Hosoya Polynomial , Wiener Index , Coloring and Planar of Annihilator Graph of Z

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 Zn denote ΓB(Zn), with n = p m or n = pq, where p,q are distinct prime numbers and m is an integer with m ≥ 1.


Introduction
Let be a commutative ring with identity the annihilator of is the set of all element ∈ satisfy ( ) = { ∈ : . = 0 , ∀ ∈ } [6], and let ( ) be the set of all annihilator in . We consider a simple graph Γ B ( ) to The notion of an annihilator graph of a commutative ring was first introduced in 1988 by Beck [5], where he was interested in colorings, this investigation was then continued by Anderson and Nasser [3] zero-divisor graph of a commutative ring, further that Anderson and Livingston [2]. They denoted that by Γ( ). It is clear that from Beck's definition of annihilator graph of a commutative ring and Anderson's definition of a zero-divisor graph of a commutative ring can be defined Annihilator graph of a commutative ring can be defined Γ B ( ) = ((Γ( ) ∪ { ( * ) − ( ) * }) + 1 ). Such that: Γ( ) zero-divisor graph of the ring, ( * ) set of all vertices in non-zero, ( ) * set of all non-zero zero-divisors in and 1 = 0.
A graph is called a connected graph if there is at least one path between any pair of vertices in , otherwise it is called disconnected [7]. For vertices , of , let ( , ) be the length of the shortest path from to (and it is called distance between two vertices , in ). The maximum distance between any two vertices , in is called the diameter graph [7], that is set of all vertices of . A graph is complete if every two of its vertices are adjacent, so the complete graph of order is denoted by . If the vertex set of a graph can be split into two disjoint sets and (such that the induced subgraph that generated by either or is null graph), then we said is a bipartite graph. This graph is also said to be a complete bipartite graph is a bipartite graph in graph if each vertex in the set has joined to every vertex in the set with just one edge. Hosoya polynomial of the graph is defined by ( ; ) = ∑ ( , ) , where ( , ) is the number of pairs of vertices of a graph , that are at distance apart, for = 0,1,2, … , ( ). The Wiener index of is defined as the sum of all distances between vertices of the graph , and denoted by ( ), we can also find this index by differentiating Hosoya polynomial with respect to at x =1, by symbols we can write: , See [8,12].
Let ( ) denote the chromatic number of vertices, i.e., the minimal number of colors, which can be assigned to the vertices of in such a way that every two adjacent vertices have different colors [7]. We let ̌( ) denote the chromatic number of edges, i.e., the minimal number of colors, which can be assigned to the edges of in such a way that every two adjacent edges have different colors [7]. And last we assumed ( ) denote the chromatic number of faces, i.e., the minimal number of colors, that can be assigned to the faces of planar graph G in such a way that every two adjacent faces have different colors [7]. A planar graph is a graph that can be drawn in the plane without crossings for any two edges in [7]. There are many studies in the graph properties and commutative ring. See [1], [4], [10]& [11].
. Some Properties of Graph ( ) We will start this item by a lemma.
Now, we find the coefficient of 2 as the diameter of the graph Γ B ( ) is two from the Lemma (2.1) and using Lemma (2.2) so we get:

Corollary 2.4: The Wiener index of graph Γ B (
) where is prime number and is an integer with ≥ 1.
Clearly the complete graph (when is an even).

Proof:
a-From the Theorem (2.5-A-1) we get the first part of the Theorem directly.
Since the multiplication of the number +1 2 or one of its complications b-From the Theorem (2.5-A-2) we get the first part of the Theorem directly.
Since the multiplication of the number +2 2 or one of its complications From Theorem (2.7-b) we get the graph Γ B ( 16 ) contains a subgraph that is homeomorphic to 4 and 6,2 then the graph Γ B ( 16 ) it is planar by kuratowski's Theorem. ∴ (Γ B ( 16 )) = 3. Now, we find the coefficient of 2 as the diameter of the graph Γ B ( ) is two from the Lemma (2.1) and using Lemma (2.2) so we get.  is the largest complete subgraph exist in the graph Γ B ( ) (when is an odd). It is also clear that the number product of multiplication the number or one of its complications in the number or one of its complications thus a new vertex will be added to the complete graph is the largest complete subgraph exist in the graph Γ B ( ) hence the chromatic number of the graph Γ B ( ) is ( 2 + 1)

The chromatic number of vertices the graph Γ B (Z ) is
B-From the Lemma (2.1) so it is the vertex (0) connect with every vertex the graph Γ B ( ) then the degree of the vertex (0) is ( − 1) so it is the chromatic number of the edges is ( − 1).
The second part, since the multiplying the number or one of its complications

Remark:
From the Theorem (3.4), the only graphs of the formula Γ B ( ) when = 2 and = 1 does not contain a subgraph homeomorphic 3,3 or 5 therefore it is planar and colorable for faces. Otherwise, the graphs of the formula Γ B ( ) contain a subgraph homeomorphic 3,3 or 5 therefore it is not planar graphs by kuratowski's Theorem.