqref{eq:T_a_a+1} and Equation~\eqref{eq:T'_a_a+1}.
\begin{align*}
f((a+1)y+b)-f(ay+b)&= y +y \int_{u=a+\frac by}^{a+1+\frac by} \left(\log(u)+\log(y)\right) \d u\\
\end{align*}
\begin{align*}
f\left(\left(a+1+\frac{b}{y}\right)(y-1)\right)-f\left(\left(a+\frac{b}{y}\right)(y-1)\right)
&= y-1+(y-1)\int_{u=a+\frac by}^{a+1+\frac by} \left(\log(u)+\log(y-1)\right) \d u.
\end{align*}
\end{proof}

Having proven this particular case in Lemma~\ref{lem:crucialcomparison}, we continue with some observations from which we can conclude that the extremal tableaux will be as given in Section~\ref{sec:denseY}.

\begin{lem}\label{lem:imbalance}
    If $G=(U \cup V,E)$ is an extremal bipartite $(n,m)$-graph, then the minimum degree of the smaller partition class is at least the maximum degree of the other partition class.
\end{lem}

\begin{proof}
    Consider the associated Young tableau $T$ of $G$.
    Let $i$ be the value for which $(i,i) \in T$ and $(i+1,i+1) \not \in T$.
    Let $S_1=\{(i',j') \in T \mid j' >i \}$ and $S_2=\{(i',j') \in T \mid i' >i \}.$
    Now if both $S_1$ and $S_2$ are non-empty, we can construct a tableau $T'$ for which the rows in $S_1$ are deleted and are added as columns, in such a way that another Young tableau $T'$ has been formed.
    It is not hard to see that the degree sequence of the graph $G'$ associated with $T'$ majorizes the degree sequence of $G$. For this, note that is sufficient to compare the degrees of $\{u_1, \ldots, u_i,v_1, \ldots, v_i\}$ (and thus the number of cells in their rows/columns).
    Thus, since $f$ is strictly convex, by Karamata's inequality we know $h(T')>h(T).$
\end{proof}

\begin{figure}[h]
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\scalebox{0.9}{
\begin{tikzpicture}
\draw [fill = dark-gray](0,0) -- (4,0) -- (4,4) --(0,4) -- cycle;
\draw [fill = medium-gray](6,0) -- (4,0) -- (4,4) -- cycle;
\draw [fill = medium-gray](0,6) -- (0,4) -- (4,4) -- cycle;
\draw [fill = half-gray](0,4.5) -- (2.5,4.5) -- (2.5,4) --(0,4) -- cycle;
\draw [fill = half-gray](0,4.5) -- (0.5,4.5) -- (0.5,5) --(0,5) -- cycle;
\draw [fill = half-gray](0,5.5) -- (0.5,5.5) -- (0.5,5) --(0,5) -- cycle;
\draw [fill = half-gray](4.5,2) -- (4.5,0) -- (4,0) --(4,2) -- cycle;
\draw [fill = half-gray](4.5,1.5) -- (4.5,0) -- (5,0) --(5,1.5) -- cycle;
\draw (1,5) node { $S_1$};
\draw (4.5,2.25) node {$S_2$};
\end{tikzpicture}}\\
\subcaption{Young tableau $T$}
\label{fig:T}
\end{center}
\end{minipage}\quad
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\scalebox{0.9}{
\begin{tikzpicture}
\draw [fill = dark-gray](0,0) -- (4,0) -- (4,4) --(0,4) -- cycle;
\draw [fill = medium-gray](8,0) -- (4,0) -- (4,4) -- cycle;
\draw [fill = half-gray](4.5,2.5) -- (4.5,0) -- (4,0) --(4,2.5) -- cycle;
\draw [fill = half-gray](4.5,2) -- (4.5,0) -- (5,0) --(5,2) -- cycle;
\draw [fill = half-gray](5.5,1.5) -- (5.5,0) -- (5,0) --(5,1.5) -- cycle;
\draw [fill = half-gray](5.5,0.5) -- (5.5,0) -- (6,0) --(6,0.5) -- cycle;
\draw [fill = half-gray](6.5,0.5) -- (6.5,0) -- (6,0) --(6,0.5) -- cycle;
\end{tikzpicture}}
\subcaption{Young tableau $T'$}
\label{fig:T'}
\end{center}
\end{minipage}
\caption{Sketch of the rearrangement in Lemma~\ref{lem:imbalance}}
\end{figure}

