Graph Irregularity Characterization with Particular Regard to Bidegreed Graphs

TL;DR

This paper investigates the relationship between two graph irregularity measures, degree deviation and degree variance, establishing bounds and comparing their effectiveness, especially in bidegreed graphs.

Contribution

It provides new bounds for irregularity measures and clarifies their equivalence in bidegreed graphs, enhancing understanding of graph irregularity characterization.

Findings

Established upper bounds for S(G) and Var(G)

Sharpened Nikiforov's inequality for connected graphs

Proved equivalence of irregularity measures in bidegreed graphs

Abstract

In this study we are interested mainly in investigating the relations between two graph irregularity measures which are widely used for structural irregularity characterization of connected graphs. Our study is focused on the comparison and evaluation of the discriminatory ability of irregularity measures called degree deviation S(G) and degree variance Var(G). We establish various upper bounds for irregularity measures S(G) and Var(G). It is shown that the Nikiforov's inequality which is valid for connected graphs can be sharpened in the form of Var(G) < S(G)/2. Among others it is verified that if G is a bidegreed graph then the discrimination ability of S(G) and Var(G) is considered to be completely equivalent.