A Cheeger inequality for the lower spectral gap

TL;DR

This paper establishes lower bounds relating spectral gaps to Cheeger constants in various classes of vertex-transitive graphs, confirming a conjecture about Cayley sum graphs.

Contribution

It proves new inequalities linking spectral gaps and Cheeger constants for Cayley and related graphs, answering a previously open question.

Findings

Edge bipartiteness constant is proportional to Cheeger constant over degree.

Vertex bipartiteness constant is proportional to Cheeger constant.

Smallest eigenvalue of normalized adjacency is bounded below by a quadratic function of the Cheeger constant.

Abstract

Let $\Gamma$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Denote the degree of $\Gamma$ by $d$, its edge Cheeger constant by $\mathfrak{h}_\Gamma$, and its vertex Cheeger constant by $h_\Gamma$. Assume that $\Gamma$ is undirected, non-bipartite. We prove that the edge bipartiteness constant of $\Gamma$ is $\Omega({\mathfrak{h}_\Gamma}/{d})$, the vertex bipartiteness constant of $\Gamma$ is $\Omega(h_\Gamma)$, and the smallest eigenvalue of the normalized adjacency operator of $\Gamma$ is $-1 + \Omega({h_\Gamma^2}/{d^2})$. This answers in the affirmative a question of Moorman, Ralli and Tetali on the lower spectral gap of Cayley sum graphs.