Volume 31, Number 2, 2019 Volume 31, Number 2, 2019
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Volume 31, Number 2, 2019 Volume 31, Number 2, 2019 ZANCO Journal of Pure and Applied Sciences The official scientific journal of Salahaddin University-Erbil https://zancojournals.su.edu.krd/index.php/JPAS ISSN (print ):2218-0230, ISSN (online): 2412-3986, DOI: http://dx.doi.org/10.21271/zjpas RESEARCH PAPER The Graph Based on a Given Ideal of a Ring Fryad H. Abdulqadr Department of Mathematics, College of Education, Salahaddin University - Erbil, Kurdistan Region, Iraq A B S T R A C T: In this paper we introduce a new kind of graph associated with a commutative ring with identity, and we discover some of its characterizations and properties. Let R be a commutative ring with identity and K be a non-trivial ideal of R. The graph-based on an ideal K, denoted by (R), is the undirected graph whose vertices are non-trivial ideals of R and two of its vertices are adjacent if and only if IJ⊂K. KEY WORDS: Graphs, Annihilating-Ideal Graphs, Zero Divisor Graphs,Graphs based on the ideals. DOI: http://dx.doi.org/10.21271/ZJPAS.31.2.1 ZJPAS (2019) , 31(2);1-8 . 1. INTRODUCTION : In the recent years, many kinds of graphs 2011), (Lalchandani, 2017) and (Selvakumar, associated with a ring were introduced. The 2018). The Girth of the annihilating-ideal graph of concept of zero divisor graphs was introduced by commutative rings was studied by (Ahrari, 2015). (Beck, 1988) but his motive was in graph In this paper, we introduce and study a new kind coloring. The diameter of zero divisor graph of a of graph called graph-based on non-trivial ideals commutative ring with identity was studied by of R. The rings considered in this paper are (Anderson, 1999). The concept of the commutative with identity. Also we use K, annihilating-ideal graph AG(R) of a commutative Max(R) and Min(R) to denote a non-trivial ideal, ring R was introduced by (Behboodi, 2011) as a the set of maximal and minimal ideals of R graph whose vertices are non-trivial ideals of R, respectively. and two distinct vertices I and J are adjacent if and only if IJ=(0). Many of its properties and 2. BACKGROUNDS characteristics were discovered by (Behboodi, In this section, we recall some definitions and theorems that we need in our work. According to (Gary, 1986), we will need the following definitions. * Corresponding Author: Frya H. Abdulqadr Definition2.1: E-mail: [email protected] Article History: 1. The degree of a vertex v in a graph G is the Received:27/06/2018 Accepted:07/02/2019 number of edges of G incident to v. The degree of Published:23/04/2019 a vertex v and the set of all vertices incident to v Abdulqadr .F. /ZJPAS: 2019, 31 (2): 1-8 2 in G are denoted by and NG(v) graph G is n-colorable is called the chromatic respectively. number, and is denoted by X(G). 2. A graph G is complete if every two of its 15. A graph G1 is isomorphic to a graph G2if there vertices are adjacent. A complete graph of order n exists a one to one mapping f, called an is denoted by Kn. A clique number cl(G) of G is isomorphism, from V(G1) onto V(G2) such that f the greatest positive integer n such that Kn is a preserves the adjacency. subgraph of G. According to (Gary, 1986), we need the 3. A vertex u is said to be connected to a vertex v following results. in a graph G if there exists a u-v path in G. A Theorem2.2: (Kuratowsky Theorem) graph G is connected if every two of its vertices A graph G is planar if and only if it does not are connected. contain a graph homeomorphic with K5 or K3, 3. 4. The distance d(x, y) between two vertices u and According to (Behboodi, 2011), we will need v in a connected graph G is the minimum of the following results. lengths of the u-v paths of G. Proposition2.3: Let R be an Artinian ring. Then 5. The eccentricity e(v) of a vertex v of a every non-zero proper ideal I of R is a vertex of connected graph G is the number AG(R). Theorem2.4: For every ring R, the annihilating- 6. The radius radG of a connected graph G is ideal graph AG(R) is connected and diam(AG(R))≤3. Moreover, if AG(R) contains a defined as . 7. A vertex v of a connected graph G is a central cycle, then g(AG(R)) ≤ 4. vertex if e(v)=radG. 8. The diameter of a connected graph G is 3. THE GRAPH BASED ON AN IDEAL K . We start this section with the following 9. A vertex v of a connected graph G is a cut- definition. vertex if its removal produces a disconnected Definition3.1: Let R be a commutative ring with graph. identity and K be a non-trivial ideal of R. The 10. An acyclic graph G has no cycles. A tree is an graph based on an ideal K, denoted by (R), is acyclic connected graph. the undirected graph whose vertices are non-trivial 11. The length of the shortest cycle in a graph G ideals of R and two of its vertices are adjacent if that contains cycles is called the girth of G and and only if IJ⊂K. denoted by g(G). 12. A graph G is embeddable on a surface S if G Before starting our main results, we give the can be drown on S so that edges intersect only at a following example. vertex mutually incident with them. A graph is Example3.2: Consider rings z24 and z64. The planar if it can be imbedded in the plane. following figures show the graphs z64) and 13. A dominating set for a graph G is a subset D (z24). 4 of a vertex set of G such that every vertex not in D 2 2 is adjacent to at least one member of D. The 3 domination number γ(G) is the number of vertices 12 in a smallest dominating set for G. 32 8 14. An assignment of colors to the vertices of a 4 8 graph G, one color to each vertex, so that adjacent vertices are assigned different colors is called a 6 16 coloring of G; a coloring in which n colors are The graph (z24) The graph z64) used is an n-coloring. The minimum n for which a Fig.1 ZANCO Journal of Pure and Applied Sciences 2019 Abdulqadr .F. /ZJPAS: 2019, 31 (2): 1-8 3 Remark3.3: 1. IJ=(0)⊂K, for every edge {I, J} of AG(R). Fig.2: The graph z30) Thus AG(R) is a subgraph of . Obviously, z30) is incomplete. 2. If P is a proper ideal of R such that K⊂P, then is a subgraph of . We are now in a position to give the following main result of this section. We begin our results of this section with the Theorem3.7: following Lemma. 1. Every two minimal ideals of R are adjacent Lemma3.4: ideal vertices in (R). 1. If R is Artinian, then every non-trivial ideal of 2. If K is a minimal ideal of R, then and R is an ideal vertex of . AG(R) are identical graphs. 2. If I is an ideal of R such that 0 I⊂K, then I is 3. Let Min(R) be finite. Then every ideal an ideal vertex adjacent to each other ideal vertex I Min(R) of (R) is adjacent to a minimal vertices in . ideal of R. Moreover, the domination of (R) is 3. If K Max(R), then degK is finite if and only if at most | |. (R) is a finite graph. Proof: Proof: The prove is trivial. 1. Let I, J with I J. If IJ (0), then the The next result illustrates the completeness of minimally of I and J gives that IJ=I=J, which is impossible. Thus IJ=(0)⊂K. Hence I and J are . Theorem3.5: Let |Max(R)| 2. If K Max(R) and adjacent ideal vertices in (R). 2. Let K and {I, J} be any edge of every ideal vertex I Max(R) of contains . It follows that IJ⊂K. Since K is a minimal properly in K, then is a complete graph. Proof: Let I and J be any two distinct ideal ideal of R, IJ=(0). Thus {I, J} is an edge of AG(R). It follows from Remark3.3 that and vertices of Then either I K or J K. Suppose that I K. If Max(R)={K}, then IJ⊆I⊂K. AG(R) are identical graphs. Suppose that Max(R)={K, M} with M K. If I, 3. Let I Min(R) be any ideal vertex of (R). If J Max(R), then the maximally of M gives that K Min(R), then there exists a minimal ideal L IJ=MK⊂K. Now assume that at least one of I and such that L⊂K. It follows from Lemma3.4 that I is J is not maximal ideal, let be I. Then IJ⊆I⊂K. adjacent to L in (R). Assume that K Min(R). From each case, we have shown that I and J are Since I is an ideal vertex of (R), there exists an ideal vertex J I of (R) such that IJ⊂K.