\begin{thr}\label{thr:finalstep_Bnmy}
For fixed integers $m$ and $n$, let $T_{*}$ be a tableau with $m$ cells for which the sum of its length $x$ and width $y$ is at most $n$. If $h(T_{*})$ is maximal among all such tableaux, then $x,y$ satisfy $xy-\min\{x,y\}<m \le xy.$
\end{thr}

\begin{proof}
    Assume $m$ cannot be written as a product $xy$ with $x+y\le n$ and
    $T_{*}$ is an extremal tableau with $m\le (x-1)y,$ where $y <x.$
    Let $x_1=x, x_2, \ldots, x_y$ be the number of cells in every row.
    By Lemma~\ref{lem:imbalance}, we know that $[y]\times [y] \subset T_{*}$ and thus $x_i \ge y$ for every $1 \le i \le y.$
    Furthermore, since $m\le (x-1)y,$ there is a smallest index $i$ such that $x_i<x-1.$
    Now, the first $i$ rows of $T_{*}$, i.e. $T_i=T_{*} \cap \left([x]\times [i]\right)$, form a tableau that has the form of the tableau $T$ in Lemma~\ref{lem:crucialcomparison}.
    Replacing $T_i$ by the corresponding $T'_i$ from Lemma~\ref{lem:crucialcomparison} (which does not increase the length of the tableau) will imply that we also form a $T'_{*}$ for which $h(T'_{*})>h(T_{*}),$ since $h(T'_{*})-h(T_{*})=h(T'_i)-h(T_i)>0.$
    For the latter, note that the only rows and columns that possibly changed, are rows $1$ till $i$ and columns $x_i+1$ till $x,$ all of which do not contain any cell outside $T_i$ or $T'_i$.
    This is the desired contradiction.
    Figure~\ref{fig:Youngtableau_conclusion} presents this final comparison.
\end{proof}

\begin{figure}[h]
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\begin{tikzpicture}
\draw [fill = medium-gray](0,0) -- (7.5,0)--(7.5,1) -- (7,1)--(7,2.5) --(4,2.5) --(4,3)--(0,3)-- cycle;
\draw [fill = light-gray](0,5)--(3,5)--(4,4)--(4,3)--(0,3)-- cycle;
\draw (2,4) node {$S$};
\draw (3.5,1.5) node {$T_i$};
\end{tikzpicture}
\subcaption{Young tableau $T_{*}$}
\end{center}
\end{minipage}
\quad
\begin{minipage}[b]{.49\linewidth}
\begin{center}
\begin{tikzpicture}
\draw [fill = light-gray](0,5)--(3,5)--(4,4)--(4,3)--(0,3)-- cycle;
\draw [fill = medium-gray](0,0) -- (7,0) -- (7,1.5) --(6.5,1.5) --(6.5,3)--(0,3)-- cycle;
\draw (2,4) node {$S$};
\draw (3.5,1.5) node {$T'_i$};
\end{tikzpicture}
\subcaption{Young tableau $T'_{*}$}
\end{center}
\end{minipage}
\caption{The local move increasing $h(T_{*})$ in Theorem~\ref{thr:finalstep_Bnmy} }\label{fig:Youngtableau_conclusion}
\end{figure}

\section{Maximum entropy and extrema of other chemical indices}\label{sec:chemicalindex}

The degree-based entropy is one example of a degree-based topological index (or chemical index), the latter defined as
$$\TI_g(G) = \sum_{uv \in E(G)} g(\deg(u), \deg(v)),$$
for some function $g$, as initially done in e.g.~\cite{gutman2013degree}.
See e.g.~\cite[Table~1]{LP22} for a clear overview of some more examples.
In this case, one can conclude that the interesting graphical function-indices, defined similarly as $\sum_{v \in V(G)} g(\deg(u))$, will essentially boil down to $g(x)=c_1x+c_2 x \log(x)$ and thus be directly related to the entropy case. So we focus on the chemical indices.
The function $h(G)$ can be written as $\TI_g$ where $g(x,y)=\log(xy)$.
Our idea of approach $2$ in Subsection~\ref{subsec:approach2} works for any $\TI_g$ whenever $g$ is an increasing, concave function in $xy$.
E.g. when $g(x,y)=xy$ or $g(x,y)=\sqrt{xy}$ (i.e. for the Second Zagreb index and Reciprocal Randic index), the graphs maximizing $\TI_g(G)$ among all bipartite graphs of size $m$ are precisely the complete bipartite graphs $K_{q,y}$ with $q \cdot y=m$.
When we know both the order and size, one can expect that a similar exposition implies that the extremal graphs are again near complete bipartite, i.e. $K_{y,x}$ with some small number $r$ of removed (or added) edges. Xu et al.~\cite{XTLW15} studied these for the
first and second
Zagreb indices for example.

On the other hand, if $g(x,y)=x+y$ or $g(x,y)=(x+y)^2$, since $\deg(u)+\deg(v) \le m+1$ for every $uv \in E(G)$ the unique extremal graph (maximizing $\TI_g(G)$) is easily seen to be the star $K_{1,m}.$
In particular, this implies that among bipartite graphs of a given size, the set of graphs maximizing the First Zagreb index and Second Zagreb index are not equal. This contrasts some of the intuition in the concluding section of~\cite[Sec.~4]{XTLW15}.
Also among bipartite graphs of a given order and size,
the bipartite graphs with the maximum first Zagreb index
might be different than the one with the maximum second Zagreb index.
For the first Zagreb index these extremal bipartite graphs were determined by Zhang and Zhou~\cite{Zhang2014}. Here the asymmetry (large degrees) play a major role and so there are examples for which the bipartite graphs maximizing the first Zagreb index are not complete bipartite, while there do exist complete bipartite graphs $K_{y,x}$ with $xy=m$ and $x+y<n$ that maximize the second Zagreb index.

Finally, we prove that the graphs with maximum entropy can be easily determined and are the graphs for which the degree sequence is as balanced as possible.
We prove these statements by showing that the latter graphs are those which minimize $\TI_g(G)$ for any increasing, strictly convex function $g$.
Thus, we give a short alternative proof for the result in~\cite{Dong21} in this more general setting and for the result of~\cite{HLP22}.

Among all graphs with fixed size $m$, without constraint on the order, $mK_2$ is the unique extremal graph maximizing the entropy (note that $h(mK_2)=0)$ and minimizing $\TI_g(G)$ as well (here the order $n \ge 2m$ and isolated vertices are taken into account as well when $g(0) \not=0$).

If there is a condition on both the order and size, the extremal graphs and bipartite graphs are precisely the nearly-regular graphs, as proven in the following two propositions.
Cheng et al.~\cite{CGZD09} determined these bipartite graphs with the minimum value of the first Zagreb index for bipartite graphs of a given order and size.
The proof is essentially a corollary of Karamata's inequality and noting that there is a degree sequence that is majorized by all other possible realizations.

