Suppose that ????????????(????, ????) is the Cayley graph whose vertices are all elements of ???? and two vertices ???? and ???? are adjacent if and only if ????????−1 ∈ ????. In this paper,we introduce the generalized Cayley graph denoted by ????????????????(????, ????) which is a graph with a vertex set consisting of all column matrices ???????? in which all components are in ???? and two vertices ???? ???? and ???????? are adjacent if and only if ????????[(????????)−1]???? ∈ ????(????), where ????????−1 is a column matrix that each entry is the inverse of the similar entry of ???????? and ????(????) is ???? × ???? matrix with all entries in ???? , [????−1]???? is the transpose of ????−1 and ???? ≥ 1 and m∈ ????. We aim to provide some basic properties of the new graph and determine the structure of ????????????????(????, ????) when ????????????(????, ????) is a complete graph ???????? for every ???? ≥ 2, n ≥ 3 and n, m∈ ???? .

Keywords