A spectral bound for vertex-transitive graphs

TL;DR

This paper derives an explicit spectral lower bound for the smallest eigenvalue of the normalized adjacency operator in finite, non-bipartite, vertex-transitive graphs, linking spectral properties to combinatorial features.

Contribution

It introduces a new spectral bound that depends solely on the degree and isoperimetric constant of the graph, advancing understanding of spectral graph theory.

Findings

Provides an explicit lower bound for the smallest eigenvalue

Connects spectral bounds to isoperimetric constants

Enhances spectral analysis of vertex-transitive graphs

Abstract

For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on the graph only through its degree and its isoperimetric constant.