Idempotent Divisor Graph of Commutative Ring

This work aims to introduce and to study a new kind of divisor graph which is called idempotent divisor graph, and it is denoted by . Two non-zero distinct vertices v1 and v2 are adjacent if and only if , for some non-unit idempotent element . We establish some fundamental properties of , as well as it’s connection with . We also study planarity of this graph.


1.
Introduction Let be a finite commutative ring with unity . We denote , and the set of zero divisors, the set of idempotent elements and the set of unit elements respectively. In [1], Beck introduced the idea that connects between ring theory and graph theory when studied the coloring of commutative ring. Later in [2], Anderson and Livingston modified this idea when studied the zero divisor graph that have vertices and for , edges if and only if . Many authors studied this notion see for examples [3], [4], [5] and [6] Recently, there are other concepts of zero divisor graph, see for examples [7], [8], [9] and In graph theory " denotes by the eccentricity of a vertex v of a connected graph G which is the number . That means is the distance between v and a vertex furthest from v. The radius of G ,which is denoted by is , while the diameter of G is the maximum eccentricity and it is denoted by . Consequently, is the greatest distance between any two vertices of G. Also, a graph G has radius 1 if and only if G contains a vertex u adjacent to all other vertices of G. A vertex v is a central ISSN: 0067-2904 vertex if and the center is the sub-graph of G that induced by its central vertices. The girth of a graph G is the length of a shortest cycle contained in G, it is denoted by . The neighborhood of x in a graph G denotes by , is the set of all such that y is adjacent to x in G. In our graph in this case, | . symbolized complete graph and complete bipartite graph respectively. we call star graph. A clique number of G symbolized is greats complete subgraph of G. If a connected graph does not contain cycle, we call tree. Let and two graphs, is a graph with and , and for , ⋃ the graph is a graph with and . A path graph of order n is denoted by is a graph with and , so that is a graph and it called a cycle graph of order for . For more details see for example" [11]. In ring theory, a ring is said to be local if has exactly one maximal ideal. Also, if finite local ring, then the cardinality of R symbolized | | equal , where prime number and , as well as the cardinality of maximal ideal , where A ring R is called Boolean, if every element is an idempotent. We denote is a field order . In section two we defined a new graph on the ring and prove some basic properties of about this graph and we give all possible graphs less than or equal 6 vertices. In section three, we give all graphs to be planer.

2.
Examples and Basic Properties In this section, we introduce a new class of divisor graph manly idempotent divisor graph, we give some of about this graph, and we also provide some examples. Definition 2.1: The undirected graph is called idempotent divisor graph, and which is symbolized by which a simple graph with vertices set in , and two nonzero distinct vertices and are adjacent if and only if , for some non-unit idempotent element .Example 1: Let , since the idempotent elements , then is:

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If has only idempotent elements 0 and 1, then . Consequently, when R local , then .

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If finite non local ring, then Since , then has idempotent element distinct .

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If non-local ring, then there are at greater than or equal two non-trivial idempotent elements in . if , then also idempotent and (because if , then and implies that which is a contradiction. Therefore, ). Hence if , then adjacent with , for every , so that ( ) . Example 2: We shall give all possible idempotent divisor graphs, with . If | | , then is local and | | , so by [12] . If | | , then is local and | | , so by [12] . If | | , and is local, then | | , so that by [12].    Recall that "a graph is said to be planar if it can be drawn in the plane in such a way that pairs of edges intersect only at vertices, if at all. If has no such representation, G is called , where and or non-planar. It we know that a graph is planar if and only if contained no sub-graph or " [11]. Proposition 3.3: For any local ring , a graph is planar if and only if is isomorphic to one of the following table: