Regular and Irreguar m-polar Fuzzy Graphs

Transcription

1 Global Journal of Matheatical Sciences: Theory and Practical. ISSN Volue 9, Nuber 07, pp International Research Publication House Regular and Irreguar -polar Fuzzy Graphs Dr. K. Kalaiarasi and L. Mahalakshi PG and research Departent of Matheatics,Cauvery College for Woen, Trichy-8, Tail Nadu, India. PG and research Departent of Matheatics,Cauvery College for Woen, Trichy-8, Tail Nadu, India. Abstract In this paper we define irregular -polar fuzzy graphs and its various classifications.size of regular -polar fuzzy graphs are derived.the relation between highly and neighbourly irregular -polar fuzzy are established.soe basic theores related to the stated graphs have also been presented. Keywords: -polar fuzzy graphs, regular -polar fuzzy graphs, irregular and degree of -polar garphs. INTRODUCTION: The origin of graph theory started with Konigsberg bridge proble in 735.This proble led to the concept of Eulerian graph.euler studied the Konigsberg proble and constructed a structure that solves the proble that is referred to as an Eulerian graph.in this paper we introduced the notation of the -polar fuzzy set as a generalization of -polar fuzzy graph. Currently,concept of graph theory are highly utilize by coputer science applications,especially in area of research, including data ining iage segentation,clustering and net working. The first basic definitions of -polar fuzzy

2 40 Dr. K. Kalaiarasi and L. Mahalakshi graph was proposed by G.Ghorai and M.pal[] fro a study on -polar fuzzy graphs. In 999,Molodtsov [9]introduced the concept of soft set theory to solve iprecise probles in the field of engineering, Social science, econoics, edical science and environent. Molodtsov applied this theory to several directions such as soothness of fuction, gae theory, operation research, probability and easureent theory.in resent ties, a nuber of research studies contributed into fuzzification of soft set theory. As a result any researchers were ore active doing research on soft set. However H.Rashantou, S.Saanta, M.pal, R.A.Borzooel [4] introduced -poar fuzzy graphs with categorical properties. Also S.Saanta, M.pal [5] defined basic definition of irregular -polar fuzzy graphs. In 05, G.Ghorai and M.pal [,,] also introduced another group of explained definition of copleent and isoorphis of -polar fuzzy graph and soe operations and density of -polar fuzzy graphs. At this sae tie they elaborated faces and dual of -polar fuzzy planar graphs. The copleent of a fuzzy graphs, Indian journal of pure and Applied atheatics was introduced by M.S.Sunitah, A.Vijayakuar[9]. In 0,Akra[3] introduced the concept of -polar fuzzy graphs and defined different operations on it. Now -polar fuzzy graph theory is growing and expanding its application. Also we refer basic definitions of fuzzy set theory and K.Kalaiarasi[0] defined optiization of fuzzy integrated Vendor-Buyer inventory odels. Soe basic Definitions and basis of regular and irregular theores are discussed. Definition. * An -polar fuzzy graph of a graph of a graph G V, E is a pair G A, where : V [0, is an -polar fuzzy set in V and set in V such that for each xy V and xy 0 eleent in [ 0,]. B for all B : V [0, ] is an -polar fuzzy i,,.... pi [ B xy] in { pi A x, pi A y for all xy V,{0 0,0,0...,0} E is the sallest A is called the -polar fuzzy vertex and B is called the polar fuzzy edge.

3 Regular and irreguar -polar fuzzy graphs 4 Exaple:. Definition. The order of the -polar fuzzy graph G V, A, is denoted by V or O G where O G V x v i pi A x Definition.3 The size of G A, is denoted by E or S G where S G = E = pi B xy i xy v Definition.4 be a -polar fuzzy graph where set in V and B : V [0, ] : V [0, is an -polar fuzzy be two -polar fuzzy sets on a non-epty finite set V and E v v If d u k, d v k for all u, v U, V. k,k are two real nubers,then the graph is called k,k regular -polar fuzzy graph.if there exists a vertex which is adjacent to a vertex with sae degree.

4 4 Dr. K. Kalaiarasi and L. Mahalakshi Exaple:.4 d v 0.4,0.8,. d v 0.4,0.8,. d v 3 0.4,0.8,. d v d v d v3 The graph is regular 3-polar fuzzy graph.he graph is said to regular. Reark: If the graph is said to regular if the odd graph or even graph of degree are sae. Exaple of odd graph and even graph are given below. Exaple of odd reguar graph:

5 Regular and irreguar -polar fuzzy graphs 43 Exaple of Even Regular graph: In even regular graph the alternative edges are equal. Definition.5 be a -polar fuzzy graph where : V [0, is an -polar fuzzy set in V and B : V [0, ] be two -polar fuzzy sets on a non-epty finite set V and E v v G is said to be totally regular -polar fuzzy graph. If there exists a vertex which is adjacent to a vertex with sae total degree td u. Definition.6 Let G be a connected -polar fuzzy graph.then G is called neighbourly totally regular -polar fuzzy graph if for every two adjacent vertices of G have sae total degrees. Definition.7 Let G be a connected -polar fuzzy graph.then G is called neighbourly regular -polar fuzzy graph if for every two adjacent vertices of G have sae degrees.

6 44 Dr. K. Kalaiarasi and L. Mahalakshi Exaple:.7 d v 0.5,0.9,0.6 d v 0.5,0.9,0. 6 d v 3 0.4,0.8,0. 4 d v d and d v d v v3 So this graph is an exaple of an irregular 3-polar fuzzy graph. Definition.8 Let G be a connected -polar fuzzy graph.then G is called neighbourly irregular -polar fuzzy graph if for every two adjacent vertices of G have distinct degrees. Exaple:.8 That is d v d v d v3

7 Regular and irreguar -polar fuzzy graphs 45 Definition :.9 be a -polar fuzzy graph where set in V and B : V [0, ] : V [0, is an -polar fuzzy be two -polar fuzzy sets on a non-epty finite set V and E v v G is said to be totally irregular -polar fuzzy graph if there exists a vertex which is adjacent to a vertex with distinct total degree td u. Exaple:.9 d v 0.5,0.9,0.6 d v 0.5,0.9,0. 6 d v 3 0.4,0.8,0. 4 td v.3,0.8,.0 td v 0.8,0.9,. 3 and td v 3 0.8,0.9,. 3 td v td and td v td. v3 v There exist a vertex which is adjacent to a vertex with distinct total degrees.so this graph is an exaple of Totally irregular -polar fuzzy graphs. Definition.0 Let G be a connected -polar fuzzy graph.then G is called neighbourly totally irregular -polar fuzzy graph if for every two adjacent vertices of G have distinct total degrees.

8 46 Dr. K. Kalaiarasi and L. Mahalakshi Reark: It is need not be highly irregular -polar fuzzy graph.that is any adjacent vertices of the graph G have the sae degree.other than any adjacent vertices of the graph G have the distinct degree. Exaple:.0 d v.,0.4,0.4 d v 0.5,0.4,0. 6 d v 3 0.8,0.4,0. 4 td v.3,0.8,.0 td v 0.8,0.9,. 3 and td v 3 0.9,0.6,0. 7 td v td v td v3 This graph is an exaple of neighbourly totally irregular -polar fuzzy graphs. Definition. Let G be a connected -polar fuzzy graph.then G is called highly irregular -polar fuzzy graph if every vertex of G is adjacent to vertices with distinctl degrees.

9 Regular and irreguar -polar fuzzy graphs 47 Exaple:. d v 0.6,0.5,0.7 d v.4,0.7,0. 7 d v 3.4,0.7,0. 7 d v.,0.9,. d v d but v d v d v3 v4 This is highiy irregular -polar fuzzy graph.it is need not be neighbourly irregular. Theore. be a -polar fuzzy graph.then G is highly irregular -polar fuzzy graph and neighbourly irregular -polar fuzzy graph iff the degrees of all vertices of G are distinct. Proof: be a -polar fuzzy graph where : V [0, is an -polar fuzzy set in V and B : V [0, ] be two -polar fuzzy sets on a non-epty finite set V and E v v Let V v, v,... v }.We assue that G is highly irregular and neighbourly irregular To prove: { n -polar fuzzy graph. If the degrees of all vertices of G are distinct. Let the adjacent vertices V be v,...,, v3 vn with degrees d v, d v,... d vn

10 48 Dr. K. Kalaiarasi and L. Mahalakshi We know that G is highly irregular -polar fuzzy graph and neighbourly irregular -polar fuzzy graph so d v d v... d v. n So it is obivious that all vertices are of distinct degrees. Conversely, We assue that the degrees of all vertices of G are distinct. This eans that every two adjacent vertices have distinct degrees and to every vertex the adjacent vertices have distinct degrees. Hence G is highly irregular and neighbourly irregular -polar fuzzy graph. Theore. Let G be a -polar fuzzy graph if G is neighbourly irregular then G is a neighbourly totally irregular -polar fuzzy graph. Proof: be a -polar fuzzy graph where set in V and V and B : V [0, ] E v v : V [0, is an -polar fuzzy be two -polar fuzzy sets on a non-epty finite set Assue that G is a neighbourly irregular -polar fuzzy graph. To prove: G is a neighbourly totally irregular -polar fuzzy graph. If the graph G is a neighbourly irregular -polar fuzzy graph.that is the degree of every two adjacent vertices are distinct.

11 Regular and irreguar -polar fuzzy graphs 49 The adjacent vertices are u and u with distinct degrees d u and d u Also let u c, c, u c3, c4 -ebership function. c,c, c,c 3 4 are constants. Clearly, td u c c d u td u c c d u td u td u. 3 4 Therefore for any two adjacent vertices v and v with distinct degrees.its total degrees are also distinct. Hence G is a neighbourly totally irregular -polar fuzzy graph. Theore:.3 Let G be a -polar fuzzy graph.if G is neighbourly totally irregular neighbourly irregular -polar fuzzy graph. Proof: be a -polar fuzzy graph where set in V and V and B : V [0, ] E v v then G is : V [0, is an -polar fuzzy be two -polar fuzzy sets on a non-epty finite set Assue that G is a neighbourly totally irregular fuzzy graph.ietotal degrees of every two adjacent vertices are distinct. To prove: G is neighbourly irregular -polar fuzzygraph. Consider two adjacent vertices v and v with degrees x, x and x, x Also assue that v c,c where c,c are constant. Then td v td v d v d v x c x c x x and c c. Hence degree of adjacent vertices of G are distinct.this is true for every pair of

12 50 Dr. K. Kalaiarasi and L. Mahalakshi adjacent vertices in G. Hence the result. Reark: Also we have to prove G is neighbourly irregular -polar fuzzy graph then G is neighbourly totally irregular fuzzy graph. Theore:.4 The size of a k, regular -polar fuzzygraph is is p -vertex. Proof: k pk, pk where p v.that be a -polar fuzzy graph where polar fuzzy set in V and B : V [0, ] finite set V and E v v The size of G is : V [0, is an - be two -polar fuzzy sets on a non-epty We ve S G i pi B xy d v S G S G k, k S G pk, pk SG pk, pk Hence the theore. Theore:.5 be a -polar fuzzy graph.then G is highly regular -polar fuzzy graph and neighbourly regular -polar fuzzy graph iff the degrees of all vertices of G are sae. Proof: be a -polar fuzzy graph where : V [0, is an -polar fuzzy

13 Regular and irreguar -polar fuzzy graphs 5 set in V and V and B : V [0, ] E v v be two -polar fuzzy sets on a non-epty finite set Let V v, v,... v }.We assue that G is highly regular and neighbourly regular { n -polar fuzzy graph. To prove: If the degrees of all vertices of G are sae. Let the adjacent vertices V be v,...,, v3 vn with degrees d v, d v,... d vn We know that G is highly regular -polar fuzzy graph and neighbourly regular - polar fuzzy graph so d v d v... d v. n So it is obivious that all vertices are of sae degrees. Conversely, We assue that the degrees of all vertices of G are sae. This eans that every two adjacent vertices have sae degrees and to every vertex the adjacent vertices have sae degrees. Hence G is highly regular and neighbourly regular -polar fuzzy graph. CONCLUSION: Graph theory is an extrely useful tool for solving cobinatorial probles in different areas, including algebra, nuber theory, Geoentry, topology,o perations research, optiisation and coputer science. Because research on odelling of real world probles often involve uliti-agent, ulti-attribute, ulti-object, ulti-index, ulti-polar inforation, uncertainty and process liits. -polar fuzzy graphs are very useful. In this paper we have described order, size of a -polar fuzzy graphs. The necessary and sufficient conditions for a -polar fuzzy graph to be the regular -polar and irregular -polar fuzzy graphs have been presented.size of a -polar fuzzy graphs and relation between size and order of a -polar fuzzy graphs have been calculated.we have define irregular,neighbourly irregular,totally and highly irregular -polar fuzzy graphs.soe relations about the defined graphs have been proved. REFERENCES [] Akra.M,Dudek.W,Regular bipolar fuzzy graphs,neural coputing and

14 5 Dr. K. Kalaiarasi and L. Mahalakshi applications. [] Al-Hawary.T,coplete fuzzy graphs,int.j.math.cob [3] M.Akra,Bipolar fuzzy graphs with categorical properties [4] Chen.J,Li.s,Ma.s,Wang.X,-polar fuzzy sets an extension of bipolar fuzzy sets,hindwai publishing Corporation,sci,world.J048,Article Id: [5] Ghorai.G and pal.m,a study on -polar fuzzy graphs. [6] Ghorai.G and pal.m, On copleent and isoorphis of -polar fuzzy graphs. [7] Ghorai.G,Pal.M,On soe operations and density of -polar fuzzy graphs,pac,sci.rev A.Nat.Sci Engg [8] Ghorai.G,Pal.M,Faces and dual of -polar fuzzy planar graphs.j.intell Fuzzy syste. [9] Kalaiarasi,K., Optiization of a Multiple Vendor Single Buyer Integrated Inventory Model with a Variable Nuber of Vendors, International journal of Matheatical Sciences and Engineering Applications IJMSEAISNN Vol.5 No.V Sep0, pp [0] Kalaiarasi.K,Optiization of fuzzy integrated Vendor-Buyer inventory odels,annala of Fuzzy atheatics and inforatics,vol.,no..oct 0,pp [] Kalaiarasi,K., Optiization Of Fuzzy integrated Two Stage Vendor- Buyer Inventory Syste, International of Matheatical Sciences and Applications,Vol.,No May 0. [] MolodtsovD.A,the theory of soft sets,urss publishers,moscow,004. [3] Shiura.MEds,Fuzzy sets and their applications,acadeic press,new York,975,pp [4] Saanta.S,Pal.M,irregular bipolar fuzzy graphs Int.J.Appl.fuzzy sets [5] Sunitha.M.S,Vijayakuar.A,Copleent of a fuzzy graphs,indian journal of pure and applied atheatics 339,pp, [6] Zadeh.L.A,Fuzzy sets,inforation and control [7] Zhang.W.R,Zhang.L,Yin Yang bipolar logic and bipolar fuzzy logic,inforation sciences