AL-Rafidain Journal of Computer Sciences and Mathematics

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^mq, where p, q are distinct prime numbers and m is an integer with m ≥ 1 .
 

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.

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.
 

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.
 

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.
 

, 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 .
 

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.
 

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.