\begin{prop}
    Let $g$ be an increasing, strictly convex function.
    Among all graphs with fixed size $m$ and order $n$, every nearly-regular graph (i.e. $\deg_{max}-\deg_{min} \le 1$) is extremal and these are the only extremal graphs.
\end{prop}
\begin{proof}
     Since the degree sequence of any nearly-regular graph is majorized by the degree sequence of any  other $(n,m)$-graph, the result follows by Karamata's inequality. The characterization of the extremal graphs is a consequence of $g$ being strictly convex.
     The existence of nearly-regular graph (even connected if $m\ge n-1$ when necessary) is immediate.
     If $n$ is even, then $K_n$ can be partitioned into $n-1$ perfect matchings and one can add edges from one perfect matching at a time.
     If $n$ is odd, then $K_n$ can be partitioned into $\frac{n-1}{2}$ $2$-regular graphs, where at least one of them is a cycle (when $n=2k+1$ and $V=\{v_1,v_2,\ldots, v_n\}$, for every $1 \le \ell \le k$, form the graph with edges $v_iv_j$ for which $ i-j \equiv \pm \ell \pmod n$).
     Now when $m=an+b,$ one can take the union of $a$ such $2$-regular graphs and a matching or its complement of the cycle.
\end{proof}

\begin{prop}
    Among all bipartite graphs with size $m$ and order $n$, every bipartite graph $G=(U \cup V,E)$ with $\lvert U \rvert = \left\lceil \frac n2 \right\rceil$ and $\lvert V \rvert = \left\lfloor \frac n2 \right\rfloor$ for which the degrees in one partition class differ by at most one attains the maximum entropy in the graph class. Furthermore, a bipartite graph is extremal if and only if it
    has the same degree sequence as this balanced example.
\end{prop}

\begin{proof}
Let $G'=(U' \cup V',E')$ be any other bipartite graph with size $m$ and order $n$ whose partition classes have size $u \ge v$.
The sum of the $i \le v$ largest degrees in $V'$ are at least equal to the sum of the $i$ largest degrees of $G$ and the sum of the $i\le u$ smallest degrees in $U'$ are at most the sum of the $i$ smallest degrees in $G$. Here we use that $u \ge \left\lceil \frac n2 \right\rceil $ and $v \le \left\lfloor \frac n2 \right\rfloor$ and the degrees in $G$ are as balanced as possible.
Remark that if $u \ge i >\left\lceil \frac n2 \right\rceil $, then the sum of the $i$ degrees in $U'$ is at most $m$.
So we conclude that the degree sequence of $G$ is majorized by any degree sequence of any other bipartite graph with order $n$ and size $m$, from which we conclude (also for  the uniqueness statement) by Karamata's inequality and $g(x)$ being strictly convex.

We remark that one can always construct at least one such a balanced bipartite graph $G$.
If $n$ is even, just partition $K_{n/2,n/2}$ in perfect matchings and add the edges from one matching at a time up to the point you selected precisely $m$ edges.
When $n$ is odd, for every $1 \le m \le \left\lfloor \frac{n^2}{4} \right\rfloor,$ one can construct a graph by adding for every $1 \le k \le m$ an edge between the vertices $a_i$ and $b_j$ for which $k \equiv i \pmod{ \left\lceil \frac n2 \right\rceil}$ and $k \equiv j \pmod{ \left\lfloor \frac n2 \right\rfloor}$ respectively.
\end{proof}

A few examples have been presented in Figure~\ref{fig:maxentropy_bipgraphs}.
Since everything boils down to having a degree sequence that is as balanced as possible, i.e. it is majorized by any other degree sequence of a graph, the result only depends on the degree sequence.

We end with a few observations.
There do exist both connected and disconnected extremal bipartite graphs (as presented for $m=n=9$).
If $n$ is odd, it is possible there is no extremal bipartite graph attaining the maximum over all graphs, since for bipartite $(n,m)$ graphs we might be forced to have $\deg_{max}-\deg_{min} \ge 2$ (equality can be attained), as is the case with $m=n=9$.
Finally, we observe that the partition itself does not necessarily has to be balanced (e.g. $m=30, n=22, \lvert U\rvert =10$ and $\lvert V\rvert =12$) .

\begin{figure}[h]
\begin{minipage}[b]{.32\linewidth}
\begin{center}
    \begin{tikzpicture}
    {

