Prime ideal graphs of commutative rings

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Introduction
In [5] Beck defined a zero divisor graph for commutative ring R with vertex set R and two vertices are adjacent if their product is zero. It is denoted by Γ(R). After this time, the study of algebraic structures using the properties of graphs has become an exciting research topic. It is leading to many fascinating results and questions. There are many papers on assigning a graph to a ring, for example see [7,1,2]. In [3], Anderson and Livingston did a little change with definition of zero diversor graph Γ(R) by replacing its vertex set with Z(R) \ {0}. In [7,4,1,2] several other type of graphs such as the zero divisor graph Γ(R), unit graph G(R) and nilpotent graph Γ N (R) has been defined for a commutative ring R.
A graph is connected if there exists a path connecting any two distinct vertices. The distance between vertices x and y is the length of the shortest path connecting them, and the diameter of a graph Γ, diam(Γ), is the supremum of these distances. The girth of a graph, denoted g(Γ) is the length of the shortest cycle in the graph G. The eccentricity of a vertex v in Γ, denoted by ecc(v), is the largest distance between v and any other vertex u in Γ. The eccentric connectivity polynomial of a graph Γ is defined by n(n−1) , where n is the order of Γ. Let R be a commutative ring with unity. The nilpotent graph of R denoted by Γ N (R), is a graph with vertex set R\{0} and two distinct vertices x and y are connected if and only if xy ∈ N , where N is the set of all nilpotent elements of R [7]. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R [4]. The zero-divisor graph of a ring R is defined as the graph Γ(R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, are adjacent if and only if xy = 0 [1].
In this paper, we introduce a new type of graph structure on a commutative ring R as follows: we take R \ {0} as a vertex set and two distinct vertices x and y are adjacent if xy ∈ P , where P is a prime (primary) ideal of R. It is called the prime (primary) ideal graph that associated to (primary) ideal of a ring R and denoted Γ P . Our definition is the generalization of the zero-divisor graph, that is the zero-divisor graph and nilpotent graph of R are the subgraphs of this a new graph.
In section 2, we give some basic results and concepts which will be used in our study. Section 3, devoted to study the relationship among prime ideal graph and other type of graphs. Furthermore, we show that the prime ideal graph Γ P is simply connected with diameter at most 2 and girth 3 if it contains cycle. Also, we prove that it has equal number of chromatic and clique. Finally, we compute the eccentric connectivity polynomial (the total eccentricity polynomial), the transmission and the mean distance of Γ P .

Preliminaries
The following result and concept are well known in [8].
Definition 2.1. Let R be a ring. An element a ∈ R is said to be nilpotent if a n = 0 for some positive integer n. The set of nilpotent elements in a ring R is called the nilradical of R. We denote the nilradical by N .
Theorem 2.1. Let N be the nilradical of a ring R. Then N is the intersection of all prime ideals in R that is, Definition 2.2. A subset X of the vertices of G is called a clique if the induced subgraph on X is a complete graph. The maximum size of a clique in a graph G is called the clique number of G. It is denoted by ω(G). Definition 2.4. A subset X of the vertices of G is called an independent set if the induced subgraph on X has no edges. The maximum size of an independent set in a graph G, denoted by α(G) is called the independence number of G.
Definition 2.5. A proper coloring of a graph G is a coloring of the vertices of G so that no two adjacent vertices receive the same color. The chromatic number χ(G) is the least integer k such that there is a proper coloring of G using k colors.

Some Properties of Prime Ideal Graphs
Throughout this paper all rings are finite commutative with identity. Now, we study the relationship among prime ideal graphs and other kind of graphs on the same ring. We will show that other type of graphs such as zero-divisor graph, nilpotent graph and primary ideal graph are subgraphs of our graph.
Definition 3.1. Let R be a commutative ring with unity. The prime ideal graph of R denoted by Γ P (R), is a graph with vertex set R \ {0} and two distinct vertices x and y are connected if and only if xy ∈ P , where P is prime ideal of R.
The prime ideals of R are P 1 = x + 1 and P 2 = x 2 + x + 1 . The prime ideal graphs of them are the following: Γ P 2 = K 1,6 and Γ P 1 = Lemma 3.1. Every zero divisor graph is subgraph of prime ideal graph.
The converse of Lemma 3.1 is not true for the following example;

