Strong Dominating Sets of Direct Product Graph of Cayley Graphs with Arithmetic Graphs

Transcription

1 International Journal of Applied Information Systems (IJAIS) ISSN : Volume No., June 01 Strong Domatg Sets of Direct Product Graph of Cayley Graphs with Arithmetic Graphs M. Manjuri Department of Applied Mathematics, Sri Padmavati Women s University, Tirupati, Andhra Pradesh, India. B. Maheswari Department of Applied Mathematics, Sri Padmavati Women s University, Tirupati, Andhra Pradesh, India. ABSTRACT Today, graph theory is one of the most flourishg branches of modern mathematics with wide applications to combatorial problems to classical algebraic problems. Graph theory has applications diverse areas such as social sciences, lguistics, physical sciences, communication enge erg etc. Because of this diversity applications it is useful to develop study this subject abstract terms of the objects of any particular system which one may be terested. Product of graphs are troduced Graph Theory very recently developg rapidly. In this paper, we consider direct product graphs of Cayley graphs with Arithmetic graphs discuss strong domation parameter of these graphs. Keywords Euler totient Cayley graph, Arithmetic ct graph, Strong domatg set. graph, direct produ Subject Classification: 8R10 1. INTRODUCTION Domation graphs is the fast growg area Graph Theory that has emerged rapidly the last four decades. Domation graphs has applications to several fields such as facility location problems, School Bus Routg, Computer Communication Networks, Radio Stations, Locatg Radar Stations Problem etc., Number Theory is one of the oldest branches of mathematics, which herited rich contributions from almost all great mathematicians, ancient modern. Nathanson [] was the pioneer troducg the concepts of Number Theory, particularly, the Theory of Congruences Graph Theory, paved the way for the emergence of a new class of graphs, namely Arithmetic Graphs. Cayley Graphs are another class of graphs associated with elements of a group. If this group is associated with some Arithmetic function then the Cayley graph becomes an Arithmetic graph. The Cayley graph associated with Euler totient function is called an Euler totient Cayley graph. Products are often viewed as a convenient language with which one can describe structures, but they are creasgly beg applied more substantial ways. Computer Science is one of the many fields which graph products are becomg common place. The direct product was troduced by Alfred North Whitehead Bertr Russell their Prcipia Mathematica [] is also called as the tensor product, categorical product, cardal product, relational product, Kronecker product, weak direct product or conjunction. Direct Product Graph be two simple graphs with their vertex sets as respectively. Then the direct product of these two graphs denoted by is defed as the graph with vertex set, is the Cartesian product of the sets such that any two distct vertices of are adjacent if is an edge of is an edge of.. EULER TOTIENT CAYLEY GRAPH For any positive teger, let Then, is addition modulo is an abelian group of order For any positive teger, let denote the set of all positive tegers less than relatively prime to That is Then is the Euler totient function. We can see that is a symmetric subset of the group The defition of Euler totient Cayley graph is as follows. The Euler totient Cayley graph graph whose vertex set V is given by the edge set is is defed as the Some properties of Euler totient Cayley graphs enumeration of Hamilton cycles triangles can be found Madhavi [1]. The Euler totient Cayley graph is a complete graph if is a prime it is regular. The strong domation parameter of these graphs are studied by the authors [] the followg results are required they are presented without proofs. Theorem.1: If is a prime, then the strong domation number of is 1. Theorem.: The strong domation number of is, if is an odd prime. Theorem.: Suppose is neither a prime nor, are primes are tegers 1. Then the strong domation number of is given by is the length of the longest stretch of consecutive tegers each of which shares a prime factor with. ARITHMETIC GRAPH be a positive teger such that. Then the Arithmetic graph is defed as the graph whose vertex set consists of the divisors of two vertices are adjacent graph if only if for some prime divisor of In this graph vertex 1 becomes an isolated vertex. 1

