w-Hosoya Polynomials for Connection for Some Special Graphs

Let and be any two distinct vertices in a connected graph . A container is a set of internally disjoint -paths. The width of is denoted by or is , and the length of is the length of the longest -path in . Then, for a given positive integer w, the width distance between any two distinct vertices u and v in a connected graph is define by: , where the minimum is taken over all containers of width w.
In this paper, we find the Hosoya polynomials, and Wiener indices of the join of two special graphs such as bipartite complete graphs, paths, cycles, star graphs and wheel graphs with respect to the width distance.