EVEN SUM CORDIAL LABELING FOR SOME NEW GRAPHS

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1 International Journal of Mechanical ngineering and Technology (IJMT) Volume 9, Issue 2, February 2018, pp Article ID: IJMT_09_02_021 Available online at ISSN Print: and ISSN Online: IAM Publication Scopus Indexed VN SUM CORDIAL LABLING FOR SOM NW GRAPHS S. Abhirami Assistant Professor, Department of Mathematics, Sona College of Technology, Salem, Tamil Nadu, India R. Vikramaprasad Assistant Professor, Department of Mathematics, Government Arts College, Salem, Tamil Nadu, India R. Dhavaseelan Assistant Professor, Department of Mathematics, Sona College of Technology, Salem , Tamil Nadu, India ABSTRACT In this paper, we investigate the subdivision graph,, Crown graph, Comb graph, Shell graph, Binary tree graph and Triangular snake graph are even sum cordial Keywords: Subdivision graph K, ; K, ; Crown graph; Comb graph; Shell graph; Binary tree graph; Triangular snake graph and Triangular book graph. AMS Classification: 05C78 Cite this Article: S. Abhirami, R. Vikramaprasad and R. Dhavaseelan, ven Sum Cordial Labeling for Some New Graphs, International Journal of Mechanical ngineering and Technology 9(2), pp INTRODUCTION By a graph G = (V, ), we mean a finite, undirected graph with neither loops nor multiple edges. For graph theoretic terminology we refer to Harary [6], the origin of graph labeling can be attributed to Rosa[8] and we refer to Gallian [5]. In [2, 3], Cahit introduced the concept of cordial labeling of graph. For special graphs we refer [9] [10]. S.Abhirami et al., was introduced ven Sum cordial graph[4]. In this paper, we investigate the subdivision graph K,, Crown graph, Comb graph,shell graph,binary tree graph and triangular snake graph are even sum cordial editor@iaeme.com

2 ven Sum Cordial Labeling for Some New Graphs 2. PRLIMINARIS Definition 2.1: [9] The complete bipartite graph with bipartition (X, Y) such that X = m and Y = n is denoted by K m,n.the graph K 1,n is called a star. The vertex of K 1,n with degree n is called the central vertex or apex. Definition 2.2: [10] The subdivision graph S(G) is obtained from G by subdividing each edge of G with a vertex. Definition 2.3: [10] Consider two stars K, and K, Then G = K, is the graph obtained by joining apex (central) vertices of stars to a new vertex x. Note that G has 2n + 3 vertices and 2n + 2 edges. Definition 2.4: [7] The corona G 1 G 2 of two graphs G 1 and G 2 is defined as the graph obtained by taking one copy of G 1 (with p 1 vertices) and p 1 copies of G 2 and then joining the i th vertex of G 1 to all the vertices in the i th copy of G 2. Remark 2.5: [7] The graph P n K 1 is called a comb and the graph C n K 1 is called a Crown. Definition 2.6: [7] A shell graph is defined as a cycle C n with (n-3) chords sharing a common end point called the apex. Shell graph are denoted as C (n, n-3). A shell Sn is also called fan f n-1. Definition 2.7: An ordered rooted tree is a binary tree if each vertex has atmost two children. Definition 2.8: A full binary tree is a binary tree in which each internal vertex has exactly two children. Remark 2.9: very full binary tree has odd number of vertices and hence has even number of edges. Definition 2.10: [7] The triangular snake T n is obtained from the path P n by replacing each edge of the path P n by a triangle. Let u 1,u 2,.u n be the vertices of the ven sum cordial path P. Let V(T n ) = V(P n ) {v i : 1 i n-1} and (T n ) = V(P n ) {u i v i,v i u i+1 : 1 i n-1} Definition 2.11: [10] One edge union of cycles of same length is called a book. The common edge is called base of the book. If we consider t copies of cycles of length n 3, then the book is denoted by. If n = 3, 4, 5 or 6, then the Book B is called book with triangular, rectangular, pentagonal or hexagonal pages respectively. Definition 2.12: [4] ven sum cordial graph Let G= (V, ) be a simple graph and f: V {1, 2, 3. V } be a bijection. For each edge uv, assign the label 1 if f (u) +f (v) is even and the label 0 otherwise. f is called an even sum cordial labeling if 0 1 1, where 1 and 0 denote the number of edges labeled with 1 and not labeled with 1 respectively. Proposition 2.13: [4] 1. Any path is an even sum cordial graph. 2. Any cycle C n is an even sum cordial graph except n = 6, 6 + d, 6 +2d,when d=4 3. MAIN RSULTS Proposition 3.1: Subdivision graph (K, )is an ven sum cordial graph Proof: Let G be the Subdivision graph (k, ). Let v 1,v 2,.v n,,v 1,v 2,.v n vertices of K 1,n, V(G) = 2n+1 and (G) = 2n. Define f: V (G) {1, 2, 3.2n} Then labeling the graph as follows. be the editor@iaeme.com

