Degree Deviation and Spectral Radius

TL;DR

This paper establishes bounds relating the spectral radius of a graph to its degree deviation, contributing to conjectures and generalizing known spectral bounds for specific graph constructions.

Contribution

It proves a new inequality connecting the spectral radius and degree deviation, and generalizes existing bounds for graphs derived from bipartite structures.

Findings

Proved $ ext{lambda} - rac{2m}{n} \

bound on spectral radius in terms of degree deviation

generalized bounds for bipartite graph modifications

Abstract

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $\lambda$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a conjecture of Nikiforov, we show $\lambda-\frac{2m}{n}\leq \sqrt{\frac{2s}{3}}$. For our result, we show that the largest eigenvalue of a graph that arises from a bipartite graph with $m_{A,B}$ edges by adding $m_A$ edges within one of the two partite sets is at most $\sqrt{m_A+m_{A,B}+\sqrt{m_A^2+2m_Am_{A,B}}}$, which is a common generalization of results due to Stanley and Bhattacharya, Friedland, and Peled.