On the Spectrum of a Class of Distance-transitive Graphs

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

Distance-regular Cayley graphs with small valency

We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most $4$, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency $5$, and the Cayley graphs among all distance-regular graphs with girth $3$ and valency $6$ or $7$. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table