On the Bipartiteness Constant and Expansion of Cayley Graphs

TL;DR

This paper establishes a quantitative relationship between the non-bipartiteness of Cayley graphs and their spectral gap, linking eigenvalues to expansion properties, which improves previous bounds and highlights structural differences from general graphs.

Contribution

The paper provides a new lower bound on the smallest eigenvalue of Cayley graphs in terms of their expansion, improving previous results and demonstrating a tight relationship specific to Cayley graphs.

Findings

Lower bound on eigenvalue gap in Cayley graphs

Improved bound over previous work by Biswas and Saha

Shows the bound does not hold for general non-bipartite graphs

Abstract

Let $G$ be a finite, undirected $d$-regular graph and $A(G)$ its normalized adjacency matrix, with eigenvalues $1 = \lambda_1(A)\geq \dots \ge \lambda_n \ge -1$. It is a classical fact that $\lambda_n = -1$ if and only if $G$ is bipartite. Our main result provides a quantitative separation of $\lambda_n$ from $-1$ in the case of Cayley graphs, in terms of their expansion. Denoting $h_{out}$ by the (outer boundary) vertex expansion of $G$, we show that if $G$ is a non-bipartite Cayley graph (constructed using a group and a symmetric generating set of size $d$) then $\lambda_n \ge -1 + ch_{out}^2/d^2\,,$ for $c$ an absolute constant. We exhibit graphs for which this result is tight up to a factor depending on $d$. This improves upon a recent result by Biswas and Saha who showed $\lambda_n \ge -1 + h_{out}^4/(2^9d^8)\,.$ We also note that such a result could not be true for general non-bipartite graphs.