The Covariance of Topological Indices that Depend on the Degree of a Vertex

TL;DR

This paper investigates the covariance properties of certain topological indices based on vertex degrees in graphs, providing conditions for when these indices are uncorrelated with the graph's edge set.

Contribution

It establishes a necessary and sufficient condition for topological indices to have zero covariance with the edge set, advancing understanding of their statistical independence.

Findings

Identifies when topological indices are uncorrelated with the edge set

Provides a mathematical condition for zero covariance

Enhances understanding of index-edge relationships in graph theory

Abstract

We consider topological indices I that are sums of f(deg(u)) f(deg(v)), where {u,v} are adjacent vertices and f is a function. The Randi{\'c} connectivity index or the Zagreb group index are examples for indices of this kind. In earlier work on topological indices that are sums of independent random variables, we identified the correlation between I and the edge set of the molecular graph as the main cause for correlated indices. We prove a necessary and sufficient condition for I having zero covariance with the edge set.