The folded hypercube $FQ_n$ is the Cayley graph Cay$(\mathbb{Z}_2^n,S)$, where $S=\{e_1,e_2,\dots,e_n\}\cup \{u=e_1+e_2+\dots+e_n\}$, and $e_i = (0,\dots, 0, 1, 0,$ $\dots, 0)$, with 1 at the $i$th position, $1\leq i \leq n$. In this paper, we show that the folded hypercube $FQ_n$ is a Distance-transitive graph. Then, we study some properties of this graph. In particular, we show that if $n\geq 4$ is an even integer, then the folded hypercube $FQ_n$ is an $automorphic$ graph, that is, $FQ_n$ is a Distance-transitive primitive graph which is not a complete or a line graph.

The reciprocal degree Distance (RDD), defined for a connected graph G as vertex-degree- weighted sum of the reciprocal Distances, that is, RDD (G) = S u, vÎV (G) dG (u)+dG (v)/dG (u, v). The reciprocal degree Distance is a weight version of the Harary index, just as the degree Distance is a weight version of the Wiener index. In this paper, we present exact formulae for the reciprocal degree Distance of join, tensor product, strong product and wreath product of graphs in terms of other graph invariants including the degree Distance, Harary index, the first Zagreb index and first Zagreb coindex. Finally, we apply some of our results to compute the reciprocal degree Distance of fan graph, wheel graph, open fence and closed fence graphs.

The Distance matrix, Distance eigenvalue, and Distance energy of a connected graph have been studied in detail in literature where as the study on Distance seidel matrix associated with a connected graph is in progress. The eigenvalues ∂ S 1 ≥ ∂ S 2 >. . . ∂S n of the Distance seidel matrix D S (G) of a graph G forms the Distance seidel spectrum of G. We describe here the Distance seidel spectrum of some types of subdivision related graphs of a regular graph in terms of its adjacency spectrum. We also derive analytic expressions for the Distance seidel energy of S¯(Cp), the partial complement of the subdivision graph of a cycle Cp and the Distance seidel energy of S(Cp), the complement of the even cycle C2p

In this paper we classify Distance-regular graphs, including strongly regular graphs, ad-mitting a transitive action of the linear groups L(3; 2), L(3; 3), L(3; 4) and L(3; 5) for which the rank of the permutation representation is at most 15. We give details about constructed graphs. In addition, we construct self-orthogonal codes from Distance-regular graphs obtained in this paper.

For a group G, and a subset S of G such that 1G∉S, let X=Cay (G, S) be the corresponding Cayley graph. Then X is said to be normal edge transitive if NAut (X) (G) is transitive on edges. In this paper, we determine all connected directed Cayley graphs of finite abelian groups with valency at most 3 which are normal edge transitive but not normal.