Maximizing the order of a regular graph of given valency and second eigenvalue

TL;DR

This paper explores the maximum size of k-regular graphs with a second eigenvalue below a certain threshold, building on foundational spectral graph theory results.

Contribution

It investigates the maximum number of vertices in k-regular graphs constrained by their second eigenvalue, extending spectral bounds to graph size limits.

Findings

Determines bounds on the number of vertices for given valency and eigenvalue constraints.

Provides new insights into the spectral properties influencing graph size.

Extends classical spectral graph theory results to finite graph enumeration.

Abstract

From Alon and Boppana, and Serre, we know that for any given integer $k\geq 3$ and real number $\lambda<2\sqrt{k-1}$, there are finitely many $k$-regular graphs whose second largest eigenvalue is at most $\lambda$. In this paper, we investigate the largest number of vertices of such graphs.