Integral Cayley graphs over a nonabelian group of order $8n$

TL;DR

This paper characterizes which Cayley graphs over a specific nonabelian group of order 8n are integral, using group representation theory to determine eigenvalue integrality and identifying families of such graphs.

Contribution

It provides necessary and sufficient conditions for integrality of Cayley graphs over the nonabelian group T_{8n}, including explicit representations and characterizations.

Findings

Derived irreducible matrix representations and characters of T_{8n}

Established criteria for Cayley graph integrality over T_{8n}

Characterized families of connected integral Cayley graphs

Abstract

A graph is called an integral graph when all eigenvalues of its adjacency matrix are integers. We study which Cayley graphs over a nonabelian group $$ T_{8n}=\left\langle a,b\mid a^{2n}=b^8=e,a^n=b^4,b^{-1}ab=a^{-1} \right \rangle $$ are integral graphs. Based on the group representation theory, we first give the irreducible matrix representations and characters of $T_{8n}$. Then we give necessary and sufficient conditions for which Cayley graphs over $T_{8n}$ are integral graphs. As applications, we also characterize some families of connected integral Cayley graphs over $T_{8n}$.