On Edge Regular Fuzzy Line Graphs

Iteratioal Joural of Computatioal ad Applied Mathematics ISSN 1819-4966 Volume 11, Number 2 (2016), pp 105-118 Research Idia Publicatios http://wwwripublicatiocom O Edge Regular Fuzz Lie Graphs K Radha 1 ad N Kumaravel 2 1 PG Departmet of Mathematics, Periar EVR College, Tiruchirappalli 620 023, Tamil Nadu, Idia E-mail: radhagac@ahoocom 2 Departmet of Mathematics, K S R Istitute for Egieerig ad Techolog, Namakkal 637 215, Tamil Nadu, Idia E-mail: kumaramaths@gmailcom Abstract I this paper, degree of a edge i fuzz lie graph is obtaied ad some properties of edge regular fuzz lie graphs are studied Fuzz lie graph of a edge regular fuzz graph eed ot be edge regular Coditios uder which it is edge regular are provided Kewords: Strog fuzz graph, complete fuzz graph, edge regular fuzz graph, totall edge regular fuzz graph, fuzz lie graph 2010 Mathematics Subject Classificatio: 03E72, 05C72 1 INTRODUCTION Fuzz graph theor was itroduced b Azriel Rosefeld i 1975 [11] Though it is ver oug, it has bee growig fast ad has umerous applicatios i various fields Durig the same time Yeh ad bag have also itroduced various coectedess cocepts i fuzz graphs [12] JNMordeso (1993) itroduced the cocept of fuzz lie graph [2] KRadha ad NKumaravel (2014) itroduced the cocept of edge regular fuzz graphs [9] I this paper, we stud about edge regular propert of fuzz lie graphs First we go through some basic defiitios i the et sectio from [1] [5], [8] ad [9] 2 BASIC CONCEPTS Let V be a o-empt fiite set ad E V V A fuzz graph G :( is a pair of fuctios : V [0, 1] ad : E [0, 1] such that (, ) ( ) ( ) for all

106 K Radha ad N Kumaravel, V The order ad size of a fuzz graph G :( are defied b O ( G) ( ) ad S ( G) ( ) A fuzz Graph G :( is strog, if V E ( ) ( ) ( ) for all E A fuzz Graph G :( is complete, if ( ) ( ) ( ) for all, V The uderlig crisp graph is deoted b G : ( V, E ) be a fuzz graph o G :( V, The degree of a verte is d ( ) ( ) If each verte i G has same degree k, the G is said to be a G regular fuzz graph or k regular fuzz graph The degree of a edge e uve is defied b d uv ) d ( u ) d ( v ) 2 ( G G G If each edge i G has same degree, the G is said to be edge regular The degree of a edge E is d ( ) ( z) ( z) 2( ) If each edge i fuzz G z z graph G has same degree k, the G is said to be a edge regular fuzz graph or k edge regular fuzz graph Give a graph G : ( V,, its lie graph L( G ) : ( Z, is a graph such that S u v Z E, u v, u, v V, W S S S S,, E, The umber of edges i L( G ) : ( Z, is half of the sum of the degree of edges i G : ( V, G : ( V, is a ccle if ad ol if L( G ) : ( Z, is also a ccle The fuzz lie graph of G :( is L( G) : ( with uderlig graph L( G ) : ( Z, where Z S u v E, u v, u, v V, W S S S S,, E,, ( ) ( ), S Z ad ( S S ) ( ) ( ) for ever S S W [2] For the sake of simplicit, the vertices of L(G) ma be deoted b istead of S ad the edges b istead of S S A homomorphism of fuzz graphs h: G G is a map h: V V which satisfies ( ) ( h( )) for V ad (, ) ( h( ), h( )) for, V A weak isomorphism h: G G is a map h: V V which is a bijective homomorphism that satisfies ( ) ( h( )) for V A co weak isomorphism h: G G is a map h: V V which is a bijective homomorphism that satisfies (, ) ( h( ), h( )) for, V A isomorphism h: G G is a map h: V V which is a bijective homomorphism that satisfies ( ) ( h( )) for V ad (, ) ( h( ), h( )) for, V We deote it as G G

