permutation representation of graphs

there are many geometric and algebraic representations of graphs. recently, we introduce a new representation of graphs by use of permutations and present some results about this representation and related parameter. let g be a graph. a k-permutation representation of g is a map p of v (g) to symmetric group s_k, such that for any two vertices v and u, v ∼ u if and only if for each i ϵ {1, 2, 3, · · · , k} we have p(v)(i)≠ p(u)(i). in other words, p(v) o p(u)⁻¹ ϵ d_k where d_k denote the set of all derangements of s_k. we define the permutation representation number pr(g) to be the minimum of k such that g has a k-permutation representation. in addition, we find upper and lower bounds for this parameter of graphs.