The Structure of the Generalized Cayley Graph Cay_m (〖 D〗_2n,S) of Valency Three

Abstract

     Suppose that G is a finite group and S is a non-empty subset of G such that e∉S and S^(-1)⊆S. Suppose that Cay(G,S) is the Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy^(-1)∈S. In this paper,we introduce the generalized Cayley graph denoted by Cay_m (G,S) that is a graph with vertex set consists of all column matrices X_m  which all components are in G and two vertices X_m  and Y_m  are adjacent if and only if X_m [(Y_m )^(-1) ]^t∈M(S), where 〖Y_m〗^(-1)  is a column matrix that each entry is the inverse of similar entry of Y_m  and M(S) is m×m matrix with all entries in S , [Y^(-1) ]^t  is the transpose of Y^(-1)  and m≥1. We aim to determine the structure of Cay_m (G,S) when G is the dihedral group of order 2n and | S |= 3 for every m≥2, n≥3.