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.
 

Properties of the width distance in graphs are given in this paper . The w-Hosoya Polynomials of straight pentagonal chains and of alternate pentagonal chains are obtained with Wiener indices of the width distance of such graphs.
 

In  this paper, we give some definitions and  properties of the width distance and find the Hosoya polynomials, Wiener indices, and the average distances of some special cog-graphs with respect  to the width distance.
 

For a connected vertex disjoint graphs G₁ and G₂ , we define G1 ☒ G2 as the graph obtained from the union of G₁ and G₂ with four edges joining  the vertices of an edge of G₁ to the vertices of an edge of G₂ .In this paper we obtain Wiener polynomials of the width distance-2 for K_s ☒ K_t   ,   K_s ☒ G_t and    G_s ☒ G_t.The Wiener index of each such composite graph is also obtained. 
 

The w-Wiener polynomials of the cartesian product of K₂ with a cycle  C_t  and with a wheel W_t are obtained in this paper , in which  w does not exceed the connectivity of the product graph . The diameter and Wiener index with respect to the width distance -w  for K₂×C_t   and K₂×W_t are also obtained .
 

, be the w- width ,   Let G be a k₀-connected graph, and let the distance between the two vertices  u,v  in G. The w-Wiener polynomial of the width distance of G is defined by:
 
 
                                    The w-Wiener polynomials of the Cartesian product of K₂with Complete graphK_p, Star S_p, Complete bipartite graph K_r,s and path  P_{r ,} are obtained in this paper. The diameter with respect to the width distance-w, and the Wiener index for each such graphs are also obtained.
 

Let G be a k₀-connected graph ,and let , ,be the w- width distance between the two vertices  u,v  in G. The w-Wiener polynomial of the width distance of G is defined by:
 
 
 
The w-Wiener polynomials of the square of a path , the square of a cycle  ,and of an m-cube  are obtained in this paper . The diameter with respect to the width distance –w ,and the Wiener index for each such graphs are also obtained .
 

Let G be a k₀-connected graph ,and let ,,be the w- width, distance between the two vertices  u,v  in G. The w-Wiener polynomial  of the width distance of G is defined by:
 
 
           
  
W_w(G;x) is obtained in this paper for  some special graphs G such as a cycle , a wheel, a theta graph , a straight hexagonal chain , and Wagner graph .The diameter with respect to the width distance – w, and the Wiener index for each such special graphs are also obtained in this paper.