Remarks on singular Cayley graphs and vanishing elements of simple groups

TL;DR

This paper explores the singularity of Cayley graphs derived from finite groups, linking graph singularity to character theory, especially focusing on vanishing elements in simple and alternating groups.

Contribution

It connects the singularity of Cayley graphs to the existence of vanishing elements in finite simple groups, providing new insights and approaches in algebraic combinatorics.

Findings

Singular Cayley graphs relate to characters with sum zero over conjugacy classes.

Vanishing elements in simple groups are key to understanding graph singularity.

Proposed methods for constructing singular Cayley graphs.

Abstract

Let $\Gamma$ be a finite graph and let $A(\Gamma)$ be its adjacency matrix. Then $\Gamma$ is {\it singular} if $A(\Gamma)$ is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs ${\rm Cay}(G,H)$ when $G$ is a finite group and when the connecting set $H$ is a union of conjugacy classes of $G.$ In this situation the singularity problem reduces to finding an irreducible character $\chi$ of $G$ for which $\sum_{h\in H}\,\chi(h)=0.$ At this stage we focus on the case when $H$ is a single conjugacy class $h^G$ of $G.$ Here the above equality is equivalent to $\chi(h)=0$. Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element $h\in G$ is called vanishing if $\chi(h)=0$ for some irreducible character $\chi$ of $G.$ We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs.