### Keywords : Width Distance

##### w-Hosoya Polynomials for Connection for Some Special Graphs

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2012,* Volume 9, Issue 2, Pages 173-184

DOI:
10.33899/csmj.2012.163726

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.

##### w-Hosoya polynomials for Pentagonal Chains

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2012,* Volume 9, Issue 1, Pages 79-91

DOI:
10.33899/csmj.2012.163689

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.

##### Hosoya Polynomials of The Width Distance of Some Cog-Special Graphs

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2011,* Volume 8, Issue 2, Pages 123-137

DOI:
10.33899/csmj.2011.163647

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.

##### Wiener Polynomials of the Width Distance for Compound Graphs of G1 ☒ G2

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2010,* Volume 7, Issue 2, Pages 31-46

DOI:
10.33899/csmj.2010.163895

For a connected vertex disjoint graphs G_{1} and G_{2} , we define G1 ☒ G2 as the graph obtained from the union of G_{1 }and G_{2 }with four edges joining the vertices of an edge of G_{1 }to the vertices of an edge of G_{2 }.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.

##### w-Wiener polynomials of Width Distance for Cartesian Product K2 with Cycle and Wheel

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2009,* Volume 6, Issue 3, Pages 37-55

DOI:
10.33899/csmj.2009.163837

The w-Wiener polynomials of the cartesian product of K_{2} 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 K_{2}×C_{t }and K_{2}×W_{t} are also obtained .

##### w-Wiener Polynomials for Width Distance of the Cartesian Product of K2 with Special Graphs

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2008,* Volume 5, Issue 2, Pages 117-133

DOI:
10.33899/csmj.2008.163989

, be the w- width , Let G be a k_{0}-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 K_{2}with 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.

##### w-Wiener Polynomials of the Width Distance of the Square of a Path and a Cycle and a m-Cubic

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2008,* Volume 5, Issue 1, Pages 11-32

DOI:
10.33899/csmj.2008.163959

Let G be a k_{0}-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 .

##### w-Wiener Polynomials for Width Distance of Some Special Graphs

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2007,* Volume 4, Issue 2, Pages 103-124

DOI:
10.33899/csmj.2007.164030

Let G be a k_{0}-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.