AL-Rafidain Journal of Computer Sciences and Mathematics

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.
 

The minimum distance of a vertex v to an set of vertices of a graph G is defined as :
      .
The n-Wiener polynomial for this distance of a graph G is defined as
      ,
where  is the number of order pairs (v,S), , such that
      ,
and  is the diameter for this minimum n-distance.
In this paper, the n-Wiener polynomials for some types of graphs such as complete graphs, bipartite graphs, star graphs, wheel graphs, path and cycle graphs are obtained .The n-Wiener index for each of these special graphs is given. Moreover, some properties of the coefficients of   are established.