On the order of regular graphs with fixed second largest eigenvalue

TL;DR

This paper investigates the maximum size of connected k-regular graphs with bounded second largest eigenvalue, establishing bounds that depend linearly on k for fixed eigenvalue constraints.

Contribution

It provides new bounds on the order of regular graphs with a fixed second largest eigenvalue, refining understanding of their size limits.

Findings

Established that v(k, λ) is bounded between 2k+2 and 2k + C(λ) for large k.

Proved the existence of a constant C(λ) depending on λ.

Extended the implications of the Alon-Boppana Theorem for graph order bounds.

Abstract

Let $v(k, \lambda)$ be the maximum number of vertices of a connected $k$-regular graph with second largest eigenvalue at most $\lambda$. The Alon-Boppana Theorem implies that $v(k, \lambda)$ is finite when $k > \frac{\lambda^2 + 4}{4}$. In this paper, we show that for fixed $\lambda \geq1$, there exists a constant $C(\lambda)$ such that $2k+2 \leq v(k, \lambda) \leq 2k + C(\lambda)$ when $k > \frac{\lambda^2 + 4}{4}$.