e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:02/February-2021 Impact Factor- 5.354 www.irjmets.com FACE MAGIC LABELING OF N NUMBER OF VERTICES OF GRAPHS Motcharakkini D *1, Stanis Arul Mary A*2 *1PG Scholar,Department of Mathematics, , Coimbatore, Tamil Nadu, India *2Assistant professor, Department of Mathematics, Nirmala college for women, Coimbatore, Tamil Nadu, India. ABSTRACT The study of graphs is called graph theory. In this dissertation, we have investigate face magic labeling of some graphs based on the magic labeling. Here we have discussed about the face magic labeling of some graphs. Keywords: Para chain Hexagon, Friendship graph, Ladder graph, Prism graph. I. INTRODUCTION The active area and one of the fast developing area in graph theory is labeling. Graph labeling is where the vertices are assigned some values subject to certain conditions. Labeling of vertices and edges play a vital role in graph theory[1]. Recently the concept of face magic labeling was introduced and many research articles are being published in this topic[7]. Labeling is the process of assigning integers to graph elements under some constraint. we have established the face magic labeling of Para chain Hexagon, Friendship graph, Ladder graph and Prism graph in this paper. II. PRELIMINARIES Definition:2.1 [1] If the weights of all face values are equal, then the graph is called face magic graph Definition:2.2 [1] A Cactus Graph G is a connected graph in which no edge lies in more than one cycle. Consequently each block of a cactus graph is either an edge or a cycle. If all blocks of G are cycle of the same size the cactus is uniform. Definition:2.3 [1] If G has at most two cut vertices and each cut vertices is shared exactly two triangles, then we can call that G is a chain triangular cactus. So here we replacing triangles in this definition by length of the cycle is 6. Therefore here we obtain cactus whose every block is C6 and We can call this cactu as hexagon cactus. If the cut vertices are adjacent then such a hexagon is an Ortho hexagon. Definition:2.4 [4] The Friendship graph (or Dutch windmill graph or n-fan) Fn is a planar undirected graph with 2n+1 vertices and 3n edges. The friendship graph Fn can be constructed by joining n copies of the cycle graph C3 with a common vertex. Definition:2.5 The ladder graph Ln is a planar undirected graph with 2n vertices and 3n-2 edges. Definition:2.6 A prism is also called by the name of a circular ladder graph .That is a graph corresponding to the skeleton of an prism. Therefore Prism graphs are planar and polyhedral. III. FACE MAGIC LABELING OF SOME GRAPHS Theorem : 3.1 Let Hn be an Para chain Hexagon graph for n ≥ 4, then Hn is face magic. Proof: Let G = Hn be a para chain Hexagon, www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1068] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:02/February-2021 Impact Factor- 5.354 www.irjmets.com The vertex set of G is represented as V(G) {Vij / 1 i 3, 1 j n} The labeling of G can be labeled as j / i 1 j is odd 2n -1- j 1 j n , i 1 j is even L(vi,j ) 3n -1- j 1 j n , i 2 3n - 2 j 1 j n , i 3 j is even th Let W j be the weight of the j face of G containing the vertices v1, j ,v1, j1,v2, j,v3, jv3, j1 . we have to prove the weight of the labels assigned to the vertices in each face of G are equal. It is enough to prove Wj is true for any 3 continuous values of j-1, j, j+1. Let W be some of the labels formed by the six vertices. j W L(v ) L(v ) L(v ) L(v ) L(v ) L(v ) j 1, j 1, j1 2, j 2, j1 3, j 3, j1 L(v1, j ) { j / i 1 and j is odd L(v1, j1 ) {2n 1 j / i 1 and j is even L(v2,j ) {3n -1- j / i 2 L(v3,j ) {3n 2 j / i 3 and j is odd L(v3,j1 ) {3n 1 j / i 3 and j is even th Let Wj be the weight of (j-1) face value of G Wj-1 L(v1, j1 ) L(v1,j-1 ) L(v2,j-1 ) L(v2,j-1 ) L(v3,j-1 ) L(v3, j1 ) www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1069] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:02/February-2021 Impact Factor- 5.354 www.irjmets.com j 2n 1 j 3n 1 j 3n 1 j 3n 2 j 3n 1 j Replacing j by (j-1) ( j 1) 2n 1 ( j 1) 3n 1 ( j 1) 3n 1 ( j 1) 3n 2 ( j 1) 3n 1 ( j 1) 2n -1 3n -1 3n -1 3n - 2 3n 1 14n - 4 ........... .......... (1) th Let Wj be the weight of j face value of G j 2n -1- j 3n -1- j 3n -1- j 3n - 2 j 3n 1 j 2n -1 3n -1 3n -1 3n - 2 3n 1 14n - 4 ...........................(2) th Let Wj1 be the weight of (j1) face value of G (j1) 2n -1- (j1) 3n -1- (j1) 3n -1- (j1) 3n - 2 (j1) 3n 1 (j1) 2n -1 3n -1 3n -1 3n - 2 3n 1 14n - 4 ..........................(3) From (1) , (2) and (3) Wj-1 W j W j1 for all j That is Wj k for all values of 1 j n Hence para chain Hexagon is a face magic graph. Theorem : 3.2 A Friendship graph Fn for n 2 is face magic. Proof Let G be a friendship graph with 2n 1 vertices and 3n edges. Then the vertex set G is represented by v 1 V(G) 0 vij 1 i n , 1 j 2 www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1070] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:02/February-2021 Impact Factor- 5.354 www.irjmets.com th Let Wj be the weight of the j face of G contain the vertices v 0 , vi,1,vi,2 . We have to prove the weight of sum of the labels assigned to the vertices in each face of G are equal. It is enough to prove Wj is true for any three continuous values of j-1 , j , j 1 . Let Wj be the sum of the labels formed by three vertices Wj L(vo ) L(vi,1 ) L(vi,2 ) Where, L(v0 ) 1 L(vi,1 ) {n i j / j 1 1 i n L(vi,2 ) {2n j 1/ j 2 1 i n th Let Wj-1 be the weight of the ( j-1) value of G Wj-1 1 n i ( j 1) 2n ( j 1) i 3n 1 ...... (1) th Let Wj be the weight of the j value of G Wj 1 n i j 2n j i 3n 1 ....... (2) th Let Wj be the weight of the (j1) value of G Wj-1 1 n i ( j 1) 2n ( j 1) i 3n 1 ........... (3) www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1071] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:02/February-2021 Impact Factor- 5.354 www.irjmets.com From equation (1) ,(2) and (3) Wj-1 W j W j1 Wj k for all values of 1 j n Hence the Friendship graph Fn is a face magic graph. Theorem : 3.3 The Ladder graph Ln is a face magic graph, for n ≥ 3 Proof : Let G = Ln be a ladder graph. The vertex set of G is represented by V (G) {vij / 1 i n , 1 j 2 The labeling of G can be represented as 2n i 6 , 1 i n, j 1,2,1,2,... L(vij ) 2n i 1 , 1 i n j 2,1,2,1... th Let Wi be the weight of the j face of G contains the vertices v1 ,v1,j1 ,v2, j ,v2, j1 We have to prove the weight of sum of the labels assigned to the vertices Let in each face of G are equal. It is enough to prove Wi is true for any three continuous values of i -1 , i , i 1 . Wi is the sum of the labels formed by 4 vertices. Wi L(v1, j ) L(v1, j1 ) L(v2, j ) L(v2, j1 ) Where L(vi, j ) {2n i 6 , 1 i n , j 1,2,1,2... L(vi,j1 ) {2n i 1 , 1 i n , j 2,1,2,1 ... www.irjmets.com @International Research Journal of Modernization in Engineering, Technology and Science [1072] e-ISSN: 2582-5208 International Research Journal of Modernization in Engineering Technology and Science Volume:03/Issue:02/February-2021 Impact Factor- 5.354 www.irjmets.com th Let Wi be the weight of i face value of G Wi 2n i - 6 2n - i 1 2n i - 6 2n - i 1 8n -12 2 8n -10 ...........