We call a Clayey graph Γ = Cay(G, S) normal for G, if the right regular representation R(G) of G is normal in the full automorphism group of Aunt(Γ). in this paper, we give a classification of all non-normal Clayey graphs of finite abelian group with valency 6. 

We study a generalization of strongly regular graphs. We call a graph strongly walkregular if there is an ` > 1 such that the number of walks of length ` from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint un...

In [3] Cameron et al. classified strongly regular graphs with strongly regular subconstituents. Here we prove a theorem which implies that distance-regular graphs with strongly regular subconstituents are precisely the Taylor graphs and graphs with a1 = 0 and ai ∈ {0, 1} for i = 2, . . . , d .