Spectral radius minus average degree: a better bound

TL;DR

This paper introduces improved lower bounds for the spectral radius minus average degree in graphs, providing a better measure of how far a graph is from being regular.

Contribution

The authors present new, tighter lower bounds on the spectral radius minus average degree, advancing the understanding of graph irregularity measures.

Findings

New lower bounds outperform previous ones

Enhanced understanding of graph irregularity

Applicable to various classes of graphs

Abstract

Collatz and Sinogowitz had proposed to measure the departure of a graph $G$ from regularity by the difference of the (adjacency) spectral radius and the average degree: $\epsilon(G)=\rho(G)-\frac{2m}{n}$. We give here new lower bounds on this quantity, which improve upon the currently known ones.