International Journal of Multidisciplinary Research and Modern Education (IJMRME) ISSN (Online): (

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1 ISSN (Onlne): ( Volume II, Issue II, 06 BALANCED HESITANCY FUZZY GRAPHS J. Jon Arockara* & T. Pathnathan** * P.G & Research Department of Mathematcs, St. Joseph s College of Arts and Scence, Cuddalore, Tamlnadu ** P.G & Research Department of Mathematcs, Loyola College, Chenna, Tamlnadu Abstract: In ths paper, a new Hestancy Fuzzy Graph Model called Balanced Hestancy Fuzzy Graphs (BHFGs) s ntroduced. Also, we dscuss the arous concepts related wth Balanced Hestancy Fuzzy Graphs wth ther graphcal representatons. Further we deelop the concept of self complementary hestancy fuzzy graphs along wth the theoretcal llustraton. Key Words: Hestancy Fuzzy Graphs, Hestancy Subgraphs, Self Complementary Hestancy Fuzzy Graph, Densty of Hestancy Fuzzy Graphs & Balanced Hestancy Fuzzy Graphs.. Introducton: Fuzzy Set Theory has ts root n the eastern logc. It manly deals wth the ambguty prealng n the system. Lotf A. Zadeh [4] n the year 965 deeloped the theoretcal framework of fuzzy set theory n order to capture the agueness that has occurred n the real lfe crcumstances. Then many deas/concepts [7] hae been deeloped and researchers throughout the world are comng out wth numerous results by ncorporatng fuzzy set theoretcal concepts n ther experments. Seeral hypotheses hae been framed and suggestons are dered wth the help of fuzzy decson makng tools [8] to mproe the decsons made by the decson makers, planners and other authortes. Also arous concepts of fuzzy set theory hae been successfully appled n other areas whch nclude Medcne, Engneerng, Economcs, Robotcs, Socal studes and so on. Fuzzy Graph s one such concept that was frst ntroduced by A. Rosenfeld [9] n the year 975 and has been much useful n the feld of operatons research, system analyss, automata theory, sgnal processng and so on. Importantly, J. N. Mordeson [8], M. S. Suntha [] [] and A. Nagoor Gan [5] defned other maor concepts n the fuzzy graph theory. T. Pathnathan and J. Jesntha Roslne ntroduced two new fuzzy graphs namely Double Layered Fuzzy Graphs (DLFGs) [5] [9] [0] [6] [7] and extensely studed ther mportant propertes wth applcaton. T. Pathnathan and J. Jon Arockara [] ntroduced a new fuzzy graph called Hestancy Fuzzy Graphs and dscussed ther arous theoretcal propertes and aldatons. In addton to ths, the concept of regularty [5], constant [3], ndex matrx representaton [4] and arous Cartesan products [] were also dered. Ths artcle presents the concepts of balanced extenson of hestancy fuzzy graphs and self complementary hestancy fuzzy graphs wth the help of theoretcal llustratons. Erdos and Reny [3] frst studed the balanced extenson on random graphs [4] n order to deal wth the complex networks. Complex networks n the sense wth more connectons and dmensons, the network become ague and t complcates the stuaton. To oercome such stuaton, Balanced Fuzzy Graphs (BFGs) s deeloped. T.AL-Hawary [] ntroduced the concept of Balanced Fuzzy Graphs (BFGs) and further extensons hae been made by Mohammed Akram and M. G. Karunambga [6]. Karunambga and others [6] defned the concepts of densty and balanced notaton for an Intutonstc Fuzzy Graphs. 46

2 ISSN (Onlne): ( Volume II, Issue II, 06 The artcle s organzed as follows. Secton focuses on the basc concepts and notatons of Fuzzy Graphs (FGs), Hestancy Fuzzy Graphs (HFGs). Secton 3 ntroduces the concept of Balanced Hestancy Fuzzy Graphs (BHFGs) along wth the llustraton. Secton 4 ges the theoretcal aldatons and proofs for the newly ntroduced Self Complementary Hestancy Fuzzy graphs (SCHFGs), whch followed by concluson n secton 5.. Basc Defntons and Termnologes: Ths secton contans some basc defntons and examples on Hestancy Fuzzy Graphs and ts related topcs. Defnton.: Fuzzy Graph (FG) Let V be a non empty set. A fuzzy graph s a par of functons G(, ) where s a fuzzy subset of V, s a symmetrc fuzzy relaton on. : V [0,] : VV [0,] such that (, ) ( ) ( ), V. The underlyng crsp graph of the fuzzy graph G(, ) s denoted as * * * * G : (, ) where s referred to as the nonempty set V of nodes and * E V V. The crsp graph (V, E) s a specal case of the fuzzy graph G wth each ertex and edge of (V,E) hang degree of membershp. Defnton.: Partal Fuzzy Subgraph: A fuzzy graph H :( V ', E ') wth ': V ' [0,] and ': V ' V ' [0,] s sad to an partal fuzzy subgraph of G : ( V, E ) f, () V V, where, for all V,,,3,..., n. () E E, where, for all, E,,,,3,..., n. Defnton.3: Fuzzy Subgraph: In Partcular, a partal fuzzy subgraph H :( V ', E ') s sad to be a fuzzy subgraph of G : ( V, E) f, () V V, where, for all V,,,3,..., n. () E E, where, for all, E,,,,3,..., n. Defnton.4: Densty of a Fuzzy Graph []: The densty of a fuzzy graph G : ( V, E ) wth : V [0,] and : V V [0,] s defned as,,, V DG ( )., V Defnton.5: Balanced Fuzzy Graph []: A fuzzy Graph G( V, E ) s sad to be balanced fuzzy graph, f D( H) D( G) for all fuzzy non-empty subgraphs H :( V ', E ') of G( V, E ). Defnton.6: Strctly Balanced Fuzzy Graph []: A fuzzy graph G ( V, E) s strctly balanced fuzzy graph f DH DG G( V, E ). for all non-empty subgraphs H :( V ', E ') of 463

3 ISSN (Onlne): ( Volume II, Issue II, 06 Defnton.7: Hestancy Fuzzy Graph (HFG) []: A Hestancy Fuzzy Graph s of the form G = (V,E), where () V {,,... n } such that : V [0,], : V [0,]and : V [0,] denote the degree of membershp, non-membershp and hestancy of the element V, Where ( ) ( ) ( ) respectely and ( ) ( ) ( ) V 0 ( ) ( ) () and () E V V where : V V [0,], : V V [0,]and : V V [0,] are such that, (, ) mn( ( ), ( )) () (, ) max( ( ), ( )) (3) (, ) mn( ( ), ( )) (4) And 0 (, ) (, ) (, ) (, ) E (5) Notatons:,,, denotes the ertex, degree of membershp, non-membershp and hestancy of the ertex. e,,, denotes the edge, degree of membershp, non-membershp and hestancy of the edge relaton e (, ) on V..8 Defnton: Complete Hestancy Fuzzy Graph [] A HFG, G = (V,E) s sad to be a complete HFG f, (, ) mn( ( ), ( )) (, ) max( ( ), ( )) (, ) mn( ( ), ( )).9 Defnton: Complement of Hestancy Fuzzy Graph [] Let G = (V,E) be an HFG, then the complement of the HFG s a HFG, G(V, E) where V V,(. e.,) ; ; and mn(, ), mn(, ) and mn(, ), V 464

4 ISSN (Onlne): ( Volume II, Issue II, Balanced Hestancy Fuzzy Graphs Ths secton presents the concept of subgraph, densty and balanced notaton n terms of Hestancy Fuzzy Graphs []. Also ths secton prodes the geometrcal representaton of the aboe mentoned concepts wth theoretcal llustraton. 3. Defnton: Partal Hestancy Fuzzy Subgraph An FG H ( V, E) s sad to be a partal hestancy fuzzy subgraph of G ( V, E) f () V V, where,, for all V,,,3,..., n. () E E, where,, E,,,,3,..., n. for all 3. Defnton: Hestancy Fuzzy Subgraph An FG H ( V, E) s sad to be a hestancy fuzzy subgraph of G ( V, E) f () V V, where,, for all V,,,3,..., n. () E E, where,, for all, E,,,,3,..., n. 3.3 Defnton: Densty of Hestancy Fuzzy Graph The densty of an ntutonstc fuzzy graph G ( V, E) s D( G) D G, D G, D G, where, D G s defned by D ( G) D G s defned by D ( G) D G s defned by D ( G) Otherwse, D( G) D G, D G, D G, V, E, V, E, E u, V,,,, for,, for,, for,,,,, V, V u, V DG ( ),,, E, E, E 3.4: Defnton: Balanced Hestancy Fuzzy Graph (BHFG) A hestancy fuzzy graph G ( V, E) D H D G, that s,, D H D G and D H D G D H D G 3.5 Illustraton: Balanced Hestancy Fuzzy Graph Let (, ) V V V s balanced f for all subgraphs H of G. G V E s a hestancy fuzzy graph wth the ertex set V V, V, V3, V4 by V,, : V [0,]; : V [0,]; : V [0,] where gen denotes the degree of membershp, non-membershp and hestancy degree. Then edge relaton between them s gen by 465

5 ISSN (Onlne): ( Volume II, Issue II, 06 V, V, V, V3, V, V4, V, V3, V, V4, V 3, V4, V V V, V, V3, V, V, V4, V, V3, V4, V, V3, V4, V, V, V3, V4 wth V V,, where : VV [0,]; : VV [0,]; : VV [0,] denotes the edge relaton of membershp, non-membershp and hestancy degree. The hestancy fuzzy graph s fgured as follows: (0.6, 0.3, 0.)V (0.5, 0.3, 0.05) (0., 0.5, 0.3)V (0.5, 0.3, 0.05) (0.5, 0.3, 0.05) V 4 (0.3, 0.5, 0.) (0.5, 0.3, 0.05) V 3 (0.4, 0.5, 0.) Fgure : Balanced Hestancy Fuzzy Graph Calculaton of densty alues for core Hestancy Fuzzy Graph G (V,E): Membershp Densty D G Non-membershp Densty D G Hestancy Densty D G Calculaton of densty alues for Hestancy Fuzzy Subgraph (H): {V, V4} (0.5, 0.3, 0.05) (0.6, 0.3, 0.)V V 4 (0.3, 0.5, 0.) Fgure : Hestancy Fuzzy Subgraph H 0.5 Membershp Densty D H Non-membershp Densty D H Hestancy Densty D H Smlarly, the densty alues of the other subgraphs are found by usng the defnton (Defnton 3.3) and t s shown n the below table (Table ). Also from the fgure (Fgure ) t s noted that the subgraphs {V, V3} and {V, V4} does not hae an edge. Therefore the densty alue s (0,0,0). Table : Densty alues of Balanced Hestancy Fuzzy Graph Not aton Subgraph (0.5, 0.3, 0.05) H (0.6, 0.3, 0.)V V 4 (0.3, 0.5, 0.) Vertex Set Dμ G,Dγ G,Dβ G {V,V 4} (.5,.,0.3) H (0.6, 0.3, 0.)V V 3 (0.4, 0.5, 0.) {V,V 3} (0,0,0) H 3 (0.6, 0.3, 0.)V (0.5, 0.3, 0.05) {V,V } (.5,.,0.3) (0., 0.5, 0.3)V 466

6 ISSN (Onlne): ( Volume II, Issue II, 06 H 4 V 4 (0.3, 0.5, 0.) (0.5, 0.3, 0.05) {V,V 3} (.5,.,0.3) V 3 (0.4, 0.5, 0.) H 5 (0., 0.5, 0.3)V4 V (0.3, 0.5, 0.) {V,V 4} (0,0,0) H 6 (0.5, 0.3, 0.05) V 3 (0.4, 0.5, 0.) (0., 0.5, 0.3)V {V,V 3} (.5,.,0.3) H 7 (0.6, 0.3, 0.)V (0.5, 0.3, 0.05) (0.5, 0.3, 0.05) V 4 (0.3, 0.5, 0.) {V,V,V 4} (.5,.,0.3) (0., 0.5, 0.3)V H 8 (0.6, 0.3, 0.)V (0.5, 0.3, 0.05) (0., 0.5, 0.3)V V 3 (0.4, 0.5, 0.) {V,V,V 3} (.5,.,0.3) (0.5, 0.3, 0.05) H 9 (0.6, 0.3, 0.)V (0.5, 0.3, 0.05) V 4 (0.3, 0.5, 0.) (0.5, 0.3, 0.05) {V,V 3,V 4} (.5,.,0.3) V 3 (0.4, 0.5, 0.) H 0 V 4 (0.3, 0.5, 0.) (0.5, 0.3, 0.05) {V,V 3,V 4} (.5,.,0.3) (0., 0.5, 0.3)V (0.5, 0.3, 0.05) V 3 (0.4, 0.5, 0.) H (0.6, 0.3, 0.)V (0.5, 0.3, 0.05) (0., 0.5, 0.3)V (0.5, 0.3, 0.05) (0.5, 0.3, 0.05) V 4 (0.3, 0.5, 0.) (0.5, 0.3, 0.05) {V,V,V 3,V 4} (.5,.,0.3) V 3 (0.4, 0.5, 0.) From the aboe table (Table ), the densty alues of all the subgraphs are less than or equal to the densty alues of the core graph. Therefore the graph (Fgure ) s sad to be a Balanced Hestancy Fuzzy Graph. 3.6 Defnton: Strctly Balanced Hestancy Fuzzy Graph A hestancy fuzzy graph G ( V, E) s sad to be strctly balanced f DG, that s, D H D G, D H D G and D H D G D H subgraphs H of G. 3.7 Illustraton: Strctly Balanced Hestancy Fuzzy Graph Let (, ) G V E s a hestancy fuzzy graph wth the ertex set V V, V, V3 for all gen by V,, and the edge relaton between them s gen by wth VV,,. For the followng hestancy fuzzy graph (Fgure 3), the densty alues are tabulated (Table ) as follows: (0.6, 0.3, 0.)V (0.5, 0.3, 0.05) (0.3, 0.3, 0.05) (0., 0.5, 0.3)V (0.5, 0.3, 0.05) V 3 (0.4, 0.5, 0.) G Fgure 3: Strctly Balanced Hestancy Fuzzy graph 467

7 ISSN (Onlne): ( Volume II, Issue II, 06 Table : Densty alues of Strctly Balanced Hestancy Fuzzy Graphs Notaton Subgraph Dμ G,Dγ G,Dβ G H {V,V } (.5,.,0.3) H {V,V 3} (.5,.,0.3) H 3 {V,V 3} (.5,.,0.3) H 4 {V,V,V 3} (.5,.,0.3) From the aboe table (Table ), the densty alues of all the subgraphs are equal to the densty alues of the core graph (Fgure 3). Therefore the graph (Fgure 3) s sad to be a Strctly Balanced Hestancy Fuzzy Graph. Based on the concept of completeness, the balanced noton of the hestancy fuzzy graphs s proed by the followng theorem. Also llustraton followed by the theorem ges the aldaton for ts proof. 3.8 Theorem: Eery complete hestancy fuzzy graph s balanced. Proof: Let G ( V, E) be a complete HFG, then by the defnton of complete HFG, we hae (, ) mn( ( ), ( )) ; (, ) max( ( ), ( )) ; (, ) mn( ( ), ( )) for eery, V. Now, the densty of a hestancy fuzzy graph (Defnton 3.3) s defned as follows;,,,, V, V, V DG,, (, ) E (, ) E (, ) E From the defnton of complete hestancy fuzzy graph (Defnton.8) the aboe equaton s wrtten as follows;, V, V, V DG,, (, ) E (, ) E (, ) E DG,, Let H be a non empty subgraphs of G. In smlar way, D( H) (,,) H G. Thus, D( G) D( H) (,,) Thus G s balanced. 3.8 Illustraton: Eery complete hestancy fuzzy graph s balanced Let G( V, E) s a complete hestancy fuzzy graph wth the ertex set V V, V, V, V gen by V,, 3 4 where : V [0,]; : V [0,]; : V [0,] denotes the degree of membershp, non-membershp and hestancy degree. Then edge relaton between them s gen by VV,, where : VV [0,]; : VV [0,]; wth : VV [0,] denotes the edge relaton of membershp, non-membershp and hestancy degree. 468

8 ISSN (Onlne): ( Volume II, Issue II, 06 (0.7, 0., 0.)V (0.5, 0., 0.) (0.4, 0., 0.) V (0.5, 0., 0.) (0.4, 0., 0.) (0.4, 0., 0.)V 4 (0.4, 0., 0.) (0.4, 0., 0.) (0.4, 0., 0.) V 3 (0.4, 0., 0.3) Fgure 4: Complete Hestancy Balanced Fuzzy Graph Calculaton of densty alues for core Hestancy Fuzzy Graph G (V,E): By usng the defnton (Defnton 3.3), the densty alues of the aboe graph are calculated. The densty alues of the graph G(V,E) s gen by; D( G) D G, D G, D G DG ( ),, The densty alues for the subgraphs are gen n the below table (Table 3). And t s noted that, the densty alues of all the subgraphs has the densty alue as (,,). Therefore t proes the aboe theorem and also the graph s sad to be a complete balanced hestancy fuzzy graph. Table 3: Densty Values of Complete Hestancy Balanced Fuzzy Graphs Nota ton Subgraph (0.5, 0., 0.) H (0.7, 0., 0.)V V (0.5, 0., 0.) Vertex Set Dμ G,Dγ G,Dβ G {V,V } (,,) H (0.7, 0., 0.)V (0.4, 0., 0.) {V,V 3} (,,) V 3 (0.4, 0., 0.3) H 3 (0.7, 0., 0.)V (0.4, 0., 0.) {V,V 4} (,,) (0.4, 0., 0.)V 4 V (0.5, 0., 0.) H 4 (0.4, 0., 0.) V 3 (0.4, 0., 0.3) {V,V 3} (,,) H 5 (0.4, 0., 0.)V 4 (0.4, 0., 0.) V (0.5, 0., 0.) {V,V 4} (,,) H 6 (0.4, 0., 0.)V 4 V 3 (0.4, 0., 0.3) (0.4, 0., 0.) {V 3,V 4} (,,) (0.7, 0., 0.)V (0.5, 0., 0.) V (0.5, 0., 0.) H 7 (0.4, 0., 0.) {V,V,V 3} (,,) V 3 (0.4, 0., 0.3) H 8 (0.7, 0., 0.)V (0.4, 0., 0.)V 4 (0.5, 0., 0.) (0.4, 0., 0.) V (0.5, 0., 0.) {V,V,V 4} (,,) 469

9 ISSN (Onlne): ( Volume II, Issue II, 06 H 9 (0.7, 0., 0.)V (0.4, 0., 0.)V 4 (0.4, 0., 0.) (0.4, 0., 0.) V 3 (0.4, 0., 0.3) {V,V 3,V 4} (,,) V (0.5, 0., 0.) H 0 (0.4, 0., 0.)V 4 (0.4, 0., 0.) (0.4, 0., 0.) {V,V 3,V 4} (,,) V 3 (0.4, 0., 0.3) (0.7, 0., 0.)V (0.5, 0., 0.) (0.4, 0., 0.) V (0.5, 0., 0.) (0.4, 0., 0.) H (0.4, 0., 0.)V 4 (0.4, 0., 0.) (0.4, 0., 0.) (0.4, 0., 0.) V 3 (0.4, 0., 0.3) {V,V,V 3,V 4} (,,) 4. Self-Complmentary Hestancy Fuzzy Graph: 4. Defnton: Self-Complmentary Hestancy Fuzzy Graph: A hestancy fuzzy graph G s self complementary fg G. 4. Proposton: Let G (V,E) wth : V [0,] and : V V [0,] be a self-complmentary fuzzy graph,. Then we hae,, mn,, V, V, max,, V, V, mn,, V, V Proof: Let G (V,E) be a self complmentary fuzzy graph, then we hae V V.e., ; ; and mn, mn, mn, mn, Smlarly; max, mn, 4.3 Illustraton: Self complmentary Hestancy Fuzzy Graph: V Let G (V,E) be a hestancy fuzzy graph wth V, V, V3,, gen by V and the edge relaton between them s gen by wth VV,,. Then G (V,E) s defned by usng the defnton (Defnton.9): 470

10 ISSN (Onlne): ( Volume II, Issue II, 06 (0.8, 0., 0.)V (0.8, 0., 0.)V (0.35, 0, 0) (0.4, 0., 0) (0.35, 0, 0) (0.4, 0., 0) (0.7, 0., 0)V (0.35, 0., 0) V 3 (0.8, 0., 0) (0.7, 0., 0)V (0.35, 0., 0) V 3 (0.8, 0., 0) G G Fgure 5: Self Complementary Hestancy Fuzzy Graph By usng the aboe proposton, we hae u, u u, V u, E = ( ) = u, u u, V u, E = ( ) 0.3 = 0.3 u, u u, V u, E = (0+0+0) That s G G. Therefore G s self complementary. 4.4 Theorem: Eery self-complmentary hestancy fuzzy graph has densty equal to. Proof: Let G(V,E) be the self-complmentary hestancy fuzzy graph. Then we hae, Also we hae, V, E,, V, E,, V, E D ( G) D ( G), V, V, V, V,, 47

11 ISSN (Onlne): ( Volume II, Issue II, 06,, V D ( G)., V The densty of a hestancy fuzzy graph s gen by,,,,, V, V, V D G, D G, D G,,, V, V, V,,,, V, V, V,,,,,, V, V, V,,,, Hence proed 4.5 Illustraton: Eery Self-Complmentary Hestancy Fuzzy Graph Has Densty Equal To For the graph defned below, the densty alues are calculated usng the defnton (Defnton 3.3). The alues n the below table (Table 4) shows the graph s self complementary hestancy fuzzy graph wth the densty alues be (,,). (0.8, 0., 0.)V (0.05, 0., 0.05) (0., 0., 0.05) V (0., 0.4, 0.) (0., 0., 0.05) (0., 0.4, 0.)V 4 (0.05, 0., 0.05) (0., 0., 0.05) (0.05, 0., 0.05) V 3 (0., 0.4, 0.) Fgure 6: Self-Complementary Hestancy Fuzzy Graph wth Densty (,,) Table 4: Self Complementary Balanced Hestancy Fuzzy Graph wth Densty Value Subgraph Vertex Set Dμ G,Dγ G,Dβ G Nota ton (0.05, 0., 0.05) H (0.8, 0., 0.)V V (0., 0.4, 0.) {V,V } (,,) H (0.8, 0., 0.)V (0., 0., 0.05) {V,V 3} (,,) V 3 (0., 0.4, 0.) (0.8, 0., 0.)V H 3 (0., 0., 0.05) {V,V 4} (,,) (0., 0.4, 0.)V 4 H 4 V (0., 0.4, 0.) (0.05, 0., 0.05) {V,V 3} (,,) V 3 (0., 0.4, 0.) 47

12 ISSN (Onlne): ( Volume II, Issue II, 06 H 5 V (0., 0.4, 0.) (0., 0.4, 0.)V 4 (0.05, 0., 0.05) {V,V 4} (,,) H 6 (0., 0.4, 0.)V 4 V 3 (0., 0.4, 0.) (0., 0., 0.05) {V 3,V 4} (,,) H 7 (0.8, 0., 0.)V (0.05, 0., 0.05) V (0., 0.4, 0.) (0.05, 0., 0.05) {V,V,V 3} (,,) V 3 (0.7, 0.3, 0.) H 8 (0.8, 0., 0.)V (0.05, 0., 0.05) (0., 0.4, 0.)V 4 (0.05, 0., 0.05) V (0., 0.4, 0.) {V,V,V 4} (,,) H 9 (0.8, 0., 0.)V (0., 0.4, 0.)V 4 (0., 0., 0.05) (0., 0., 0.05) V 3 (0., 0.4, 0.) {V,V 3,V 4} (,,) V (0., 0.4, 0.) H 0 (0.05, 0., 0.05) {V,V 3,V 4} (,,) (0., 0.4, 0.)V 4 (0., 0., 0.05) V 3 (0.7, 0.3, 0.) (0.8, 0., 0.)V (0.05, 0., 0.05) (0., 0., 0.05) V (0., 0.4, 0.) H (0., 0., 0.05) (0., 0.4, 0.)V 4 (0.05, 0., 0.05) (0., 0., 0.05) (0.05, 0., 0.05) V 3 (0., 0.4, 0.) {V,V,V 3,V 4} (,,) 5. Concluson: Through ths paper, the Balanced Hestancy Fuzzy graphs (BHFGs) s ntroduced and studed brefly wth the sgnfcant theoretcal results. The concept of Hestancy Fuzzy Subgraph, Partal Hestancy Fuzzy Subgraph and balanced extenson of hestancy fuzzy graphs are deeloped. In addton to that, Self Complementary Hestancy Fuzzy Graphs (SCHFGs) s dscussed and the specal case where t has densty (,, ) s theoretcally proed and erfed wth the proper llustraton. 6. References:. AL-Hawary, T., Complete Fuzzy Graphs, Internatonal Journal of Math. Combn., 0, 4(): Atanasso, K., Intutonstc Fuzzy Sets, Fuzzy Sets and Systems, Elseer Scence Publcatons, North-Holland, 986, 0: Erdos, P., and Reny, A., On the eoluton of random graphs, Publ. Math. Inst. Hung. Acad. Sc, 960, 5: Glbert, E, N., Random Graphs, Annals of Mathematcal Statstcs, 959, 30: Jesntha Roslne, J., and Pathnathan, T., Intutonstc Double Layered Fuzzy Graph and ts Cartesan Product Vertex Degree, Internatonal Conference on Computng and ntellgence Systems, 05, 4:

