### Main Subjects : Graph Theory

##### Schultz and Modified Schultz Polynomials for Vertex – Identification Chain and Ring – for Hexagon Graphs

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2021,* Volume 15, Issue 1, Pages 25-38

DOI:
10.33899/csmj.2021.168251

The aim of this paper is to find polynomials related to Schultz, and modified Schultz indices of vertex identification chain and ring for hexagonal rings (6 – cycles). Also to find index and average index of all of them.

##### The n-Hosoya Polynomials of the Square of a Path and of a Cycle

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2021,* Volume 15, Issue 1, Pages 13-24

DOI:
10.33899/csmj.2021.168250

The n-Hosoya polynomial of a connected graph G of order t is defined by:

Hn (G;x) = ∑ Cn (G;x) xk, where, Cn(G,k) is the number of pairs (v,S), in which |S| = n -1, 3 ≤ n ≤ t, v ∈ V(G) , S ⊆ V (G) , such that dn(v,S) = k , for each 0 ≤ k ≤ δn. In this paper, we find the n-Hosoya polynomial of the square of a path and of the square of a cycle. Also, the n-diameter and n-Wiener index of each of the two graphsare determined

##### The Basis Number of Symmetric Difference of K2 with Some Ladder Graphs

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2021,* Volume 15, Issue 1, Pages 13-22

DOI:
10.33899/csmj.2021.168256

The basis number of a graph G is defined to be the least integer k such that G has a k-fold cycle basis. We investigate the basis number of symmetric difference of K_{2} with a ladder graph L_{m }, acircular ladder, and a Möbius ladder.

##### 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.

##### The Restricted Detour Polynomial of the Theta Graph

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2020,* Volume 14, Issue 1, Pages 13-20

DOI:
10.33899/csmj.2020.164664

The restricted detour distance D*(u,v) between two vertices u and v of a connected graph G is the length of a longest u - v path P in G such that <V(P)> = P. The main goal of this paper is to obtain the restricted detour polynomial of the theta graph. Moreover, the restricted detour index of the theta graph will also be obtained.

##### Weiner Polynomials for Generalization of Distance for Some Special Graphs

*AL-Rafidain Journal of Computer Sciences and Mathematics*,
*2006,* Volume 3, Issue 2, Pages 103-120

DOI:
10.33899/csmj.2006.164061

The minimum distance of a vertex v to an set of vertices of a graph G is defined as :

.

The n-Wiener polynomial for this distance of a graph G is defined as

,

where is the number of order pairs (v,S), , such that

,

and is the diameter for this minimum n-distance.

In this paper, the n-Wiener polynomials for some types of graphs such as complete graphs, bipartite graphs, star graphs, wheel graphs, path and cycle graphs are obtained .The n-Wiener index for each of these special graphs is given. Moreover, some properties of the coefficients of are established.