Strongly cospectral vertices in normal Cayley graphs

TL;DR

This paper establishes bounds on the number of strongly cospectral vertices in normal Cayley graphs, with explicit results for certain groups and constructions of infinite families with multiple such vertices.

Contribution

It provides an upper bound on strongly cospectral vertices in normal Cayley graphs and constructs infinite families with multiple such vertices.

Findings

Upper bound on pairwise strongly cospectral vertices in normal Cayley graphs

Explicit bounds for Cayley graphs of ^d and ^d

Existence of infinite families with four pairwise strongly cospectral vertices

Abstract

We prove an upper bound on the number of pairwise strongly cospectral vertices in a normal Cayley graph, in terms of the multiplicities of its eigenvalues. We use this to determine an explicit bound in Cayley graphs of $\mathbb{Z}_2^d$ and $\mathbb{Z}_4^d$. We also provide some infinite families of Cayley graphs of $\mathbb{Z}_2^d$ with a set of four pairwise strongly cospectral vertices and show that such graphs exist in every dimension.