Automorphism group of a family of distance-regular graphs which are not distance-transitive

Let G_n=\mathbb{Z}_n\times \mathbb{Z}_n for n\geq ۴ and S=\{(i,۰),(۰,i),(i,i): ۱\leq i \leq n-۱\}\subset G_n. Define \Gamma(n) to be the Cayley graph of G_n with respect to the connecting set S. It is known that \Gamma(n) is a strongly regular graph with the parameters (n^۲, ۳n-۳, n, ۶) \cite{۱۹}. Hence \Gamma(n) is a distance-regular graph. It is known that every distance-transitive graph is distance-regular, but the converse is not true. In this paper, we study some algebraic properties of the graph \Gamma(n). Then by determining the automorphism group of this family of graphs, we show that the graphs under study are not distance-transitive.

کلیدواژه ها:

strongly regular graph ، distance-transitive graph ، graph automorphism ، Clique