### Keywords : Width Distance

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

Ahmed M. Ali; Assma Salah Aziz; Hafen Jalal Ahmed

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

Ali Aziz Ali; Shwan Salou Abdul Ilyas

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

Tahir H. Ismail; Ahmed M. Ali

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

Ali A. Ali; Asma S. Aziz

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 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 Ks ☒ Kt   ,   Ks ☒ Gt and    Gs ☒ Gt.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

Ali Aziz Ali; Asma Salah Aziz

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 K2 with a cycle  Ct  and with a wheel Wt 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×Ct   and K2×Wt are also obtained .

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

Ali A. Ali; Asma S. Aziz

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 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 graphKp, Star Sp, Complete bipartite graph Kr,s and path  Pr , 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

Ali A. Ali; Asma S. Aziz

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 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 .

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

Ali A. Ali; Asma S. Aziz

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 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:

Ww(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.