O Edge Regular Fuzz Lie Graphs 107 21 Theorem [9]: be a fuzz graph o a k regular graph G : ( V, The is a costat fuctio if ad ol if G is both regular ad edge regular 22 Theorem [5]: be a fuzz graph o a odd ccle G : ( V, The G is regular iff is a costat fuctio 23 Theorem [5]: be a fuzz graph where G : ( V, is a eve ccle The G is regular iff either is a costat fuctio or alterate edges have same membership values 24 Theorem [8]: be a fuzz graph o G : ( V, If is a costat fuctio, the G is edge regular if ad ol if G is edge regular 25 Theorem [9]: Let be a costat fuctio i G :( o G : ( V, If G is regular, the G is edge regular 26 Theorem [10]: be a fuzz graph o a odd ccle G : ( V, The G is edge regular if ad ol if is a costat fuctio 3 DEGREE OF AN EDGE IN FUZZY LINE GRAPH For a ab W, dlg ( )( ab) ( ac) ( cb), ab W LG ( ) acw, cb cbw, ca (31) d ( ab) ( a) ( c) ( b) ( c), ab W ce, ce, cb ca 31 Theorem: be a fuzz graph such that is a costat fuctio The d ( ) ( ) ( ) 2,, LG ( ) ab c d a d b a be G G Let ( a) c, a E From (31), dlg ( )( ab) ( a) ( c) ( b) ( c), ab W ce, ce, cb ca

108 K Radha ad N Kumaravel Therefore, for all ab W, dlg ( )( ab) c c c c ce, ce, cb ca c d a c d b ( ) 1 ( ) 1 G G ( ) ( ) 2 G G c d a d b 4 EDGE REGULAR PROPERTY OF FUZZY LINE GRAPH OF FUZZY GRAPH 41 Remark: If G : ( is a edge regular fuzz graph, the L ( G) : ( eed ot be edge regular fuzz graph For eample, i figure 41 G : ( is 18 edge regular fuzz graph, but L( G) : ( is ot a edge regular fuzz graph 42 Remark: If L ( G) : ( is a edge regular fuzz graph, the G : ( eed ot be edge regular fuzz graph For eample, i figure 42 L ( G) : ( is 07 edge regular fuzz graph But G : ( is ot a edge regular fuzz graph

O Edge Regular Fuzz Lie Graphs 109 43 Theorem: be a fuzz graph such that is a costat fuctio If G :( is a edge regular fuzz graph, the L ( G) : ( is a edge regular fuzz graph Sice G :( is a edge regular fuzz graph ad is a costat fuctio, b theorem 24, G is edge regular Let G be m edge regular The b theorem 31, for a ab W, d ( ) ( ) ( ) 2 LG ( ) ab c d a d b G G cm m 2 2cm 1 Hece L ( G) : ( is a edge regular fuzz graph 44 Theorem: be a fuzz graph such that is a costat fuctio If G :( is a edge regular fuzz graph, the L ( G ) is a edge regular fuzz graph, where is a positive iteger B theorem 43, L ( G) : ( is a edge regular fuzz graph Sice is a costat fuctio, is a costat fuctio Agai b theorem 43, L( L( G )) is a edge regular fuzz graph Proceedig i this same wa, we get L ( G ) is a edge regular fuzz graph, for ever positive iteger 45 Theorem: be a strog fuzz graph such that is a costat fuctio If G :( is a edge regular fuzz graph, the L ( G) : ( is a edge regular fuzz graph B the hpothesis of this theorem, is a costat fuctio Therefore the result follows from theorem 43 46 Theorem: be a strog fuzz graph such that is a costat fuctio If G :( is a edge regular fuzz graph, the L ( G) : (, ) is a edge regular fuzz graph B the hpothesis of this theorem, is a costat fuctio Therefore the result follows from theorem 44

110 K Radha ad N Kumaravel 47 Theorem: be a fuzz graph such that is a costat fuctio If G :( is a regular fuzz graph, the L ( G) : ( is a edge regular fuzz graph If G :( is a regular fuzz graph such that is a costat fuctio, the b theorem 25, G :( is edge regular The b theorem 43, L ( G) : ( is a edge regular fuzz graph 48 Remark: The coverse of above theorem eed ot be true From the followig figures, L ( G) : ( is 16 edge regular fuzz graph ad is a costat fuctio But, G :( is ot a regular fuzz graph 49 Theorem: be a fuzz graph such that is a costat fuctio If G :( is a regular fuzz graph, the L ( G) : (, ) is a edge regular fuzz graph Proof is similar to proof of the theorem 44 410 Theorem: m If G : ( V, is a k regular graph, the L ( G ) is graph m m1 2(2 k 2 1) edge regular Let us prove this theorem b mathematical iductio Sice G : ( V, is a k regular graph, G is 2( k 1) edge regular Therefore L ( G ) is 2( k 1) regular