	\foreach \x in {1,2,3,4}{\draw[thick] (\x,2) -- (\x,0);}
	\foreach \x in {1,2,3}{\draw[thick] (\x,2) -- (\x+1,0);}
	\draw[thick] (4,2) -- (2,0);
	\draw[thick] (1,0) -- (4,2);
	\foreach \x in {1,2,3,4}{\draw[fill] (\x,0) circle (0.1);\draw[fill] (\x,2) circle (0.1);}
	}
	\end{tikzpicture}\\
	\subcaption{$n=8$}\label{fig:1a}
\end{center}
\end{minipage}\quad\begin{minipage}[b]{.32\linewidth}
\begin{center}
     \begin{tikzpicture}
    {
	\foreach \x in {1,2,3,4}{\draw[thick] (\x,5) -- (\x+0.5,3);}
	\foreach \x in {1,2,3}{\draw[thick] (\x,5) -- (\x+1.5,3);}
	\draw[thick] (5,5) -- (1.5,3);
	\draw[thick] (4,5) -- (1.5,3);
	\foreach \x in {1,2,3,4,5}{\draw[fill] (\x,5) circle (0.1);}
	\foreach \x in {1,2,3,4}{\draw[fill] (\x+0.5,3) circle (0.1);}
    }
    \end{tikzpicture}\\
    \subcaption{$n=9$ (connected)}\label{fig:1b}
\end{center}
\end{minipage}\quad\begin{minipage}[b]{.32\linewidth}
\begin{center}
     \begin{tikzpicture}
    {
	\foreach \x in {1,2}{\foreach \y in {1,2}{\draw[thick] (\x,5) -- (\y,3);}}
	\foreach \x in {3,4,5}{\draw[thick] (\x,5) -- (3.5,3);}
	\foreach \x in {4,5}{\draw[thick] (\x,5) -- (4.5,3);}
	\foreach \x in {1,2,3,4,5}{\draw[fill] (\x,5) circle (0.1);}
	\foreach \x in {1,2}{\draw[fill] (\x,3) circle (0.1);}
	\foreach \x in {3,4}{\draw[fill] (\x+0.5,3) circle (0.1);}
    }
    \end{tikzpicture}\\
    \subcaption{$n=9$ (disconnected)}\label{fig:1c}
\end{center}
\end{minipage}
\caption{Bipartite graphs of size $9$ with maximum entropy}\label{fig:maxentropy_bipgraphs}
\end{figure}

\section{Conclusion}\label{sec:conc}

In this paper, we proved that the bipartite graphs of size $m$ that maximize the entropy are complete bipartite graphs.
If there is a restriction on the order (at most $n$) and $m$ cannot be written as $x\cdot y$ where $x+y \le n$, then the extremal graphs are of the form $B(n,m,y),$ i.e. a complete bipartite graph minus a star.
Nevertheless, it seems impossible to determine $y$ with a general formula, as we will explain here.

Using a computer program\footnote{\url{https://github.com/MatteoMazzamurro/extrema-graph-entropy/blob/main/bipartite_graphs_entropy.R}}, we could find all extremal bipartite graphs for $ n \le 50$ and $m \le \left\lfloor \frac{n^2}{4} \right\rfloor.$
The output is presented in\footnote{\url{https://github.com/MatteoMazzamurro/extrema-graph-entropy/blob/main/B_n_m_50.csv}}.
For every pair $(n,m)$ and value $1 \le y \le \sqrt{m}$, we computed the values     $$q=\left\lfloor\frac{m}{y}\right\rfloor,\;  x=\left\lceil\frac{m}{y}\right\rceil,\; m-qy,\; xy-m,\; h=h(B(n,m,y))$$
and three boolean values expressing whether $B(n,m,y)$ achieved the maximum value of $h$ among all choices of $y$ and whether $m-qy$ or $xy-m$ were minimal among all possible choices for $y$ for that pair $(n,m).$
Among all extremal graphs which are not complete bipartite, we found $547$ cases for which both $xy-m$ and $m-qy$ are minimal among all possibilities, $375$ cases where only $m-qy$ is minimal, $3635$ cases where only $xy-m$ is minimal, and no cases for which neither $xy-m$ nor $m-qy$ is minimal.

Intuitively, when $xy-m$ or $m-qy$ is small, the graph $B(n,m,y)$ is nearly complete bipartite (up to the edges of a small star which has been added or removed) and so it might be extremal.
Nevertheless, this was found not to be the case in general.
When $n=17726$ and $m=318728$, the extremal graph is $B(n,m,18)$, which satisfies $q=17707$, $y=18$ and $r=m-qy=2,$ even though $y=139$ and $q=2293$ would give $r=1$.
Taking into account the estimates in Proposition~\ref{prop:denseTbq}, one can expect that there are more such examples since the difference with the upper bound depends on both $r$ and $y$, i.e. one would like $y$ and $r$ to be both small and the ratio has an influence on this. A precise statement here would be difficult, as it is related to some hard number theoretic questions.

Comparing with Proposition~\ref{prop:denseTxy}, whenever $m$ is large, one would wish that there was a small value of $r$ for which $m+r$ could be written as a product $xy$ with $x+y \le n.$
So one may expect from these observations that in most cases, the extremal graph will be $B(n,m,y)$ with $y$ chosen in such a way that $y\left\lceil \frac m y \right\rceil -m$ is minimized under the constraint that $y+\left\lceil \frac m y \right\rceil \le n,$ and so the extremal graph is a ``complete bipartite graph minus a small star".
Furthermore, when there are multiple choices for $y$, we need to choose the smallest $y$, i.e. for fixed $r$, the complete bipartite graph will be as asymmetric as possible.
To note the latter, one can compare the computations in Proposition~\ref{prop:denseTxy} for some $x'>x>y>y'$ with $xy=x'y'$.
For a fixed $1 \le k \le y-1$, we have $\Delta(x)+\Delta(y+1-k) \ge \Delta(x')+\Delta(y'+1-k)$ since $(x-u)(y+1-k-u) \ge (x'-u)(y'+1-k-u)$ for $u \in [0,1].$

We can prove that the intuition that $r=xy-m$ is minimal for the extremal $B(n,m,y)$ is true in many cases, using the notion of $M$-friable numbers.
A positive integer is called $M$-smooth or $M$-friable if its largest prime divisor is at most $M.$
At this point, the state-of-the-art on smooth/friable numbers~\cite{MR16_smooth, E17_smooth} does not address the question if for fixed constants $0<u<1$ and $c>0$, the interval $[m,m+c \log(m)]$ necessarily needs to contain a $m^u$-smooth number whenever $m$ is sufficiently large, but it does so for almost all $m$ by~\cite[Cor.~6]{MR16_smooth}.

Write $n=(m+\log(m))^u+(m+\log(m))^{1-u}$ for some real number $\frac 23 \le  u < 1$.
Now for almost all $m$, we know that there does exist a number $r \le 0.5 (1-u) \log(m)$ such that $m+r$ is $m^u$-smooth (by~\cite[Cor.~6]{MR16_smooth}).
Now $m+r$ being $m^u$-smooth implies that all its prime divisors are bounded by $m^u$ and thus also $(m+r)^u.$
Thus $m+r$ has two divisors $x,y$ for which $(m+r)^{1-u}\le y \le x \le  (m+r)^u$.
For this, note that if the largest prime factor $p$ of $m+r$ satisfies $(m+r)^{1-u}\le p \le (m+r)^u$, we can take $\{x,y\}$ to be equal to $\left\{p, \frac {m+r}p \right\}$ and otherwise we can take a product $P$ of prime divisors of $m+r$ which is between $(m+r)^{1-u}$ and $(m+r)^{2(1-u)} \le (m+r)^u$ and take $\{x,y\}$ to be equal to $\left\{P, \frac {m+r}P \right\}$.
Since the function $P+\frac{m+r}{P}$ is increasing for $P \ge \sqrt{m+r}$ and $r < \log m$, we have $x+y <n.$
Comparing Proposition~\ref{prop:denseTbq} and Proposition~\ref{prop:denseTxy} we conclude that the extremal graph is equal to the graph $B(n,m,y)$ where $y$ has been chosen in such a way that $xy-m$ is minimized.
The latter is because $y+\frac{m}{y} \le n$ and $y< \sqrt{m}$ implies that $y>> m^{0.5(1-u)}$ and thus $\log(y)-r$ is large.

