A spectral bound for graph irregularity

TL;DR

This paper introduces a new spectral upper bound for graph irregularity based on the Laplacian spectral radius, improving previous bounds and enhancing understanding of graph imbalance measures.

Contribution

The paper presents a novel spectral bound for graph irregularity that surpasses earlier bounds, linking irregularity to the Laplacian spectral radius.

Findings

New upper bound involving Laplacian spectral radius

Improves upon Zhou and Luo's 2011 bound

Provides tighter constraints on graph irregularity

Abstract

The imbalance of an edge $e=\{u,v\}$ in a graph is defined as $i(e)=|d(u)-d(v)|$, where $d(\cdot)$ is the vertex degree. The irregularity $I(G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$. This concept was introduced by Albertson who proved that $I(G) \leq \frac{n^{3}}{27}$ (where $n=|V(G)|$) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2011. Our bound involves the Laplacian spectral radius $\lambda$.