Hosoya Z Index and Correlation with Boiling Point I have recently read about the Z topological index found by Hosoya. The papers/sources The Z index is the sum of the coefficients of the matching polynomials (no. of ways to choose an independent edge from the set of edges in the graph - if we let each carbon atom be a vertex and each bond between a pair of carbon atoms is an edge.). The paper states that in particular for alkanes or saturated hydrocarbons, there is a correlation found between the Z index (which measures the extent of branching - in what way??) and the boiling point (bp) of these alkanes. 

I know that the bp of alkanes increases with its size (due to strong force of attractions). However, I am confused about how the Z index measures the extent of branching, and any underlying reasons that the Z index will correlate with the bp and other physiochemical properties. 

The papers I have read so far fail to enlighten me on these questions above. Please help. 
Thank you. 


  Some ideas I have so far: 
  Since Z index measures branching (or at least correlates to branching since more compacted graph will have a smaller sum of matching coefficient?), and branching correlates to the bp of alkanes (due to less surface area sometimes if you have more branching), perhaps thats why the Z index correlates to bp... 


Refer to Why do branched chain compounds have lower boiling points than the corresponding straight chain isomers?
For a given hydrocarbon C$_n$H$_{n+2}$ more branched hydrocarbons have lower boiling point. This is because they have less surface area to have Van der Waals interactions to entangle with each other (ropes have more chances to entangle with each other than balls). For example C(CH3)4 (n-pentane) is a gas, but all other pentanes are liquids. 

Their method is qualitative only. For instance, o-p-nitrophenol can be steam distilled and has much higher vapor pressure at a given temperature than p-nitrophenol, but the table would predict that they have the same properties. The difference is due to intramolecular h-bond present in o-, but ansent in p- isomer.