O Edge Regular Fuzz Lie Graphs 111 For a edge ab i L ( G ), d ( ab) d ( a) d ( b) 2 L( G ) L( G ) L( G ) 2( k1) 2( k1) 2 2(2k 3) Therefore the result is true for 1 m Assume that L ( G m m1 ) is a 2(2 k 2 1) edge regular graph The L m 1 ( G ) is m m1 2(2 k 2 1) regular For a edge i L ( G ), d m1 ( ) d m1 ( ) d m1 ( ) 2 L ( G ) L ( G ) L ( G ) 1 1 2(2 m 2 m 1) 2(2 m 2 m k k 1) 2 m1 m2 2(2 k 2 1) L m 1 ( G ) is m1 m2 2(2 k 2 1) edge regular B the priciple of mathematical iductio, we have L ( G ) edge regular, for ever positive iteger is a 1 2(2 k 2 1) 411 Theorem: be a fuzz graph such that ( e) c, e E, where c is a costat If G : ( V, is a k regular graph, the L ( G) : (, ) is a regular fuzz graph, where is a positive iteger 1 2 c(2 k 2 1) edge B the defiitio of fuzz lie graph, all the edge membership values is equal to c i 2 L( G), L ( G),, L ( G ), sice ( e) c, e E, where c is a costat (41) To prove L ( G) : (, ) is a 1 2 c(2 k 2 1) edge regular fuzz graph B (41) ad theorem 31, d ( ab) c d 1 ( a) d 1 ( b) 2 ab W L ( G) L ( G ) L ( G ) Usig the theorem 410, d 1 1 ( ab ) c 2(2 k 2 1) 2(2 k 2 1) 2 ab W L ( G) c k k 1 1 2 2 2 2 2 2 2 1 d ( ab) 2c 2 k 2 1, ab W L ( G) Therefore L ( G) : (, ) is a positive itegers 1 2c 2 k2 1 edge regular fuzz graph, for all 412 Theorem: be a strog fuzz graph such that is a costat fuctio ad let G : ( V, be a regular graph The L ( G) : (, ) is a edge regular fuzz graph

112 K Radha ad N Kumaravel Proof is similar to proof of theorem 411 413 Theorem: Let G : ( V, be a edge regular graph If membership value of atleast oe of the two edges of a pair of adjacet edges is equal to { ( e) e E}, the lowest edge membership value, the L ( G) : ( is a edge regular fuzz graph Let G be k edge regular ad let { ( e) e E} c The b hpothesis, ( ) ( ) c for a pair of adjacet edges ad From (31), for all W, dlg ( )( ) ( ) ( z) ( ) ( z) ze, ze, z z ( ) 1 ( ) 1 G G ck 1 ck 1 c d c d Hece L ( G) : ( is a edge regular fuzz graph c ze, ze, z z 2ck 1 c 414 Remark: The coverse of above theorem eed ot be true From the figure 42, L ( G) : ( is 07 edge regular fuzz graph But, membership value of atleast oe of the two edges of a pair of adjacet edges is ot equal to { ( e) e E}, the lowest edge membership value 5 EDGE REGULAR PROPERTY OF FUZZY LINE GRAPH OF FUZZY GRAPH ON A CYCLE 51 Theorem: be a fuzz graph ad let L ( G) : ( be a fuzz lie graph of G :( The G :( is a fuzz graph o a ccle G : ( V, if ad ol if L ( G) : ( is a fuzz graph o the ccle L( G ) : ( Z, Sice G : ( V, is a ccle if ad ol if L( G ) : ( Z, is also a ccle, G :( is a fuzz graph o a ccle G : ( V, if ad ol if L ( G) : ( is a fuzz graph o a ccle L( G ) : ( Z, 52 Theorem: If G : (, is a strog fuzz graph o a ccle G : ( V, such that is a costat fuctio, the G is isomorphic to LG ( )

