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A Note on Fuzzy Vertex Graceful Labeling on Double Fan Graph and Double Wheel Graph

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A Note on Fuzzy Vertex Graceful Labeling on Double Fan Graph and Double Wheel Graph International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 3 (2017), pp. 729-736 © Research India Publications http://www.ripublication.com A Note on Fuzzy Vertex Graceful labeling on Double Fan Graph and Double Wheel Graph K.Ameenal Bibi 1 and M.Devi2 2,1 PG and Research Department of Mathematics, D.K.M College for Women (Autonomous), Vellore-632001, India. Abstract A labeling on a Graph G which can be gracefully numbered is called a graceful labeling. If a fuzzy graph G admits a graceful labeling and if all the vertex labelings are distinct then G is called a fuzzy vertex graceful graph. In this paper, we discussed fuzzy vertex graceful labeling on double Fan Graph and Double Wheel Graph. Keywords: Fuzzy labeling, Graceful labeling, Fuzzy graceful labeling, Fuzzy Double Fan graph, Fuzzy Double Wheel graph. AMS Mathematics Subject Classification: 05C, 94D 1. INTRODUCTION Fuzzy is a newly emerging mathematical framework to exempt the phenomenon of uncertainty in real life tribulations. It was introduced by Zadeh in 1965, and the concepts were pioneered by various independent researchers, namely Rosenfeld, Kauffmann, etc. A Fuzzy set is defined mathematically by assigning a value to each possible individual in the universe of discourse, representing its grade of membership which corresponds to the degree, to which that individual is similar or compatible with the concept represented the fuzzy set. Based on Zadeh’s Fuzzy relation the first definition of a fuzzy graph was introduced by Kauffmann in 1973. The concept of a graceful 730 K.Ameenal Bibi and M.Devi labeling has been introduced by Rosa in 1967.This note is a further contribution on fuzzy graceful labeling. Fuzzy labeling on Fan graph and Wheel graph are called a Fuzzy Fan graph and Fuzzy Wheel graph .[2,6] 2. PRELIMINARIES AND OBSERVATIONS Definition 2.1[2] A Fan graph Fm,n is defined as the join of two graphs KPm n where Km is the empty set on m vertices and Pn is the path graph on n vertices. The case m=1 Corresponds to the usual Fan graph while m=2 corresponds to the double Fan graph, etc. A Double Fan graph with fuzzy labeling is called a fuzzy Double Fan graph. Definition 2.2[2] A Wheel graph Wn is a graph with n vertices (n 4), formed by connecting a single vertex to all the vertices of an (n-1) cycle. A Double Wheel graph DWn of order n can be composed to 2CN+K1. (ie)., it consists of order N, where the vertices of the two cycles are all connected to a common center. A Double wheel graph with fuzzy labeling is called a fuzzy Double wheel graph. Definition 2.3 ~ In a fuzzy Double Fan graph( F ,2 n ) if all the vertices are distinct and graceful labeling exists, then it is called a fuzzy vertex graceful labeling Double Fan graph. * A fuzzy graceful Double Fan graph consists of two vertex sets (F,F ) and Fn with * * F ,1 F =1 and Fn >1 such that FF,i 0 and FF,i 0 where i=1 to n. Proposition 2.4 In a fuzzy Double Fan graph ( ), the vertex labeling :F [0,1] and :F * [0,1] which satisfy the conditions that if the values of F only starts from A Note on Fuzzy Vertex Graceful labeling on Double Fan Graph and Double… 731 n 1 n and F* starts from then the Double Fan graph is a fuzzy vertex graceful 10 10 labeling Double Fan graph. Definition 2.5 ~ In a fuzzy Double Wheel graph DWn if all the vertices are distinct and graceful labeling exists, then it is called a fuzzy vertex graceful labeling Double Wheel graph. Proposition 2.6 A Double wheel in a fuzzy graph consists of two vertex sets V and U with V ,1 and U 1, such that (v , ui ) 0 where i=1 to n and (ui , u i1 ) 0 where i=1 to n-1. 3. MAIN RESULTS: Theorem 3.1 Every Fuzzy Double Fan graph is a fuzzy vertex graceful Double Fan graph. Proof: * Consider the Double Fan graph F2,n when m=2. Here F and F are the central vertices (m=2) and F1,F2,F3,….,Fn are other vertices in the graph. * A Fuzzy Double Fan graph consists of two vertex sets(F,F ) and Fn with =1 and >1 such that and where i=1 to n and FF, 0 where i=1 to n-1. i i1 ~ F If the fuzzy Double Fan graph is ,2 nvertex graceful, then FFFFFF,(,)(,)n1 n n n1 Where n=1,2,…,etc. ------------------(1) * * FFFFFF,(,)(,)n1 n n1 n * * F ,1 F Fn FF,i 0 FF,i 0 * * (ie)., FFFFFF,(,)(,) 1 1 Where n=1,2,…,etc. ------------------(2) n n n n Also ()()(,)FFFFn n (or) * * ()()(,)FFFFn n . 