Remarks on singular Cayley graphs and vanishing elements of simple groups

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.