International Journal of Mathematical Archive-5(9), 2014, Available online through ISSN

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1 International Journal of Mathematical Archive-5(9), 2014, Available online through wwwijmainfo ISSN ON D RULAR FUZZY RAPHS K Radha 1 and N Kumaravel 2 1 P Department of Mathematics, Periyar VR College, Tiruchirappalli , Tamil Nadu, India 2 Department of Mathematics, Paavai College of ngineering, Pachal, Namakkal , Tamil Nadu, India (Received On: ; Revised & Accepted On: ) ABSTRACT In this paper, totally edge regular fuzzy graphs, partially edge regular fuzzy graphs and full edge regular fuzzy graphs are introduced dge regular fuzzy graphs and totally edge regular fuzzy graphs are compared through various examples and various properties between degree and edge degree are provided A necessary and sufficient condition under which they are equivalent is provided Some properties of edge regular fuzzy graphs are studied and they are examined for totally edge regular fuzzy graphs Keywords: Degree of a vertex, degree of an edge, edge regular fuzzy graph, totally edge regular fuzzy graph, total degree of an edge, totally edge regular fuzzy graph 2010 Mathematics Subject Classification: 0372, 05C72 1 INTRODUCTION In 1736, uler first introduced the concept of graph theory In the history of mathematics, the solution given by uler of the well-known Konigsberg bridge problem is considered to be the first theorem of graph theory The graph theory is a very useful tool for solving combinatorial problems in different areas such as operations research, optimization, topology, geometry, number theory, algebra and computer science Fuzzy set theory, introduced by Zadeh in 1965 is a mathematical tool for handling uncertainties like vagueness, ambiguity and imprecision in linguistic variables [12] Research on theory of fuzzy sets has been witnessing an exponential growth; both within mathematics and in its application Fuzzy set theory has emerged as a potential area of interdisciplinary research and fuzzy graph theory is of recent interest The first definition of fuzzy graph was introduced by Haufmann in 1973, based on Zadeh s fuzzy relations in 1971 In 1975, Rosenfeld introduced the concept of fuzzy graphs [10] The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs using fuzzy relations, obtaining analogs of several graph theoretical concepts During the same time Yeh and Bang have also introduced various connectedness concepts in fuzzy graph [11] Now, fuzzy graphs have been witnessing a tremendous growth and finds application in many branches of engineering and technology A Nagoorani and K Radha introduced the concept of regular fuzzy graphs in 2008 [5] K Radha and N Kumaravel introduced the concept of edge degree, total edge degree and discussed about the degree of an edge in some fuzzy graphs [8] In this paper we discuss edge regular fuzzy graphs and totally edge regular fuzzy graphs through various examples We provide a necessary and sufficient condition under which they become equivalent Also we study some properties of edge regular fuzzy graphs and totally edge regular fuzzy graphs First we go through some basic definitions and results with various examples in the next chapter Corresponding Author: N Kumaravel 2 2Department of Mathematics, Paavai College of ngineering, Pachal, Namakkal , Tamil Nadu, India International Journal of Mathematical Archive- 5(9), Sept

2 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept BASIC DFINITIONS 21 Fuzzy graph [6]: Let V be a non-empty finite set and µ: [0, 1] such that µ ( x, y) σ ( x) σ ( y) Note 22: The underlying crisp graph of : ( σ, µ ) 23 Degree of a vertex [4]: Let : ( σ, µ ) V V A fuzzy graph : ( σ, µ ) for all x, y V is a pair of functions σ: V [0, 1] and is denoted by : ( V, ) where V V be a fuzzy graph on : ( V, ) The degree of a vertex u is d = µ The minimum degree of is δ ( ) = { d ( v), v V} and the maximum degree of is ) = { d ( v), v V} 24 Total degree of a vertex [6]: Let (, ) td u v ( : σ µ be a fuzzy graph on : ( V, ) The total degree of a vertex u V = µ + σ u v is defined by 25 Degree of an edge in a graph [1]: Let : ( V, ) by ) ( ) ( ) 2 be a graph and let e = be an edge in d = d u + d v ( Then the degree of an edge e = is defined 26 Degree of an edge in a fuzzy graph [8]: Let : ( σ, µ ) be a fuzzy graph on : ( V, ) The degree of an edge is d = d + d ( v) 2 This is equivalent to d () µ = uw µ ( uw) + wv w u µ ( wv) = uw µ (uw) + µ ( wv) 2µ The minimum edge degree and maximum edge degree of are δ ( ) = { d, } and ( ) = { d, } 27 Total degree of an edge in a fuzzy graph [8] : σ µ be a fuzzy graph on : ( V, ) The total degree of an edge = d + d ( v) This is equivalent to td () = µ ( uw) + µ ( wv) + () Let (, ) td µ d () + () xample 28: µ wv uw w wv u is defined by µ = Fig-21: Fuzzy graph : (σ, µ) 2014, IJMA All Rights Reserved 101