732 K.Ameenal Bibi and M.Devi Examplen 1 3.2 Case(i)10 The value of F starts from Here, , 0.01, FF1 , 0.03 (,)(,) FF2 FFFF1 1 2 , 0.06 (,)(,) , 0.10 (,)(,) FF3 FFFF2 2 3 FF4 FFFF3 3 4 Therefore, , ( , ) (0.01)( 1) Where n=1,2,…,etc ------------(1) FFFFn1 n n Similarly, FF1, 2 0.02 FF2, 3 0.03 FF3, 4 0.04 and etc , (0.01)( 1) Where n=1,2,…,etc. --------------------------(2) FFn n1 n Using (2) in (1), we get ,(,)(,) Where n=1,2,…,etc. FFFFFFn1 n n n1 Also n The value of F* starts from 10 FF*, 0.20 (FF*) () 4 4 FF*, 3 0.16 FFFF *, 4 3, 4 FF*, 2 0.13 FFFF *, 3 2, 3 and so ()()(,)on.FFFFn Therefore n * * (FFFF ,n1 ) ( ,n ) (0.01)n ----------------- (1) FF1, 2 0.02, FF2, 3 0.03, and etc A Note on Fuzzy Vertex Graceful labeling on Double Fan Graph and Double… 733 , (0.01) Where n=1,2,…,etc. ------ (2) FFn1 n n Using (2) in (1), we get (ie). *, *, , FFFFFFn n1 n1 n ()()(,) * * FFFFn n Case (ii) n 1 The value of F starts from 100 Here, , 0.001, , 0.003 (,)(,) and so on. FF1 FF2 FFFF1 1 2 Therefore, FFFF, ( , ) (0.001)(n )1 Where n=1,2,…,etc n1 n Similarly, FF1, 2 0.002 FF2, 3 0.003 and etc , (0.001)( )1 Where n=1,2,…,etc. FFn n1 n ,(,)(,) Where n=1,2,…,etc. FFFFFFn1 n n n1 n The value of F* starts from 100 FF*, 4 .0 20 (FF*) ()4 FF*, 3 .0 016 FFFF *, 4 3, 4 FFi, i1 0 and so on. Therefore FFFF*, n *, n1 .0( 001)n FF1, 2 0.002 FF2, 3 0.003and etc FF, 0 FF* , 0 , (0.001) Where n=1,2,…,etc. i i FFn1 n n Since all the membership value of edges and where i=1 to n and where i=1 to n-1 and (,)()()u v u v . Since all the membership values of the vertices are distinct, this graph is a fuzzy vertex graceful labeling Double Fan graph. 734 K.Ameenal Bibi and M.Devi Theorem 3.3 For some n 4, the Double Wheel graph DWn is a fuzzy vertex graceful wheel graph, where the central vertex :V [0,1]and the vertices from the inner and outer cycles are :wi [0,1] and (v , wi ) 0.001 i for inner cycle and (v , wi ) 0.001 ()i n for outer cycle where i=1 to n. Proof: A Double Wheel graph DWn is a graph with n vertices exists only if n 4. A Wheel in a fuzzy graph consists of two node sets V and W with V 1and W 1 such that (v , wi ) 0where i=1 to n and (wi , w i1 ) 0 where i=1 to n-1. In the Double Wheel graph, V is the central vertex, wi denotes the vertices in the inner and outer cycles. Here and : [0,1] wi . ()()(,)wi v v wi where (v , wi ) 0.001 i for inner cycle and ()()(,)wi v v wi where (v , wi ) 0.001 ()i n for outer cycle where i=1 to n and (,)()()wi w i1 vwi vwi1 where i=1 to n-1 (or) (,)()()wn1 w n vwn1 vwn while (wn , w1 ) (vwn ) ( vw1 ) (0.001)(n 2) Example: 3.4 Case (i): n 1 When ()v starts from 100 Here, when and : [0,1] wi . For Inner cycle ( ) ( ) 0.001 ( ) ( ) 0.002 w1 v w2 v ( ) ( ) 0.003 ( ) ( ) 0.004 and etc w3 v w4 v (ie)., ()()(,) where ( , ) 0.001 , i=1 to n wi v v wi v wi i A Note on Fuzzy Vertex Graceful labeling on Double Fan Graph and Double… 735 Also ( , ) ( ) ( ) 0.001, w1 w 2 vw1 vw2 :V [0,1] (w2 , w 3 ) (vw2 ) ( vw3 ) 0.001, (w3 , w 4 ) (vw3 ) ( vw4 ) 0.001and etc (,)()() where i=1 to n-1 (or) wi w i1 vwi vwi1 (,)()() while ( , ) ( ) ( ) (0.001)( 2) wn1 w n vwn1 vwn wn w1 vwn vw1 n For outer cycle (w ) ( v ) (0.001)( i n) i ()()(,)wi v v wi where (v , wi ) 0.001 ()i n where i=1 to n (ie)., ( ) ( ) 0.005, ( ) ( ) 0.006, ( ) ( ) 0.007 w1 v w2 v w3 v and etc Case (ii) n 1 When starts from 1000 Here, when and : [0,1] wi . For Inner cycle (w1 ) ( v ) 0.0001, (w2 ) ( v ) 0.0002 and etc (ie).,()()(,)wi v v wi where (v , wi ) 0.0001 i , i=1 to n Also (,)()()wi w i1 vwi vwi1 where i=1 to n-1 (or) ()v (,)()()wn1 w n vwn1 vwn While (wn , w1 ) (vwn ) ( vw1 ) (0.0001)(n 2) For outer cycle (w ) ( v ) (0.0001)(i n) ()()(,)w v v w i i i where (v , wi ) 0.0001 ()i n where i=1 to n In the Similar way, for any value of the labeling of all vertices 736 K.Ameenal Bibi and M.Devi in the inner and outer cycles wi are distinct where i=1 to n.
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