A formula for eigenvalues of integral Cayley graphs over abelian groups

TL;DR

This paper derives an algebraic formula for the eigenvalues of integral Cayley graphs over abelian groups, using an analogue of the Möbius function, to better understand their spectral properties.

Contribution

It provides a new explicit formula for eigenvalues of integral Cayley graphs with connection sets as unions of conjugacy classes in abelian groups.

Findings

Derived an algebraic eigenvalue formula involving a Möbius function analogue.

Characterized when Cayley graphs over abelian groups are integral.

Enhanced understanding of spectral properties of integral Cayley graphs.

Abstract

Let $Z$ be an abelian group, $ x \in Z$, and $[x] = \{ y : \langle x \rangle = \langle y \rangle \}$. A graph is called integral if all its eigenvalues are integers. It is known that a Cayley graph is integral if and only if its connection set can be express as union of the sets $[x] $. In this paper, we determine an algebraic formula for eigenvalues of the integral Cayley graph when the connection set is $ [x]$. This formula involves an analogue of M$\ddot{\text{o}}$bius function.