### Keywords : Wiener index

##### Hosoya Polynomial, Wiener Index, Coloring and Planar of Annihilator Graph of Zn

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 2, Pages 41-52

DOI:
10.33899/csmj.2020.167337

Let R be a commutative ring with identity. We consider Γ_{B}(R) an annihilator graph of the commutative ring R. In this paper, we find Hosoya polynomial, Wiener index, Coloring, and Planar annihilator graph of Z_{n} denote Γ_{B}(Z_{n}) , with n= p^{m} or n=p^{m}q, where p, q are distinct prime numbers and m is an integer with m ≥ 1 .

##### Zero Divisor Graph Of ZpM qr with Applications

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 2, Pages 13-23

DOI:
10.33899/csmj.2019.167334

In this paper, we study zero-divisor graph of the ring Z_{p}^{m}_{qr} and give some properties of this graph. Also, we find the chromatic number, Hosoya polynomial and Wiener index of this graph.

##### On Annihilating - Ideal Graph of Zn

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2018,* Volume 12, Issue 2, Pages 31-42

DOI:
10.33899/csmj.2018.163576

In this paper, we study and give some properties of annihilating-ideal graphs of Z_{n}, also we find Hosoya polynomial and Wiener index for this graph.

##### Hosoya Polynomials of Generalized Vertex Identified and Edge Introducing Graphs

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2013,* Volume 10, Issue 3, Pages 77-88

DOI:
10.33899/csmj.2013.163528

The vertex identified and edge introducing graphs of two disjoint connected graphs are extended to n, , disjoint connected graphs. Hosoya polynomials and Wiener indices of such composite graphs are obtained by applying Gutman's Theorem and using mathematical induction on n.

##### 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.

##### 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 .

##### Hosoya Polynomials of Steiner Distance of Complete m-partite Graphs and Straight Hexagonal Chains(*)

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2008,* Volume 5, Issue 1, Pages 115-126

DOI:
10.33899/csmj.2008.163953

The Hosoya polynomials of Steiner distance of complete m-partite graphs and Straight hexagonal chains are obtained in this paper. The Steiner *n*-diameter and Wiener index of Steiner n-distance of and 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.