G(R)
Hence Γ(R) and G(R) are subgraphs of Γ 2 In general, there is no relationship between unit graph and prime ideal graph. For instance in Z 77 , there is an edge between 3 and 5 in G(Z 77 ) because 3+5=8 which is unit in Z 77 . However 3 * 5 = 15 which is also unit in Z 77 . This edge does not belong to prime ideal graph because P does not contain unit elements.  Proof. By definitions both graph have the same number of vertices. Let xy ∈ E(Γ N (R)), then by Theorem 2.1, we have xy ∈ P prime ideal P , so xy ∈ P for all prime ideal P of R. Therefore E(Γ N (R)) ⊆ E(Γ P ). We are done.  Proof. Clear. Now we present our main results which are related to Γ P . The following lemma tell us our graph is always bi-regular. Throughout this paper, we assume that P * = P \ {0} and |P | = h. Lemma 3.5. Let P be a prime ideal of a finite ring R and x ∈ Γ P . Then Proof. It is obvious from the definition of ideal rp ∈ P for any r ∈ R and p ∈ P . Then all elements in R \ {0} connected to p. As p is not connected to itself, therefore degree of each element in P is |R| − 2.
Moreover, for x / ∈ P , there are two cases, case one if x is connected to all elements in P \ {0}. Thus we have h − 1 edges and the other case come from definition of prime ideal graph.
Theorem 3.2. Let R be a ring and P be a prime ideal of R with h elements. Then the prime ideal graph has |R| − 1 vertices and (h − 1)(|R| − 1 − h 2 ) edges. Proof. We know that each element either in P or not. By Lemma 3.5, we have Now there are h − 1 elements in P * = P \ {0}. Also there are (|R| − 1) − (h − 1) = |R| − h elements which are not in P * . From the handshaking lemma, we have x∈V (Γ P ) deg(x) = 2|E(Γ P )|, that is . Lemma 3.6. Let P be a prime ideal of a finite ring R and Γ P be the prime ideal graph. Then d(x, y) = 1 if x ∈ P or y ∈ P 2 if x / ∈ P and y / ∈ P .
Proof. Let x, y ∈ Γ P . For x, y ∈ R \ {0}, if x or y in P , then xy ∈ P . So there is an edge between of them, that is d(x, y) = 1. Let x / ∈ P and y / ∈ P , then xp ∈ P and yp ∈ P , for all p ∈ P , that is there are edges between x and p (y and p). Thus d(x, y) ≤ d(x, p) + d(p, y) = 1 + 1 = 2 and the result follows. Theorem 3.3. For any ring R, Γ P is a simple, connected graph with diameter two.
Proof. The connectedness and simplicity follow from Definition 3.1. The rest follows from Lemma 3.6 Proposition 3.1. Let R be a ring and P be a prime ideal of R. The following hold: 1. If |P | = 2, then gr(Γ P ) = ∞. 2. If |P | ≥ 2, then gr(Γ P ) = 3.

Proof.
1. It is clear that x is adjacent to y and x or y is adjacent to all elements in R. Furthermore, far any r 1 in R is not adjacent to r 2 because r 1 , r 2 / ∈ P . Hence Γ P has no cycle and the result follows. 2. Since |P | > 2, then we can assume that without loss of generality P = {x, y, z}. All elements in P are adjacent together. Therefore Γ P contains a cycle of length 3, as required.
Lemma 3.7. Let R be a ring and let P be a primary ideal of R with h elements such that it is not prime ideal, then degree of each vertex of the primary ideal graph is defined by if xy ∈ P, then neither x nor y in P h − 1 otherwise Proof. Clear. We observe that Γ P is a subgraph of Γ <2> .
Theorem 3.4. Let R be a ring and P be a primary ideal of R with h elements, which is not prime ideal, then the primary ideal graph has order |R| − 1 and size h−1 2 ( 3|R| 2 − 2) + 2 h 4 ( |R| 2 − h).