On the other hand, there will be infinity many examples which are not of this form.
A simple example for this, where $m-qy$ can be arbitrarily large, can be constructed as follows.
Fix $c>0$ and let $n=2a+1$ and $m=a^2+c$,
where $a=b^2+b$.
Now $B(n,m,a)$ is an example for which $m-qy=c$ is minimized (note that $q+y=2a$ is needed).
On the other hand $B(n,m,y)$ will always satisfy $xy-m \ge 2b-c$ since (note that $x+y=n$ is needed) $$(a-b+1)(a+b)-m=a^2+a-b^2+b>m>a^2+a-b^2-b=(a-b)(a+b+1).$$
For fixed $c$ and $b$ sufficiently large, we conclude as then $2b-c>c \log\left( \frac{2b-c}{c} \right).$

\section*{Acknowledgement}
Thanks to Bart Michels and Régis de la Bretèche for sharing their expertise on smooth/friable numbers.

\paragraph{Open access statement.} For the purpose of open access,
a CC BY public copyright license is applied
to any Author Accepted Manuscript (AAM)
arising from this submission.

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


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Cervical screening in Australia is a successful public health initiative.
Since the introduction of the National Cervical Screening Program in 1991, there has been a significant fall in incidence of and mortality from cervical cancer.
Laboratory quality procedures are critical to ensuring optimal outcomes.
Laboratory accreditation procedures are being reviewed in line with recent government recommendations.
For a sustainable program, cost-containment issues need to be considered; screening interval, management of screen-detected abnormalities, and new technologies are the critical drivers of cost.
Cervical screening in Australia is one of the great public health success stories, as witnessed by a continuing dramatic fall in the incidence of carcinoma of the cervix and mortality from this disease (Box) since the introduction of the National Cervical Screening Program (NCSP).1,2 Here, we highlight two areas of current interest — quality assurance and cost control.
A critical aspect of the NCSP is that screening tests are performed at an optimal level.3 Asymptomatic women and their health carers who participate in screening assume that this is the case.
Since 1987, Australia has had compulsory laboratory accreditation. The National Pathology Accreditation Advisory Committee (NPAAC) sets quality standards for patho-logy laboratories, and The National Association of Testing Authorities and the Royal College of Pathologists of Australasia inspect individual laboratories triennially to ascertain standards compliance.
For gynaecological cytology, these quality standards include performance standards, which are numerical outcome measures set by the NCSP, in conjunction with the NPAAC. Although the performance standards comprise only one part of an inspection, if a laboratory falls outside these standards this may indicate that the overall reporting quality is substandard.
A recent independent audit of the NCSP highlighted that the standards which had been set were valid and the inspection process was sound.4 However, the 3-year interval between inspections and the lengthy appeal process meant that Medicare was continuing to pay for smears from laboratories with questionable quality for excessive periods of time. In response to this audit, a number of initiatives have been undertaken or are currently in progress.
The Department of Health and Ageing commissioned a review of laboratory accreditation and has endorsed its numerous recommendations.5 Significantly, the timely management of laboratories that fall outside quality guidelines is being addressed.
The NPAAC has undertaken this review and the requirements remain substantially unchanged, as they have proven to be robust. Issues pertaining to qualifications and quality audits of staff, both scientific and medical, are being addressed by the NPAAC Review Committee.
A technical subcommittee of the NPAAC is currently reviewing the performance standards and will examine their appropriateness as well as the numerical values. The review is also addressing the minimum number of Pap smears a laboratory reports annually; both New Zealand and the United Kingdom set minimum annual numbers as part of their quality standards. Minimum numbers of Pap smears may also be critical when assessing staff performance.
