New measures of graph irregularity

TL;DR

This paper introduces four new measures of graph irregularity, compares them, and applies them to derive bounds on graph parameters, strengthen classical theorems, and develop new network heterogeneity indices.

Contribution

It presents novel measures of graph irregularity and demonstrates their applications in bounding graph parameters and enhancing classical graph theorems.

Findings

Derived upper bounds for chromatic number and Colin de Verdiere parameter.

Strengthened Turan's theorem for irregular and r-partite graphs.

Connected new irregularity measures to the Randic index and network heterogeneity.

Abstract

In this paper, we define and compare four new measures of graph irregularity. We use these measures to prove upper bounds for the chromatic number and the Colin de Verdiere parameter. We also strengthen the concise Turan theorem for irregular graphs and investigate to what extent Turan's theorem can be similarly strengthened for generalized r-partite graphs. We conclude by relating these new measures to the Randic index and using the measures to devise new normalised indices of network heterogeneity.