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Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 496~504 ISSN: 2455-7102 SOME DOMINATION PARAMETERS OF ZERO DIVISOR GRAPHS K.BUDADODDI R. RAMAKRISHNA PRASAD & A.MALLIKARJUNA REDDY Assistant Professor, Department of Mathematics, Sri Krishnadevaraya University College of Engineering & Technology, Ananthapuramu, Andhra Pradesh, India-515003. Research Scholar, Department of Mathematics, Sri Krishnadevaraya University, Ananthapuramu, Andhra Pradesh, India-515003. Professor, Department of Mathematics, Sri Krishnadevaraya University, Ananthapuramu, Andhra Pradesh, India-515003. [email protected] Article Info ABSTRACT Graph theory is one of the most developing branches of mathematics with wide applications to various branches of science and technology. A graph can be used to represent almost any physical situation Keyword: Dominating set, Domination number, zero divisors, divisor graphs. . Zero involving discrete objects and a relationship among them. The origin of graph theory started with the problem of Konigsberg bridge problem in 1735 by the great mathematician Leonard Euler. The theory of domination in graphs introduced by C. Berge and O. Ore which is rapidly growing area of research in graph theory today. The study of domination in graphs originated around 1850 with the problem of determining the minimum number of queens that can be placed on an nxn chess board so as to cover or dominate every square. The concept of zero divisor graphs of a ring was introduced by I. Beck in which the coloring of a Journal homepage: http://ojal.us/ojatm/ 496 Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 496~504 ISSN: 2455-7102 commutative ring is mainly discussed. In this work, the set of vertices is taken to be R and two vertices x and y are adjacent if their product should be zero. The investigation of colourings of a commutative ring was then continued by D. D. Anderson and M. Naseer. In this paper, I investigate some domination parameters of zero divisor graphs. Copyright © 2015 Open Journal of Applied & Theoretical Mathematics (OJATM) All rights reserved. 1. INTRODUCTION Let R be a commutative ring with unity of R and , Z(R) be the set of all zero-divisors be the set of nonzero zero-divisors of R. The zero –divisor graph of R denoted by (Z ( R) * ) is usually written (R) whose vertices are the elements of the and distinct vertices are adjacent if and only if . This definition defined by David F.Anderson and Philip S. Livingston in [3]. The main object of this unit is to study the inter relation of ring theoretic properties of R with graph theoretic properties of (R) . This study helps illuminate the structure of . For , define if or . The relation is always reflexive and symmetric, but is usually not transitive. The zero divisor graph (R) measures this lack of transitivity in the sense that is transitive if and only if (R) is complete. The idea of a zero-divisor graph of a commutative ring was introduced by I. Beck in [5], where he was mainly interested in colourings. This investigation of colourings of a commutative ring was then continued by D.D.Anderson and M.Naseer in [4]. Their definition was slightly different than the previous definition. They let all elements of R be vertices and has distinct and are adjacent if and only if Journal homepage: http://ojal.us/ojatm/ 497 . They denote their Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 496~504 ISSN: 2455-7102 zero-divisor graph of R by 0 ( R) . In 0 ( R) , the vertex 0 is adjacent to every other vertex, but nonzero zero-divisor are adjacent only to 0. Note that (R) is a sub graph of 0 ( R) . 2. PROPERTIES OF ZERO DIVISOR GRAPH 1. (R) is the empty graph if and only if R is an integral domain. 2. Let A and B be integral domains and let R A B . Then (R) is a complete bipartite graph with ( R) A B 2 . 3. Let R be a commutative ring then (R) is connected. Let R be a commutative ring. Then (R) is complete if and only if either or , for all . 4. Let R be finite commutative ring. If (R) is complete then either local with char R= or and ( R) p n 1 where In this work the commutative ring R is is prime and or R is . and zero-divisor graph (R) is ( Z n ) . 