On Cayley graphs over generalized dicyclic groups

TL;DR

This paper investigates the spectral properties of Cayley graphs over generalized dicyclic groups, providing conditions for their integrality and extending previous results on related group structures.

Contribution

It offers a complete characterization of when Cayley graphs over generalized dicyclic groups are integral, and conditions for their distance powers to be integral, generalizing prior work on dicyclic groups.

Findings

Necessary and sufficient condition for Cayley graph integrality over generalized dicyclic groups

Sufficient conditions for integrality of all distance powers of these graphs

Extension of previous results on dicyclic groups to generalized cases

Abstract

Recently, several works by a number of authors have studied integrality, distance integrality, and distance powers of Cayley graphs over some finite groups, such as dicyclic groups and (generalized) dihedral groups. Our aim is to generalize and/or to give analogues of these results for generalized dicyclic groups. For example, we give a necessary and sufficient condition for a Cayley graph over a generalized dicyclic group to be integral (i.e., all eigenvalues of its adjacency matrix are in $\mathbb{Z}$). We also obtain sufficient conditions for the integrality of all distance powers of a Cayley graph over a given generalized dicyclic group. These results extend works on dicyclic groups by Cheng--Feng--Huang and Cheng--Feng--Liu--Lu--Stevanovic, respectively.