On the Wiener-like root-indices of graphs

TL;DR

This paper studies the roots of specific graph polynomials, analyzing their properties and potential as structural graph measures, with applications in graph characterization and comparison of topological indices.

Contribution

It provides analytical and numerical insights into the roots of modified graph polynomials and introduces a new measure for comparing topological indices based on structure sensitivity.

Findings

Roots of graph polynomials have high discrimination power on trees.

Modified polynomial roots can characterize graph topology.

A new measure compares topological indices in terms of sensitivity.

Abstract

In this paper, we examine roots of graph polynomials where those roots can be considered as structural graph measures. More precisely, we prove analytical results for the roots of certain modified graph polynomials and also discuss numerical results. As polynomials, we use, e.g., the Hosoya, the Schultz, and the Gutman polynomial which belong to an interesting family of degree-distance-based graph polynomials; they constitute so-called counting polynomials with non-negative integers as coefficients and the roots of their modified versions have been used to characterize the topology of graphs. Our results can be applied for the quantitative characterization of graphs. Besides analytical results, we also investigate other properties of those measures such as their degeneracy which is an undesired aspect of graph measures. It turns out that the measures representing roots of graph polynomials possess high discrimination power on exhaustively generated trees, which outperforms standard versions of these indices. Furthermore, a new measure is introduced that allows us to compare different topological indices in terms of structure sensitivity and abruptness.