Journal of Scientific Research and Reports

For a simple connected undirected graph G = (V;E), the Wiener index W(G) of G is defined as half the sum of the shortest-path distances between all pairs of vertices u; v of G. The kth power of a graph G, denoted by G^k, is a graph with the same vertex set as G such that two vertices are adjacent in G^k if and only if their distance is at most k in G. Let P_n be a path on n vertices. In this paper, for the graph G = P_n2P_n, we obtain a closed form expression for W(G²). In addition, a correct closed form expression is stated forW (P³_n). But we are unable to provide a proof forW (P³_n) of how such expression has arrived. This may be compared with the existing result: for a graph G = P_n2P_n, W(G²) can be computed by an algorithm in linear time.