A tight upper bound of spectral radius in terms of degree deviation

TL;DR

This paper proves a conjecture relating the spectral radius of a graph to its degree deviation, providing a tight upper bound without restrictions on the size of the graph.

Contribution

It establishes a precise upper bound on the spectral radius deviation in terms of degree deviation, confirming Nikiforov's conjecture for all graph sizes.

Findings

Proves the conjecture for all graphs, removing size restrictions.

Provides a tight upper bound on spectral radius deviation.

Enhances understanding of spectral graph theory.

Abstract

Let $G$ be a graph with $n$ vertices and $m$ edges. The spectral radius $\rho(G)$ of $G$ is the largest eigenvalue of the adjacency matrix of $G$. As is well known, $\rho(G)\geq\frac{2m}{n}$ with equality if and only if $G$ is regular. To bound $\rho(G)-\frac{2m}{n}$, Nikiforov (2006) introduced the degree deviation of $G$ as $$s(G)=\sum_{1\leq i\leq n}|d_{i}-\frac{2m}{n}|,$$ where $d_{1},d_{2},\ldots,d_{n}$ are the degrees of the vertices of $G$. Nikiforov conjectured that $\rho(G)-\frac{2m}{n}\leq\sqrt{\frac{1}{2}s(G)}$ for sufficiently large $m$ and $n$. In this paper, we settle this conjecture without the assumption that $m$ and $n$ are large.