2 International Journal of Applied Information Systems (IJAIS) ISSN : Volume No., June 01 we consider the graph without vertex 1, as the contribution of this isolated vertex is nothg when we study the domation parameters. Clearly, graph is a connected graph. If is a prime, then graph consists of a sgle vertex. it is connected. In other cases, by the defition of adjacency there exist edges between prime numbers, their prime powers also to their prime products. each vertex of is connected to some vertex The strong domation number of graph is obtaed by the authors the proof of the followg theorem can be found []. Theorem.1: If are primes are tegers 1, then the strong domation number of is given by is the core of.. DIRECT PRODUCT GRAPH OF WITH In this paper the direct product graph of Euler totient Cayley graph with Arithmetic graph is considered. The properties some domation parameters of these graphs can be found []. denote the Euler Totient Cayley graph denote the Arithmetic graph. Sce are two simple graphs, by the defition of adjacency the direct product, the graph is a simple graph. Further the graph is a completely disconnected graph, if is a prime the degree of a vertex is given by domatg set. But the given graph does not satisfy the conditions of a strong domatg set. the strong domation number does not exist for the graph Theorem.: If are distct primes, then the strong domation number of is Proof: are distct primes. Consider the graph be the vertex sets of the graphs respectively. Consider the set We now prove that is a domatg set of. Consider the vertices which are the vertices are consecutive tegers So, their difference is, which is the set hence they are adjacent. Thus the vertices becomes a domatg set of are adjacent with the vertices respectively as Now we check whether is a strong domatg set or not. We know from the properties of Euler totient Cayley graph that is regular, is the cardality of the set S. So the degree of each vertex is For the graph contas the vertices as there is an edge between there is no edge between. STRONG DOMINATING SETS OF DIRECT PRODUCT GRAPH In this section mimum strong domatg sets of direct product graph of with are discussed obtaed its strong domation number various cases. Strong domation be a graph Then, strongly domates if (i) (ii) A set is called a strong domatg set of if every vertex is strongly domated by at least one vertex The strong domation number of is the mimum cardality of a strong domatg set. Some results on strong domation for general graphs can be seen []. The results on strong domation of direct product graph are as follows. Theorem.1: If is a prime, then the strong domation number does not exist for the graph Proof: Then the graph is a completely disconnected graph on vertices. So, all these vertices form a If is any vertex of then we know that From this, we have That is (1) By equation (1), we see that every vertex domated by some vertices is a strong domatg set of. is strongly Now we show that is mimal. Suppose we delete a vertex say, from Then we see that does not form a domatg set of because the vertex is not adjacent with any vertex by the 1

3 International Journal of Applied Information Systems (IJAIS) ISSN : Volume No., June 01 defition of direct product. Similar is the case by the deletion of any other vertex In similar les we cannot get any strong domatg set mimal than the set is a mimal strong domatg set of with cardality Theorem.: If of is Proof: respectively., then the strong domation number Consider the graph be the vertex sets of By the properties of Euler totient Cayley graph, the graph is regular. So, each vertex has same degree For the graph is regular hence each vertex has degree The graph contas two vertices these two vertices are joed by an edge as By the properties of direct product, the graph is regular, so that the degree of every vertex is. Now we show that is a domatg set of. be any vertex of Then the vertex is adjacent with either or as is a domatg set of The vertices are adjacent as Thus by the defition of direct product a vertex is adjacent with either is a domatg set it also becomes a strong domatg set of as the graph is regular. Now we show that is mimal. That is deletion of any vertex does not make a domatg set any more. Suppose the vertex is deleted from. We know that the degree of a vertex is So let be adjacent with the vertices Consider the set of vertices Sce is all the vertices are not domated by Otherwise the vertices of are domated neither by nor by because it is a null graph. So all the vertices S are not domated by Sce there is no edge between to itself all the vertices S are not domated by Thus no vertex can domate the vertices of S. Thus is not a domatg set of Similar is the case with the deletion of any other vertex Thus becomes a mimal strong domatg set of with cardality Theorem.: If is neither a prime nor nor are distct primes then the strong domation number of is given by is the length of the longest stretch of consecutive tegers of each of which shares a prime factor with n, k is the core of n, is the number of primes with exponent >1 is the number of primes with exponent Proof: be neither a prime nor nor Suppose = denote the vertex sets of respectively. By Theorem., the strong domation number of is be a strong domatg set of, Theorem.1, is the core of. Now we have the followg possibilities. are consecutive tegers. Aga by Case 1: Suppose for all. By Theorem.1 of Case 1, the strong domatg set of with mimum cardality is given by. That is Further Consider the set Then a vertex. for any is a domatg set of is not adjacent with any vertex no edges between the primes can t be a domatg set of as there are So we have to extend the set let be the entire vertex set of vertices, the are consecutive order. Then for some Sce is a domatg set of, the vertex is adjacent with some vertex, say Then the vertex is adjacent with the vertex becomes a domatg set of We now show that is a strong domatg set of be any vertex Then This implies that every vertex by some vertex. is strongly domated 1

