Spectrum of twists of Cayley and Cayley sum graphs

TL;DR

This paper investigates the spectral properties of twisted Cayley and Cayley sum graphs, establishing bounds on their spectra based on automorphisms, graph degree, and Cheeger constants, with extensions to anti-automorphisms and Schreier graphs.

Contribution

It provides new bounds on the spectra of twisted Cayley and Cayley sum graphs, depending only on degree, automorphism order, and Cheeger constants, extending to anti-automorphisms and Schreier graphs.

Findings

Spectra are bounded away from -1 depending on degree, automorphism order, and Cheeger constant.

Results apply to both Cayley and Cayley sum graphs under undirected and connected conditions.

Similar spectral bounds are obtained for graphs with anti-automorphisms and Schreier graphs.

Abstract

Let $G$ be a finite group with $|G|\geq 4$ and $S$ be a subset of $G$. Given an automorphism $\sigma$ of $G$, the twisted Cayley graph $C(G, S)^\sigma$ (resp. the twisted Cayley sum graph $C_\Sigma(G, S)^\sigma$) is defined as the graph having $G$ as its set of vertices and the adjacent vertices of a vertex $g\in G$ are of the form $\sigma(gs)$ (resp. $\sigma(g^{-1} s)$) for some $s\in S$. If the twisted Cayley graph $C(G, S)^\sigma$ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from $-1$ and this bound depends only on its degree, the order of $\sigma$ and the vertex Cheeger constant of $C(G, S)^\sigma$. Moreover, if the twisted Cayley sum graph $C_\Sigma(G, S)^\sigma$ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from $-1$ and this bound depends only on its degree and the vertex Cheeger constant of $C_\Sigma(G, S)^\sigma$. We also study these twisted graphs with respect to anti-automorphisms, and obtain similar results. Further, we prove an analogous result for the Schreier graphs satisfying certain conditions.