In the program, the main pressure points for cost control include: participation, the proportion of women requiring further investigation, and new technologies.
The 2-year screening policy is under review. It is a short rescreening interval by international standards.8 Alternatively, significant cost savings could be achieved if Medicare were to pay for only one screening test every 2 years. We estimate that the potential savings to government from reduced claims to Medicare could be of the order of $32 million per year (calculations available on request).
The lifetime risk of a woman being diagnosed with cervical cancer in a developed country like Australia, in the absence of any screening, has been estimated as 1.58%.9 By contrast, the lifetime risk of a woman having a colposcopy in Australia is 76.8%.10 These figures suggest significant overinvestigation, with most investigated abnormalities having a very low probability of becoming malignant.
A range of new approaches for screening for the prevention of cervical cancer have been developed, including new methods of collecting the cell sample, entirely new screening tests, the use of computers in reporting the cell sample, and testing for the presence of human papillomavirus DNA. None of the new technologies are currently listed on the Medicare Benefits Schedule. On the advice of the Medical Services Advisory Committee, the Minister for Health and Ageing decided, in October 2002, that liquid-based cytology or human papillomavirus testing for triage of women with abnormal smears would not be included on the Medicare Benefits Schedule.
New technologies in cervical screening could be expensive. It is essential that an evidence-based approach be used to determine their possible role in cervical screening in Australia.
*Age-standardised death rate (Australian Standard Population 1991).
1 Douglass Hanly Moir Pathology, North Ryde, NSW.
2 Victorian Cytology Service, Carlton South, VIC.
1. Australian Institute of Health and Welfare (AIHW). Cervical screening in Australia 1997-1998. AIHW Catalogue No. 9. Canberra: AIHW, 2000: 36. Available at: http: //www.aihw.gov.au/publications/can/csa97-8/csa97-8.pdf (accessed May 2003).
2. A decade of change. A report on Australia's National Cervical Screening Program 1989-1999. Canberra: Commonwealth Department of Health and Aged Care, 2000.
3. Report of the Steering Group on Quality Assurance in Screening for the Prevention of Cancer of the Cervix. Making the Pap smear better. Canberra: AGPS, 1993.
4. Australian National Audit Office. The National Cervical Screening Program. Audit Report No. 50 2000-2001. Canberra: Australian National Audit Office, 2001.
5. Health Access and Financing, Department of Health and Ageing. Report of the evaluation of the pathology laboratory accreditation arrangements, 2002. Available at: http: //www.health.gov.au/haf/branch/dtb/evaluation.htm (accessed May 2003).
6. Report of the Evaluation Steering Committee. The interim evaluation of the organised approach to preventing cancer of the cervix 1991-95. Canberra: AGPS, 1995: 27. Available at: http: //www.health.gov.au/pubhlth/publicat/document/eval.pdf (accessed May 2003).
7. Australian Health Ministers' Advisory Council. Cervical Cancer Screening Evaluation Committee. Cervical cancer screening in Australia: options for change. Australian Institute of Health: Prevention Program Evaluation Series No 2. Canberra: AGPS, 1991: 90.
9. IARC Working Group on Evaluation of Cervical Cancer Screening Programmes. Screening for squamous cervical cancer: duration of low risk after negative results of cervical cytology and its implication for screening policies. BMJ 1986; 293: 659-664.
10. Kavanagh AM, Santow G, Mitchell H. Consequences of current patterns of Pap smear and colposcopy use. J Med Screen 1996; 3: 29-34.
11. National Cervical Screening Program Guidelines Review Group. Available at: www.csp.nsw.gov.au/nhmrc (accessed May 2003).