3. DOMINATION NUMBER OF ZERO DIVISOR GRAPH We know that a dominating set of a graph G(V,E) is a subset D of V(G) such that every vertex in V-D is adjacent to at least one vertex in D. The minimum cardinality of a dominating set of G is called the domination number of G and it is denoted by . In this section we study the domination number of zero divisor graphs. Theorem 3.1: If n=2 where is an odd prime, then the non-zero zero divisor graph is star graph and it is also K 1, p 1 complete bipartite graph. Proof: Let n=2 , where graph is Z(R)*=V= { is an odd prime. Then the vertex set of non-zero zero divisor multiplication modulo distinct even integers which is in V. The product of which is not divisible by n where n=2 , is an odd prime. Journal homepage: http://ojal.us/ojatm/ 498 }. Let be two will become an even integer Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 496~504 ISSN: 2455-7102 The product vertex is is divisible by n=2 , if one vertex is 2 or multiples of 2 and another then only and are adjacent. Two distinct even integers Let us take and are not adjacent. is an even integer and is a prime number in V. The product of even and prime is an even integer and it is divisible by n where n=2 , is a prime and . Every even integer and prime number is adjacent. is adjacent to 2,4,6,….2( -1) It is clear that K P is a star graph. The graph K P is classified into two non-empty subsets V. Where X={ } and Y={2,4,6,….2( -1)}. The vertex and X is adjacent to all the vertices of Y. K 1, p 1 is a complete bipartite graph. =2 where is a prime and is a star graph and K 1, P 1 is a complete bipartite graph. Theorem 3.2: If =2 where is an odd prime, then the domination number of non- zero zero divisor graph is 1. Proof: By theorem 3.1 ( Z n ) is a star graph. D= { } forms a minimum dominating set with cardinality1. Then the domination number of non-zero zero divisor graph is 1, when n=2 , where is an odd prime. ( ( Z n ) ) = 1. Theorem 3.3: If = , where is a prime and >3. Then the non-zero zero divisor graph is K 2, P 1 complete bipartite graph. Proof: Let = 6,…, -3, , , where is a prime number and >3.Then the vertex set V= Z(R)*= {3, }. Decompose the vertex set V in to the following disjoint subsets. 1. The set S of non-zero multiples of 3 which are less than . Journal homepage: http://ojal.us/ojatm/ 499 Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 496~504 ISSN: 2455-7102 2. The set M of multiples of prime which are less than . The edge set E(G) defined by E(G)={( )/ =0 (multiplication modulo n) Z(R)*}. Let us consider two distinct vertices only multiples of 3. The product of , where the vertices of S are is divisible by 3 but not 3 . Hence two distinct vertices of S are not adjacent. The product of is divisible by , when one vertex is 3 or multiples of 3 and another vertex is prime or multiples of prime. Now let us take prime. The product of and , where the vertices of M are the multiples of is divisible by . Hence one vertex is in S and other in M are adjacent. The vertices of M are adjacent to all the vertices of S and vice-versa. The vertex set V can be decomposed into the two disjoint subsets S and M. The edges of two vertices are partition into the two sets one end in S and other end in M. n= where is a prime and >3 is a K 2, P 1 complete bipartite graph. Theorem 3.4: If = where is a prime number and >3. Then the domination number of non-zero zero divisor graph is 2. Proof: By theorem 3.3 ( Z n ) is a complete bipartite graph, if = where is a prime number and >3. The set D= { , } is a dominating set, since every vertex of V-D is adjacent to at least one vertex of D. Continuing like this D= { , ,3,6,….. } is also dominating set. D= { , } is a minimum dominating set with cardinality 2. The domination number of ( Z n ) is 2. ( Z n ) . Theorem 3.5: If n= , where is a prime and >5. Then the domination number of non-zero zero divisor graph is 2. Journal homepage: http://ojal.us/ojatm/ 500 Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 496~504 ISSN: 2455-7102 Proof: Let n= 15,…., , where , , , , is a prime and >5. Then the vertex set V= = {5, 10, }. Decompose the vertex set V into the following disjoint subsets. 1. The set S of non-zero multiples of 5 which are less than . 