content_image/388104.jpg content_image/388103.jpg content_image/388101.jpg content_image/388102.jpg content_image/388105.jpg content_image/388116.jpg content_image/388117.jpg content_image/388108.jpg content_image/388106.jpg content_image/388107.jpg
which is equal to the row sum ∑ N j $_{=1}$T$_{ij}$ , since T T = T . We also define the sum of all entries (SE) of T as 

Note that SE ( T ) / N is the average row (= column) sum. Next, we observe that 

<img src='content_image/388104.jpg'>

To obtain our lower bound, we then use the r = 1 special case of the theorem that for a non-negative sym- 

<img src='content_image/388103.jpg'>

metric matrix T and r ∈ N + [19] 

The lower bound is then 

where 

<img src='content_image/388101.jpg'>

<img src='content_image/388102.jpg'>

<img src='content_image/388105.jpg'>

## III. RESULTS FOR SLAB OF THICKNESS L z = 2 WITH FBC$_{z}$ 

We now evaluate our general lower bound in Eqs. (2.13) and (2.14) for a slab of the simple cubic lattice with thickness L$_{z}$ = 2 and FBC$_{z}$, denoted S$_{2}$$_{F}$ . In this case the transverse slice is the graph sq [2$_{F}$ × ( L$_{y}$ )$_{BCy}$]. For FBC$_{y}$, an elementary calculation yields 

<img src='content_image/388116.jpg'>

with a single λ$_{ts}$$_{(2}$$_{F}$$_{)}$ = λ$_{ts}$$_{(2}$$_{P}$$_{)}$ ≡ λ$_{ts}$$_{(2)}$ , and this is also the maximal λ for PBC$_{y}$ [10, 20], so that 

We next use the calculation of 

<img src='content_image/388117.jpg'>

in Ref. [34] (where each of the 2$_{F}$ BC’s is equivalent to 2$_{P}$ ), from which we calculate the maximal λ$_{tube}$$_{(2)}$$_{,max}$ to 

<img src='content_image/388108.jpg'>

be 

<img src='content_image/388106.jpg'>

where 

<img src='content_image/388107.jpg'>

We then substitute these results for λ$_{ts}$$_{(2)}$$_{,max}$ and λ$_{tube}$$_{(2)}$$_{,max}$ into the L$_{z}$ = 2 special case of (2.14) to obtain W ( S$_{2}$, q )$_{ℓ}$, and thus the resultant lower bound on W ( S$_{2}$$_{F}$ , q ) = W ( S$_{2}$$_{P}$ , q ) ≡ W ( S$_{2}$, q ): W ( S$_{2}$, q ) ≥ W ( S$_{2}$, q )$_{ℓ}$. 

## IV. COMPARISON WITH LARGE- q SERIES EXPANSIONS 

One way to elucidate how this lower bound W ( S$_{2}$, q )$_{ℓ}$ compares with the exact W ( sq, q ) and W ( sc, q ) is to com- pare the large- q series expansions for these three func- tions. For this purpose, it is first appropriate to give some relevant background on large- q series expansions for W ( { G } , q ) functions. Since there are q n possible color- ings of the vertices of an n -vertex graph G with q colors 

