The Largest Eigenvalue and Bi-Average Degree of a Graph

TL;DR

This paper establishes bounds relating the largest eigenvalue of a graph to its bi-average degree, providing new insights into spectral graph theory with specific focus on bipartite graphs and potential improvements to existing bounds.

Contribution

It introduces a new bound connecting the largest eigenvalue and bi-average degree, including a logarithmic factor, and explores the tightness and limitations of these bounds.

Findings

The lower bound is tight for regular and bi-regular graphs.

The upper bound involves a logarithmic factor in the number of vertices.

An example shows the logarithmic factor cannot be significantly improved.

Abstract

We show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $\lambda_{\max}(G)$, letting $$ M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}} $$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$), we have $$ M(G) \le \lambda_{\max}(G) \le \big(\frac14 \log|V| + 1 \big) \M(G). $$ Here the lower bound is attained if $G$ is regular or bi-regular, whereas the logarithmic factor in the upper bound, conjecturally, can be improved --- although we present an example showing that it cannot be replaced with a factor growing slower than $(\log |V|/\log\log|V|)^{1/8}$. Further refinements are established, particularly in the case where $G$ is bipartite.