4 International Journal of Applied Information Systems (IJAIS) ISSN : Volume No., June 01 Thus is a strong domatg set of We now show that is mimal. Suppose we delete a vertex, say from Obviously this vertex is not domated by any vertex as are primes, there is no adjacency between these vertices can t be a domatg set of Similar is the case with the deletion of any other vertex Thus is a mimal strong domatg set of Case : Suppose for all. That is By Theorem.1 of Case, we know that, k is the core of a strong domatg set is given by Further we know that domatg set of is a If we consider the set then we can t prove that is a domatg set of, because the vertices are not adjacent with any vertex because we modify the set. by cludg a vertex viz., Then Obviously is a domatg set of which is also strong.. be any vertex of Then the vertex is adjacent with some vertex vertex is adjacent with either or as are domatg sets of respectively. That is any vertex is adjacent with or.thus the vertices are adjacent with at least one vertex is a domatg set of We now show that is a strong domatg set be any vertex Then we have This implies that every vertex by some vertex Thus is a strong domatg set of is strongly domated We now show that is mimal. Suppose we delete a vertex, say from Then the vertex is not adjacent with any vertex, because Similar is the case with the deletion of any other vertex Thus is a mimaml strong domatg set of with cardality ( Case : Suppose for only one. By Theorem.1 of Case, a strong domatg set of is given by, Now we show that is a domatg set of the vertices are consecutive order The vertices of are domated by some vertex as So the vertices of are domated by some vertices Thus every vertex one vertex Now we show that is domatg set of is adjacent with at least is a strong domatg set of be any vertex of Then we have Thus every vertex of is strongly domated by at least one vertex of Now we show that vertex, say we can see that because the vertex only with the vertex is a strong domatg set of is mimal. Suppose we delete some from is no more a domatg set of is adjacent as this vertex is not., Then Thus is a mimal strong domatg set of with cardality Case : Suppose for some for That is 18

5 International Journal of Applied Information Systems (IJAIS) ISSN : Volume No., June 01 By Theorem.1 of Case, we know that k is the core of the above Cases 1, it follows that domatg set of Then by is a strong We show that is mimal. If we delete a vertex, say from then can be no more a domatg set, as there is no vertex domatg the vertex, sce. ILLUSTRATIONS strong domatg set of CONCLUSION is a with mimum cardality The strong domatg sets of Euler totient Cayley graphs Arithmetic graphs are studied by the authors. This study is motivated to fd the strong domatg sets of Direct product graph of Euler totient Cayley graphs with Arithmetic graph. Further the Strong domatg sets of strong product graph Lexicographic product graph of these graphs are also studied Fig. Fig.1 (1,1) (0,1) (1,1) (11,1) (10,1) (,1) (,1) (,1) (9,1) (8,1) (,1) (,1) (,1) Fig. Strong domatg set does not exist Fig. x Fig. 19

6 International Journal of Applied Information Systems (IJAIS) ISSN : Volume No., June 01 (9,) (9,) (9,10) (0,) (0,) (0,10) (1,) (1,) (8,10) (1,10) (8,) (,) (8,) (,) (,10) (,10) (,) (,) (,) (,) (,10) (,10) (,) (,) (,) (,10) (,) Fig. (,) (,10) (,) Strong domatg set {(0,10),(1,10),(,10),,(9,10)} 0 1 Fig. Fig.8 (,) (,) (,8) (,) (,) (,8) (,) (,8) (0,) (0,) (0,8) (1,) (1,) (1,8) (,) (,) (,8) (,) (,8) (,) (,) Fig.9 (,8) (,) (,) Strong domatg set {(0,),(1,),..,(,)} Fig.10 Fig.11 0

7 International Journal of Applied Information Systems (IJAIS) ISSN : Volume No., June 01 (8,) (8,9) (0,) (0,9) (1,) (,) (,9) (,9) (1,9) (,) (,9) (,) (,) (,9) (,) (,9) Fig.1 (,) (,9). REFERENCES Strong domatg set {(0,), (1,),(0,9),(1,9)} [1] Madhavi, L. - Studies on domation parameters enumeration of cycles some Arithmetic graphs, Ph. D. Thesis submitted to S.V.University, Tirupati, India (00). [] Manjuri, M. Maheswari, B.- Strong domatg sets of Euler totient Cayley graph Arithmetic Vn graphs, International Journal of Computer Applications (IJCA), Volume 8, No (01), -0. [] Nathanson B.Melvyn, -Connected components of arithmetic graphs, Monat.fur.Math, 9, (1980), [] Sampathkumar, E. Pushpa Latha, L.-, Strong weak domation domation balance graph, Discrete Mathematics, 11 (199), -. [] Uma Maheswari, S.- Some studies on the product graphs of Euler totient Cayley graphs Arithmetic Vn graphs, Ph. D. Thesis submitted to S.P.Women s University, Tirupati, India (01). [] Whitehead, A.N., Russel, B.- Prcipia Mathematica, Volume, Cambridge University Press, Cambridge (191). 1