O Edge Regular Fuzz Lie Graphs 113 Let ( u) c, for all u V, where c is a costat Sice G is a strog fuzz graph, ( e) c, e E B the defiitio of fuzz lie graph, ad are also costat fuctios of same costat value c Let G be a fuzz graph o a ccle v1e 1v2e2 vev 1, vi V & ei E, where v 1 v The ( ) 1 LG is a fuzz lie graph o the ccle e1 1e 22 ee 1, ei Z & i W, where e 1 e 1 Now, defie a mappig h: V Z b h( v ) e, v V i i i The ( h( v )) ( e ) c ( v ), v V ad i i i i ( h( vi ) h( vi 1)) ( h( vi )) ( h( vi 1)) ( ei ) ( ei 1) c ( ei ) ( vivi 1), ei E Therefore G is isomorphic to LG ( ) 53 Theorem: If G : (, is a fuzz graph o a ccle G : ( V,, the LG ( ) is homomorphic to G Let G be a fuzz graph o a ccle v1e 1v2e2 vev 1, vi V & ei E, where v 1 v The ( ) 1 LG is a fuzz lie graph o the ccle e1 1e 22 ee 1, ei Z & i W, where e 1 e 1 Now, defie a mappig h: Z V betwee LG ( ) ad G b h( e ) v, e Z i i i Therefore ( h( e )) ( v ) ( e ) ( e ), e Z ad i i i i i ( h( ei ) h( ei 1)) ( vivi 1) ( ei ) ( ei ) ( eiei 1), ei, ei 1 Z Therefore LG ( ) is homomorphic to G 54 Theorem: be a fuzz graph o a eve ccle G : ( V, If either is a costat fuctio of costat value k or each edge i a oe set of alterate edges has same lowest membership value k i G, the L ( G) : ( is 2k edge regular fuzz graph If is a costat fuctio, the there is othig to prove Let e1, e2, e3,, e 2m be the edges of the eve ccle G i that order Suppose a oe set of the alterate edges have same lowest membership value k i G

114 K Radha ad N Kumaravel The either ( e ) ( e ) ( e ) ( e ) ( e ), e E, i 2,4,,2m (or) 1 3 5 2m1 i i ( e ) ( e ) ( e ) ( e ) ( e ), e E, i 1,3,5,,2 m 1 2 4 6 2m i i For a two adjacet edges e i ad ei 1, ( ei) ( ei 1) k is a costat fuctio with ( eiei 1) k, i 1,2,,2 m, where e 2m1 e1 L( G) ( i i1) ( i) ( i1) 2 ( i i1), i i1 d e e d e d e e e e e W 2k 2k 2k 2k Therefore L (G) is 2k edge regular fuzz graph 55 Remark: The coverse of above theorem eed ot be true From the figure 42, L ( G) : ( is 07 edge regular fuzz graph But, alterate edges do ot has same lowest membership values i G : ( 56 Theorem: be a fuzz graph o a eve ccle G : ( V, If each edge i a oe set of alterate edges has same lowest membership value k i G, the L ( G) : (, ) is 2k edge regular fuzz graph From the theorem 54, L (G) is 2k edge regular fuzz graph Sice is a costat fuctio, b the defiitio of fuzz lie graph, all edge membership values i L( L( G )) is equal to k B the theorem 21, L( L( G )) is 2k edge regular fuzz graph Proceedig i the same wa, fiall, we get L ( G) : (, ) is 2k edge regular fuzz graph 57 Theorem: be a fuzz graph o a odd ccle G : ( V, The L ( G) : ( is a edge regular fuzz graph if ad ol if either is a costat fuctio or a two adjacet edges ad all the edges that lie alterativel from them receives lowest membership value As i the proof of theorem 54, L (G) is 2k edge regular fuzz graph Coversel, assume that L (G) is a edge regular fuzz graph Sice L (G) is a fuzz graph o a odd ccle G : ( V,, is a costat fuctio (usig theorem 26)

O Edge Regular Fuzz Lie Graphs 115 ( ) ( ) ( ) k, for all W, where k is a costat (51) If ( ) ( ), for all W, the is a costat fuctio Suppose ot, the for a two adjacet edges ad such that ( ) ( ) Without loss of geeralit, assume that ( ) ( ) Hece the result follows from (51) 58 Theorem: be regular fuzz graph o a odd ccle G : ( V, The L ( G) : ( is a edge regular fuzz graph Usig theorem 22, is a costat fuctio Therefore is also a costat fuctio i L (G) Also LG ( ) is a ccle Hece b theorem 21, L (G) is edge regular 59 Remark: The coverse of above theorem eed ot be true From the followig figures, L ( G) : ( is 08 edge regular fuzz graph But, G :( is ot a regular fuzz graph 510 Theorem: be regular fuzz graph o a eve ccle G : ( V, The L ( G) : ( is a edge regular fuzz graph Usig theorem 23, is a costat fuctio or alterate edges have same membership values I both cases, is a costat fuctio i LG ( ) Hece b theorem 21, L (G) is a edge regular

