# Some algebraic properties of a class of integral graphs determined by their spectrum

## TL;DR

This paper studies the algebraic properties of a specific class of integral Cayley graphs on cyclic groups of prime power order, showing they are determined by their spectrum and analyzing their automorphism groups.

## Contribution

It proves that certain Cayley graphs on cyclic groups of prime power order are integral, determines their automorphism groups, and shows they are uniquely identified by their spectrum.

## Key findings

- The Cayley graph $ ext{Cay}(bZ_{n}, S)$ is integral for $n=p^m$.
- The automorphism group of the graph is explicitly determined.
- The graph and its join with the complete graph are uniquely identified by their spectrum.

## Abstract

Let $\Gamma=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $\Gamma$ are integers, then we say that $\Gamma$ is an integral graph. A graph $\Gamma$ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph $\Gamma=Cay(\mathbb{Z}_{n}, S)$, where $n=p^m$, ($p$ is a prime integer, $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\, (a, n)=1\}$. First, we show that $\Gamma$ is an integral graph. Also we determine the automorphism group of $\Gamma$. Moreover, we show that $\Gamma$ and $K_v \bigtriangledown\Gamma$ are determined by their spectrum.