Statistical–mechanical theory of topological indices

Topological indices (TI) are algebraic invariants of molecular graphs representing the
topology of a molecule, which are very valuable in quantitative structure–property
relations (QSPR). Here we prove that TI are the partition functions of such molecules
when the temperature of the thermal bath at which they are submerged is very high.
These partition functions are obtained by describing molecular electronic properties
through tight-binding Hamiltonians (TBH), where the hopping parameters are topological
properties describing atom–atom interactions. We prove that the TBH proposed here
are non-Hermitian diagonalizable Hamiltonians which can be replaced by symmetric
ones. In this way we propose a statistical–mechanical theory for TI, which is exemplified
by deriving the Randić, Zagreb, Balaban, Wiener and ABC indices. The work also
illuminates how to improve QSPR models using the current theoretical framework as
well as how to derive statistical–mechanical parameters of molecular graphs.