3 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept-2014 d = µ + µ ( uw) = = 04, td = d + = = 0 6 δ ( ) = { d ( v), v V} = {04, 05, 11, 06} = 04 = d (u) ( ) = { d ( v), v V} = {04, 05, 11, 06} = 11 = d (w) d () = µ ( uw) + µ ( wv) = = 05 uw wv w u σ td () = d () + µ () = = 07 δ ( ) = { d, } = {05, 11, 10, 05} = 05 = d () = (wx) ( ) = d, = {05, 11, 10, 05} = 11 = d (uw) { } 29 Order and size of a fuzzy graph [2] The order and size of a fuzzy graph are defined by 211 Regular fuzzy graph [5] d ( ) = σ ( u and ) = O ) u V S ( µ Let : (σ, μ) be a fuzzy graph on : (V, ) If each vertex in has same degree k, then is said to be a regular fuzzy graph or k regular fuzzy graph 212 Totally regular fuzzy graph [5]: Let : (σ, μ) be a fuzzy graph on : (V, ) If each vertex in has same total degree k, then is said to be a totally regular fuzzy graph or k totally regular fuzzy graph v V Theorem 213 [2] Let : (σ, µ) be a fuzzy graph on : (V, ) Then d ( v) 2S( ) = Theorem 214 [5] Let : (σ, µ) be a fuzzy graph on : (V, ) Then td ( v) = 2S( ) O( ) Theorem 215 [9]: Let : (σ, µ) be a fuzzy graph on a cycle : (V, ) Then Theorem 216 [9]: Let : (σ, µ) be a fuzzy graph on : (V, ) Then v V + Theorem 217 [9]: Let : (σ, µ) be a fuzzy graph on a k regular graph : (V, ) Then d 2( k 1) S( ) = Theorem 218: Let : (σ, µ) be a fuzzy graph on : (V, ) Then Proof: The size of is S() = µ () td = ( d + µ ) = ) + d ( µ v V d v) = d ( d = d µ td = d S( ) µ + Hence td = d S( ) µ (By theorem 214) Partially regular fuzzy graph [7]: If the underlying graph is regular, then is said to be a partially regular fuzzy graph 220 Full regular fuzzy graph [7]: If is both regular fuzzy graph and partially regular fuzzy graph, then is said to be a full regular fuzzy graph 2014, IJMA All Rights Reserved 102

4 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept D RULAR FUZZY RAPHS AND TOTALLY D RULAR FUZZY RAPHS 31 dge regular fuzzy graph [9]: Let : (σ, μ) be a fuzzy graph on : (V, ) If each edge in has same degree k, then is said to be an edge regular fuzzy graph or k edge regular fuzzy graph 32 Totally edge regular fuzzy graph: Let : (σ, μ) be a fuzzy graph on : (V, ) If each edge in has same total degree k, then is said to be a totally edge regular fuzzy graph or k totally edge regular fuzzy graph Remark 33: 1 is a k edge regular fuzzy graph if and only if δ ( ) = ( ) = k 2 is a k totally edge regular fuzzy graph if and only if t ( ) = t ( ) = k is the maximum total edge degree of minimum total edge degree of and ( t ) δ, where δ ) Remark 34: In crisp graph theory, any complete graph is edge regular But this result does not carry over to the fuzzy case A complete fuzzy graph need not be edge regular For example, in fig32, is not edge regular, but it is a complete fuzzy graph xample 35: t ( is the Fig-31 In the above figure 31, is 14 edge regular fuzzy graph, but is not a totally edge regular fuzzy graph and also is not a regular fuzzy graph xample 36: Consider the following fuzzy graph : (σ, µ) Fig32 Here, is a 13 totally edge regular fuzzy graph, but not an edge regular fuzzy graph xample 37: Consider the following fuzzy graph : (σ, µ) Fig , IJMA All Rights Reserved 103

5 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept-2014 Here, is neither edge regular fuzzy graph nor totally edge regular fuzzy graph But is a 09 regular fuzzy graph xample 38: Consider the following fuzzy graph : (σ, µ) Fig-34 In the figure 34, is both edge regular fuzzy graph and totally edge regular fuzzy graph Also is regular fuzzy graph xample 39: Fig-35 Here, is both 09 edge regular fuzzy graph and 12 totally edge regular fuzzy graph But is not regular fuzzy graph Remark 310: From the above examples, it is clear that in general there does not exist any relationship between edge regular fuzzy graphs, totally edge regular fuzzy graphs, regular fuzzy graphs and totally regular fuzzy graphs However, a necessary and sufficient condition under which two types of fuzzy graphs, edge regular fuzzy graphs and totally edge regular fuzzy graphs are equivalent in some particular case is provided in the following theorem Theorem 311: Let : (σ, µ) be a fuzzy graph on : (V, ) Then µ is a constant function if and only if the following are equivalent: (1) is an edge regular fuzzy graph (2) is a totally edge regular fuzzy graph Proof: Suppose that µ is a constant function Let µ() = c, for every, where c is a constant Assume that is a k 1 edge regular fuzzy graph Then d() = k 1, for all td() = d() + µ(), for all td() = k 1 + c, for all Hence is a (k 1 + c) totally edge regular fuzzy graph Thus (1) (2) is proved Now, suppose that is a k 2 totally edge regular fuzzy graph 2014, IJMA All Rights Reserved 104

6 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept-2014 Then td() = k 2, for all d() + µ() = k 2, for all d() = k 2 µ(), for all Ε d() = k 2 c, for all Ε Hence is a (k 2 c) edge regular fuzzy graph Thus (2) (1) is proved Hence (1) and (2) are equivalent Conversely, assume that (1) and (2) are equivalent ie, is edge regular if and only if is totally edge regular To prove that µ is a constant function Suppose µ is not a constant function Then µ() µ(xy) for atleast one pair of edges, xy Let be a k edge regular fuzzy graph Then d() = d(xy) = k td() = d() + µ() = k + µ() and td(xy) = d(xy) + µ(xy) Since µ() µ(xy), we have td() td(xy) Hence is not a totally edge regular, which is a contradiction to our assumption Now, let be a totally edge regular fuzzy graph Then td() = td(xy) d() + µ() = d(xy) + µ(xy) d() d(xy) = µ(xy) µ() d() d(xy) 0, (since µ(xy) µ()) d() d(xy) Thus is not an edge regular fuzzy graph This is a contradiction to our assumption Hence µ is a constant function Theorem 312: If a fuzzy graph is both edge regular and totally edge regular, then µ is a constant function Proof: Let be a k 1 edge regular and k 2 totally edge regular fuzzy graph Then d() = k 1, for all and td() = k 2, for all Now, td() = k 2, for all, for all 2014, IJMA All Rights Reserved 105

7 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept-2014 d() + µ() = k 2, for all µ() = k 2 k 1, for all Hence µ is a constant function Remark 313: The converse of theorem 312 need not be true It can be seen from the following example Consider the fuzzy graph : (σ, µ) given in figure 36 Fig-36 Here, µ is a constant function, but is not an edge regular and also not a totally edge regular fuzzy graph Theorem 314 [7]: Let : (σ, µ) be a fuzzy graph such that µ is a constant function Then is a regular fuzzy graph if and only if is a partially regular fuzzy graph Theorem 315: Let : (σ, µ) be a fuzzy graph on : (V, ) If is both edge regular and totally edge regular, then is a regular fuzzy graph if and only if is a regular graph Proof: If is both edge regular and totally edge regular, then µ is a constant function By theorem 314, is a regular fuzzy graph if and only if is a regular graph Remark 316: Converse part of theorem 315 need not be true Fig-37 From the above figure, is regular, but is neither edge regular nor totally edge regular Theorem 317 [9]: Let µ = c be a constant function in : (σ, µ) on : (V, ) If is regular, then is edge regular Theorem 318: Let µ = c be a constant function in : (σ, µ) on : (V, ) If is regular, then is totally edge regular Proof: Let µ = c be a constant function in : (σ, µ) Assume that is regular fuzzy graph with d(u) = k, for all u V To prove that is totally edge regular fuzzy graph By definition of total edge degree, td () = d (u) + d (v) µ(), for all = k + k c, for all td () = 2k c, for all Hence is totally edge regular fuzzy graph Remark 319: The converse of above theorem 318 need not be true from the fig35 in example , IJMA All Rights Reserved 106

8 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept-2014 Theorem 320 [9]: Let : (σ, µ) be a fuzzy graph on : (V, ) with is k regular Then µ is constant iff is both regular and edge regular Theorem 321: Let : (σ, µ) be a fuzzy graph on : (V, ) with is k regular Then µ is constant iff is both regular and totally edge regular Proof: Let : (σ, µ) be a fuzzy graph on : (V, ) and let be a k regular graph Assume that µ is a constant function To prove that is both regular and totally edge regular Let µ() = c, d (u) = µ (), for all u V d (u) = c, for all u V = c d (u), for all u V d (u) = ck, for all u V Hence is a regular fuzzy graph Now, td () = uw = µ ( uw) + wv w u c + c + c uw wv w u µ ( wv) + (uw), for all µ, for all Ε = c(d (u) 1) + c(d (v) 1) + c, for all = c(k 1) + c(k 1) + c, for all td () = 2c(k 1) + c, for all Hence is also totally edge regular Conversely, assume that is both regular and totally edge regular To prove that µ is a constant function Since is regular Then d (u) = c (say), for all u V Also, since is totally edge regular Then td () = k (say), for all By definition of total edge degree, td() = d(u) + d(v) µ(), for all Hence µ is a constant function k = c + c µ(), for all µ() = 2c k, for all 2014, IJMA All Rights Reserved 107

9 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept-2014 Remark 322: Let : (σ, µ) be a fuzzy graph on : (V, ) If µ is a constant function if and only if the following are equivalent: (1) is regular, (2) is edge regular, (3) is totally edge regular Proof: Its proof is similar to theorem 311 Definition 323: Let : (V, ) be a graph Then is said to be edge regular, if each edge in has same degree Theorem 324: Let : (σ, µ) be a fuzzy graph on : (V, ) If µ is a constant function, then is edge regular if and only if is edge regular Proof: iven that µ is a constant function Then µ(xy) = c, where c is constant Assume that is edge regular To prove is edge regular Suppose that is not an edge regular Then d () d (xy) for atleast one pair of edges, xy By the definition of edge degree of a fuzzy graph, d () = µ ( uw) + µ ( wv), for all uw wv = uw c + w u w u c, for all wv = c(d (u) 1) + c(d (v) 1), for all d () = c(d (u) + d (v) 2), for all By the definition of edge degree in a graph, d () = c(d ()), for all d (xy) = c(d (xy)) Since d () d (xy) d () d (xy) Thus is not an edge regular This is a contradiction to our assumption Hence is edge regular Conversely, Assume that µ is a constant function and is edge regular To prove is edge regular Suppose that is not an edge regular Then d () d (xy) for atleast one pair of edges, xy 2014, IJMA All Rights Reserved 108

10 uw K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept-2014 µ ( uw) + wv uw w u µ ( wv) xz z y µ (xz) + zy c + c c + w wv u xz z zy y z x z x µ (zy) c(d (u) 1) + c(d (v) 1) c(d (x) 1) + c(d (y) 1) c c(d (u) + d (v) 2) c(d (x) + d (y) 2) By the definition of edge degree in a crisp graph, we have c d () c d (xy) Thus is not an edge regular d () d (xy) This is a contradiction to our assumption Hence is edge regular Remark 325: The above theorem need not be true, when µ is not a constant function Fig-37 From the above figure, is edge regular, but is not an edge regular and also µ is not a constant function Theorem 326: Let : (σ, µ) be a regular fuzzy graph on : (V, ) Then is edge regular if and only if µ is a constant function Proof: Let : (σ, µ) be a k regular fuzzy graph Then d (u) = k, for all u V Assume that µ is a constant function Then to prove that is edge regular Let µ = c By the definition of edge degree, d () = d (u) + d (v) 2µ(), for all = k + k 2c d () = k 1, for all, where k 1 = k + k 2c Hence is edge regular Conversely, assume that is edge regular Then to prove that µ is a constant function Let d () = k 1, for all 2014, IJMA All Rights Reserved 109

11 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept-2014 By the definition of edge degree, d () = d (u) + d (v) 2µ(), for all k 1 = k + k 2µ() µ() = (2k k 1 )/2 = c, for all, where c = (2k k 1 )/2 Hence µ is a constant function 4 PARTIALLY D RULAR FUZZY RAPH AND FULL D RULAR FUZZY RAPH 41 Partially edge regular fuzzy graph: If the underlying graph is edge regular, then is said to be a partially edge regular fuzzy graph 42 Full edge regular fuzzy graph: If is both edge regular and partially edge regular, then is said to be a full edge regular fuzzy graph Remark 43: Using the definition of partially edge regular fuzzy graph, theorem 324 can be restated as follows: Let : (σ, µ) be a fuzzy graph on : (V, ) such that µ is a constant function Then is edge regular if and only if is partially edge regular Theorem 44: Let : (σ, µ) be a fuzzy graph on : (V, ) such that µ is a constant function If is full regular fuzzy graph, then is full edge regular fuzzy graph Proof: iven µ is a constant function Let µ ( ) = c, for all Assume that is full regular fuzzy graph, where c is a constant Then d = k and d = r, for all u V, where k and r are constants ) ( ) ( ) 2 2 2, a constant d ( = d u + d v = r Hence is edge regular graph Now, d = d + d 2µ, = k + k 2c = 2( k c), a constant Hence is edge regular fuzzy graph Thus is full edge regular fuzzy graph Remark 45: The converse of theorem 44 need not be true Consider the following fuzzy graph : (σ, µ) Fig-41 Here, µ is a constant function and is full edge regular fuzzy graph But is not full regular fuzzy graph 2014, IJMA All Rights Reserved 110

12 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept PROPRTIS OF D RULAR FUZZY RAPHS Theorem 51 [9]: The size of a k edge regular fuzzy graph : (σ, µ) on a k 1 edge regular : (V, ) is where q = qk, k 1 Theorem 52: Let : (σ, µ) be a r totally edge regular and r 1 partially edge regular fuzzy graph Then S() = qr r +1 1 Proof: The size of is S() = µ () Since is r totally edge regular and is r 1 edge regular Therefore td () = r, d () = r 1, for all Thus td = 1 qr = r 1 S() + S() Hence S() = qr r 1 +1 r + S( ) µ (by theorem 218) Theorem 53: If is a k edge regular and r totally edge regular fuzzy graph, then S() = q(r k) Proof: Let be a k edge regular and r totally edge regular Then d () = k and td () = r, for all d = qk and td ) Since td () = d () + µ(), td = d + qr = qk + S() ( = qr µ () S() = q(r k) RFRNCS 1 S Arumugam and S Velammal, dge domination in graphs, Taiwanese Journal of Mathematics, Volume 2, Number 2, June 1998, A Nagoorgani and M Basheer Ahamed, Order and Size in Fuzzy raph, Bulletin of Pure and Applied Sciences, Volume 22, Number 1, 2003, A Nagoor ani and V T Chandrasekaran, A First Look at Fuzzy raph Theory, Allied Publishers, A Nagoor ani and J Malarvizhi, Properties of μ-complement of a Fuzzy raph, International Journal of Algorithms, Computing and Mathematics, Volume 2, Number 3, 2009, A Nagoorani and K Radha, On regular fuzzy graphs, Journal of Physical Sciences, Volume 12, 2008, A Nagoorgani and K Radha, The degree of a vertex in some fuzzy graphs, International Journal of Algorithms, Computing and Mathematics, Volume 2, Number 3, August 2009, A Nagoorani and K Radha, Regular Property of Fuzzy raphs, Bulletin of Pure and Applied Sciences, Volume 27, Number 2, 2008, K Radha and N Kumaravel, The degree of an edge in Cartesian product and composition of two fuzzy graphs, International Journal of Applied Mathematics & Statistical Sciences(IJAMSS) IAST, Vol 2, Issue 2, May 2013, K Radha and N Kumaravel, Some properties of edge regular fuzzy graphs, Jamal Academic Research Journal (JARJ), Special issue, 2014, , IJMA All Rights Reserved 111

13 K Radha 1 and N Kumaravel 2 / On dge Regular Fuzzy raphs / IJMA- 5(9), Sept A Rosenfeld, Fuzzy graphs, in: LA Zadeh, KS Fu, K Tanaka and M Shimura, (editors), Fuzzy sets and their applications to cognitive and decision process, Academic press, New York, 1975, R T Yeh and S Y Bang, Fuzzy relations, fuzzy graphs, and their applications to clustering analysis, in: LA Zadeh, KS Fu, K Tanaka and M Shimura, (editors), Fuzzy sets and their applications to cognitive and decision process, Academic press, New York, 1975, L A Zadeh, Fuzzy Sets, Information and control 8, 1965, Source of support: Nil, Conflict of interest: None Declared [Copy right 2014 This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited] 2014, IJMA All Rights Reserved 112