Note on the second eigenvalue of regular graphs

TL;DR

This paper provides a concise, self-contained proof establishing nearly optimal lower bounds for the second eigenvalue of the adjacency matrix in regular graphs, contributing to spectral graph theory.

Contribution

It offers a new, simplified proof of bounds on the second eigenvalue, enhancing understanding of spectral properties of regular graphs.

Findings

Derived nearly optimal lower bounds for the second eigenvalue

Provided a self-contained proof accessible to a broad audience

Clarified spectral properties relevant to regular graphs

Abstract

The goal of this expository note is to give a short, self-contained proof of nearly optimal lower bounds for the second largest eigenvalue of the adjacency matrix of regular graphs.