Computing Wiener Index of (Pn Box Pn)2

Given a simple connected undirected graph G = (V, E), the Wiener index W(G) of G is defined as half the sum of the distances of the form d(u,v) between all pairs of vertices u,v of G, where d(u, v) denotes the distance of a shortest u - v path in 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. We first outline an algorithm for computing W(P_n^k□P_n^k) in linear time. Then we obtain an expression for W ((P_η□P_η)²). This suggests an algorithm for computing W((P_η□P_η)²) in linear time. This may be compared with the existing result: for a graph G with v vertices and e edges, W(G) can be computed by an algorithm in time O(ve).