On the spectrum of a class of distance-transitive graphs

Seyed Morteza Mirafzal (Department of Mathematics Lorestan University, Khoramabad, Iran)
Ali Zafari (Department of Mathematics, Lorestan University, Khoramabad, Iran)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 5, No 1 (2017): Electronic Journal of Graph Theory and Applications

Let $\Gamma=Cay(\mathbb{Z}_n, S_k)$ be the Cayley graph on the cyclic additive group $\mathbb{Z}_n$ $(n\geq 4),$  where  $S_1=\{1, n-1\}$, \dots , $S_k=S_ {k-1}\cup\{k, n-k\}$ are the inverse-closed subsets of $\mathbb{Z}_n-\{0\}$ for any $k\in \mathbb{N}$, $1\leq k\leq [\frac{n}{2}]-1$. In this paper,  we will show that $\chi(\Gamma) = \omega(\Gamma)=k+1$ if and only if $k+1|n$. Also, we will show that if $n$ is an even integer and $k=\frac{n}{2}-1$ then $Aut(\Gamma)\cong\mathbb{Z}_2 wr_{I} {Sym}(k+1)$ where $I=\{1, \dots , k+1\}$ and in this case, we show that $\Gamma$ is an  integral graph.