Eigenvalues of graphs and spectral Moore theorems

TL;DR

This paper explores spectral Moore theorems that establish bounds on the maximum size of graphs with given properties, linking spectral graph theory to classical extremal graph problems.

Contribution

It presents recent spectral Moore theorems and demonstrates their applications in Alon-Boppana bounds and degree-diameter problems.

Findings

Spectral Moore theorems provide bounds on graph order based on eigenvalues.

Applications include improved bounds in regular graph theory.

Connections to classical degree-diameter and Moore problems are established.

Abstract

In this paper, we describe some recent spectral Moore theorems related to determining the maximum order of a connected graph of given valency and second eigenvalue. We show how these spectral Moore theorems have applications in Alon-Boppana theorems for regular graphs and in the classical degree-diameter/Moore problem.