2. The set M of multiples of prime which are less than . The edge set E(G) defined by E(G)={( )/ =0 (multiplication modulo n) Z(R)*}. Let us consider two distinct vertices multiples of 5. The product of , where the vertices of S are (mod n), is an integer. The product of . So the distinct vertices of S are not adjacent. Let , and divisible by or . The product of but is not divisible by i.e., which is (mod n). The distinct vertices of M are not adjacent. The distinct vertices of M are not adjacent. Let us consider and where the vertices of S are multiples of 5 and the vertices M are multiples of prime . The product of is divisible by . Every vertex in M are adjacent to all the vertices of S and vice-versa. D= {( )/ and } is a dominating set because every vertex in V-D is adjacent to at least one vertex in D. Continuing like this D={5, 10, 15,….5p-5, , , , } is also dominating set. D= { ( Z n ) } is a minimum dominating set with cardinality 2. . Theorem 3.6: If n= , where is a prime and >5, then the non-zero zero divisor graph is K 4, P 1 complete bipartite graph. Journal homepage: http://ojal.us/ojatm/ 501 Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 496~504 ISSN: 2455-7102 Proof: Let n= 15,….,5 -5, , , where , , is a prime and >5.Then vertex set V= = {5, 10, }. Decompose the vertex set V in to the following disjoint subsets. By Theorem 3.5 ( Z n ) is a K 4, P 1 complete bipartite graph. Theorem 3.7: If n= , where are primes and > . Then the domination number of non-zero zero divisor graph is 2. Proof: Let n= { , , , where ,……, ( are primes and > . Then the vertex set V= = ), , 2 , 3 , …..,( -1) }. Decompose the vertex set V in to the following disjoint subsets. 1. The set S of non-zero multiples of 2. The set M of multiples of which are less than n. which are less than n. The edge set E(G) defined by E(G)={( )/ =0 (multiplication modulo n) Z(R)*}. Let us consider is an integer. where the vertices of S are multiples of is divisible by , but is not divisible by . The product of where n= . Since every integer can be written as product of primes but it is not in the form . So the distinct vertices of S are not adjacent. Let us take product of is an integer. , where the vertices of M are multiples of prime . The is divisible by but is not divisible by where n= . The distinct vertices of M are not adjacent. Let us consider and , where the vertices of S are multiples of prime and the vertices of M are multiples of prime . The product is divisible by . Every vertex in M is adjacent to all the vertices of S and vice-versa. D= {( )/ and } is a dominating set, because every vertex in V-D is adjacent to exactly one vertex in D. Continuing like this D={ , 3 , …..,( -1) } is also dominating set. D= { } is a minimum dominating set with cardinality 2. Journal homepage: http://ojal.us/ojatm/ 502 , ,….., ( ), , 2 , Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 496~504 ISSN: 2455-7102 ( Z n ) . Theorem 3.8: If n= where are primes and > . Then the non-zero zero divisor graph is K p1,q1 Complete bipartite graph. Proof: By theorem 3.7, ( Z n ) is a K p1,q1 complete bipartite graph. Theorem 3.9: If where an odd prime, then the domination number of non- zero zero divisor graph is 1. Proof: Let where graph is V= an odd prime. Then the vertex set of non-zero zero divisor = { , 2 , 3 ,….., ( )}. The edge set E(G) defined by E(G)={( =0 (multiplication modulo n) complete graph on )/ Z(R)*}. When we draw the graph we get a vertices. If graph is complete then a single vertex can dominate remaining all the vertices. D= { } is a dominating set with cardinality 1. It is clear that D= { } is a minimum dominating set. The domination number of ( Z n ) is 1, when ( Z n ) where an odd prime. . Theorem 3.10: If where is an odd prime and . Then the domination number of non-zero zero divisor graph is 1. Proof: Let where { , 2 , 3 ,…, ,…...., of ( Z n ) . Hence D= { is an odd prime and }. If then .Then the vertex set V= = is adjacent to every other vertex } is a minimum dominating set. It is clear that D= { } is a minimum dominating set with cardinality 1. The domination number of non-zero zero divisor graph is 1, when an odd prime and ( ( Z n ) ) . Journal homepage: http://ojal.us/ojatm/ 503 where is Open Journal of Applied & Theoretical Mathematics (OJATM) Vol. 2, No. 4, December 2016, pp. 496~504 ISSN: 2455-7102 REFERENCES: [1] Akbari.S and Mohammadian.A, “On the zero divisor graph of a commutative ring”, J. Algebra, 274, 847-855, 2004. [2] Anderson.D.F, Axtell.M.C and Stickles.J “Zero divisor graphs in commutative rings” Comm.Algebra, 33, 2043-2050, 2005. [3] Anderson.D.F and Livinston.P.S “The zero divisor graph of a ring” J. Algebra, 217, 434-447,1999. [4] Anderson.D.D and Naseer.M “Beck’s coloring of commutative ring” J. Algebra, 159, 500-514, 1993. [5] Beck.I “coloring of commutative rings”, J. Algebra 116, 208-226, 1988. [6] Kulli.V.R “Theory of domination in graphs” Vishwa Int. Publications, Gulbarga, India, 2010. Journal homepage: http://ojal.us/ojatm/ 504