13 ISSN (Onlne): ( Volume II, Issue II, Karunambga, M. G., Akram, M., Sasankar, S., and Palael, K., Balanced Intutonstc Fuzzy Graphs, Appled Mathematcal Scences, 03, 7(5): Kerre, E. E., and Mordeson, J. N., A Hstorcal Oerew of Fuzzy Mathematcs, New Mathematcs and Natural Computaton, World Scentfc Publshng Company, 005, (): Mordeson, J. N and Nar, P.S., Fuzzy Graphs and Fuzzy Hypergraphs, Physca Verlag Publcatons, Hedelbserg, Second edton, Pathnathan, T, and Jesntha Roslne, J, Intutonstc Double Layered Fuzzy Graph, ARPN Journal of Engneerng and Appled Scences, 05, 0(): Pathnathan, T., and Jesntha Roslne, J., Double Layered Fuzzy Graph, Annals of Pure and Appled Mathematcs, 04, 8(): Pathnathan, T., and Jon Arockara, J, Varous Cartesan Products of ertex degree and edge degree n Hestancy Fuzzy Graphs, Internatonal Journal of Multdscplnary Research and Modern Educaton (IJMRME), 06, (): Pathnathan, T., and Jon Arockara, J., and Jesntha Roslne, J., Hestancy Fuzzy Graphs, Indan Journal of Scence and Technology, 05, 8(35): Pathnathan, T., and Jon Arockara, J., and Mke Dson, E., Constant Hestancy Fuzzy Graphs, Global Journal of Pure and Appled Mathematcs (GJPAM), 06, (): Pathnathan, T., and Jon Arockara, J., Index matrx representaton and arous operatons on Hestancy Fuzzy graphs, (accepted) 5. Pathnathan, T., and Jon Arockara, J., On regular hestancy fuzzy graphs Global Journal of Pure and Appled Mathematcs (GJPAM), 06, (3): Pathnathan. T, and Jesntha Roslne. J, Intutonstc Double Layered Fuzzy Graph, ARPN Journal of Engneerng and Appled Scences, 05, 0(): Pathnathan. T and Jesntha Roslne. J., (04)., Double Layered Fuzzy Graph, Annals of Pure and Appled Mathematcs, Vol. 8, No., pp Ra Kumar, and Pathnathan, T., Seng out the Poor usng Fuzzy Decson Makng Tools, Indan Journal of Scence and Technology, 05, 8 (): Rosenfeld, A, Fuzzy Graphs, In Fuzzy Sets and ther Applcatons to Cognte and Decson Processes, Zadeh. L.A., Fu, K.S., Shmura, M., Eds; Academc press, New York, 975: Rucnsk, A., and Vnce, A., Balanced Extensons of Graphs and Hypergraphs, Combnatora, Akadema Kado Sprnger Verlag, 988, 8(3): Suntha, M. S., and Mathew, S., Fuzzy Graph Theory: A Surey, Annals of Pure and Appled Mathematcs, 03, 4(): Suntha, M.S., and Vaya Kumar, A., Complement of a Fuzzy Graph, Indan Journal of Pure and Appled Mathematcs, 00, 33(9): Torra. V, Hestant fuzzy sets, Internatonal Journal of Intellgent Systems, 00, 5(6): Zadeh. L. A, Fuzzy sets, Informaton and Control, 965, 8: