Topological Atomic Displacements, Kirchhoff and Wiener Indices of Molecules

TL;DR

This paper offers a physical interpretation of Kirchhoff and Wiener indices in molecules using atomic displacements from vibrations, linking topological indices to experimental B-factors in crystallography.

Contribution

It introduces a local vertex invariant based on atomic vibrations to interpret topological indices and correlates these with experimental B-factors in molecules.

Findings

Kirchhoff index interpreted via atomic displacements

Wiener index related to atomic vibrations in acyclic molecules

Strong correlation between atomic displacements and B-factors

Abstract

We provide a physical interpretation of the Kirchhoff index of any molecules as well as of the Wiener index of acyclic ones. For the purpose, we use a local vertex invariant that is obtained from first principles and describes the atomic displacements due to small vibrations/oscillations of atoms from their equilibrium positions. In addition, we show that the topological atomic displacements correlate with the temperature factors (B-factors) of atoms obtained by X-ray crystallography for both organic molecules and biological macromolecules.