Ela the Path Polynomial of a Complete

When the graph is a path with n vertices, we simply call PG the path polynomial and denote it by Pn. Define An as the adjacency matrix of a path on n vertices. For several interesting classes of graphs, A(Gi) is a polynomial in A(G), where Gi is the ith distance graph of G ([5]). Actually, for distance-regular graphs, A(Gi) is a polynomial in A(G) of degree i, and this property characterizes these kind of graphs ([14]). In [4], Beezer has asked when a polynomial of an adjacency matrix will be the adjacency matrix of another graph. Beezer gave a solution in the case that the original graph is a path. Theorem 1.1 ([4]). Suppose that p(x) is a polynomial of degree less than n. Then p(An) is the adjacency matrix of graph if and only if p(x) = P2i+1(x), for some i, with 0 ≤ i ≤ n2 − 1. In the same paper, Beezer gave an elegant formula for Pk (An) with k = 1, . . . , n, and Bapat and Lal, in [1], completely described the structure of Pk (An), for all integers k. This result was also reached by Fonseca and Petronilho ([10]) in a noninductive way. Theorem 1.2 ([1],[4],[10]). For 0 ≤ k ≤ n− 1, n being a positive integer,