116 K Radha ad N Kumaravel 511 Remark: The coverse of above theorem eed ot be true From the figure 42, L ( G) : ( is 07 edge regular fuzz graph But, G :( is ot a regular fuzz graph 512 Theorem: be a fuzz graph o a ccle G : ( V, such that ( e) c, e E, where c is a costat The G :( is a 2 c edge regular fuzz graph ad L ( G) : ( is a 2 c edge regular fuzz graph Its proof is trivial 513 Remark: The coverse of above theorem eed ot be true From the followig figures, G :( is 10 edge regular fuzz graph ad L ( G) : ( is 08 edge regular fuzz graph, but is ot a costat fuctio 6 PROPERTIES OF EDGE REGULAR FUZZY LINE GRAPHS 61 Theorem: be a fuzz graph ad let L ( G) : ( be a fuzz lie graph of G :( The (i) S( G) O( L( G)) ad (ii) S( L( G)) ( ) ( ), where ad are adjacet, E (i) Sice ( ) ( ), S Z, ( ) ( ) S( G) O( L( G (ii) B the defiitio of lie graph, are adjacet )) E Z S( L( G)) ( ) ( ), where ad, E

O Edge Regular Fuzz Lie Graphs 117 62 Theorem: be a fuzz graph such that c, where c is a costat If G : ( V, be a k regular graph, the the size of L ( G) : ( is qc ( k 1), where q E Give G : ( V, is a k regular graph From the proof of theorem 24, G is a 2( k 1) edge regular, the umber of 1 edges i L( G ) : ( Z, = W q(2( k 1)) q( k 1) Therefore the size of 2 L ( G) : ( is qc ( k 1) 63 Theorem: be a fuzz graph such that ( e) c, e E, where c is a costat If G : ( V, be a k edge regular graph, the the size of L ( G) : ( is q E qck, where 2 Proof is similar to proof of the theorem 62 64 Remark: The above theorems 62 ad 63 are true, whe is a costat fuctio of costat value c REFERENCES [1] S Arumugam ad S Velammal, Edge Domiatio i Graphs, Taiwaese Joural of Mathematics, Volume 2, Number 2, 1998, 173 179 [2] J N Mordeso, Fuzz Lie Graphs, Patter Recogitio Letters, Volume 14, 1993, 381 384 [3] ANagoor Gai ad JMalarvizhi, Isomorphism o Fuzz Graphs, Iteratioal Joural of Computatioal ad Mathematical Scieces, Volume 2, Issue 4, 2008, 190 196 [4] ANagoorgai ad JMalarvizhi, Properties of μ-complemet of a Fuzz Graph, Iteratioal Joural of Algorithms, Computig ad Mathematics, Volume 2, Number 3, 2009, 73 83 [5] ANagoor Gai ad KRadha, O Regular Fuzz Graphs, Joural of Phsical Scieces, Volume 12, 2008, 33 40 [6] ANagoor Gai ad KRadha, Some Sequeces i Fuzz Graphs, Far East Joural of Applied Mathematics, Volume 31, Number 3, 2008, 321 335

118 K Radha ad N Kumaravel [7] A NagoorGai ad K Radha, Regular Propert of Fuzz Graphs, Bulleti of Pure ad Applied Scieces, Volume 27E, Number 2, 2008, 411 419 [8] K Radha ad N Kumaravel, O Edge Regular Fuzz Graphs, Iteratioal Joural of Mathematical Archive, Volume 5, Issue 9, 2014, 100 112 [9] K Radha ad N Kumaravel, Some Properties of Edge Regular Fuzz Graphs, Jamal Academic Research Joural (JARJ), Special issue, 2014, 121 127 [10] K Radha ad N Kumaravel, The Edge Degree ad The Edge Regular Properties of Trucatios of Fuzz Graphs, Bulleti of Mathematics ad Statistics Research (BOMSR), Volume 4, Issue 3, 2016, 7 16 [11] A Rosefeld, Fuzz graphs, i: LA Zadeh, KS Fu, K Taaka ad M Shimura, (editors), Fuzz sets ad their applicatios to cogitive ad decisio process, Academic press, New York, 1975, 77 95 [12] R T Yeh ad S Y Bag, Fuzz relatios, fuzz graphs, ad their applicatios to clusterig aalsis, i: LA Zadeh, KS Fu, K Taaka ad M Shimura, (editors), Fuzz sets ad their applicatios to cogitive ad decisio process, Academic press, New York, 1975, 125 149