3 S. Abhirami, R. Vikramaprasad and R. Dhavaseelan (!) " 2$%1;! & " 2' ; 1 ' $;! & " 2' 1 ; 1 ' $ Hence we get e * 0) " e * 1). Thus, we get e * (0) e * 1) 1.Hence G is an ven sum cordial graph. xample 3.1: Subdivision graph (K 1,7 ) and (K 1,8 ) Proposition 3.2: The graph K, is an ven sum cordial graph. Proof: Let G be the K, graph. Let u 1,u 2,.u n be the vertices of K, and!,!,.! be the vertices of K,. Let u and v be the apex vertex of K, and K, which are adjacent to a common vertex x. Then VG = 2n+3 and G = 4n+2. Define f: V(G) {1,2,3,.,2n+3} as follows (/) " 1; / & " '%1 ; 1 ' $ ;! " $%2 ;! & " $%2%' ; 1 ' $ ; 0 " 2$%3. Construct the graph K, and K, by f(u), f(/ & ), f(v) and f(! & ). Then join the vertex u and v to x by an edge. Hence we get e * 0) " e * 1). Thus, we get e * (0) e * 1) 1. Hence G is an ven sum cordial graph. xample 3.2: Graph K,2 2 Proposition 3.3: The Crown graph C n K 1 graph is an ven sum cordial graph Proof: Let G be the C n K 1 graph and VG = 2n. First we construct the even sum cordial cycle C by Proposition Then assign the label of vertices / &, 1 i n as follows (/ & ) " $%' ; 1 ' $. Then join the vertices / & to! & in cycle C n. Thus we obtained the Crown graph, with e * (0) e * 1) 1. Hence G is an ven sum cordial graph editor@iaeme.com

4 ven Sum Cordial Labeling for Some New Graphs xample 3.3: Crown graph C 5 K 1 and C 10 K 1 Proposition 3.4: The Comb graph P n K 1 is an ven sum cordial graph Proof: Let G be the P n K 1 graph with VG = 2n and!,!,.,! be the vertices of the ven sum cordial path P. First we construct the even sum cordial path P by proposition[4]. Then assign the label of vertices / &, 1 i n as follows (/ & ) " $%' ; 1 ' $. Then join the vertices u i to! & in path P. Thus we obtained the Comb graph, with e * 0) e * 1) 1.Hence G is an ven sum cordial graph. xample 3.4: Comb Graph P 4 K 1 and P 5 K 1 Proposition 3.5: The Shell graph C(n,n-3) is an ven sum cordial graph. Proof: Let G= C(n,n-3) be a graph and v 1,v 2,.v n be the vertices of the ven sum cordial cycle C. First we construct the even sum cordial cycle C by Proposition Choose! as the apex vertex from C n, to join the non-adjacent vertices in C n. We get $ 3 chords from that graph. We get e * 0) e * 1) 1. Hence G is an ven sum cordial graph. xample 3.5: Shell graph C(5,2) Proposition 3.6: very full binary tree is an ven sum cordial graph Proof: Let T be a full binary tree and let v be a root of T which is called zero level vertex. Clearly the i th level of T has 2 i vertices. If T has m levels, then the number of vertices of T is 2 m+1 1 and the number of edges 2 m+1 2. Now, we assign the label 2 m+1 1 to the root v and assign the labels for the remaining vertices as below. Here i indicates that the level of T editor@iaeme.com

5 S. Abhirami, R. Vikramaprasad and R. Dhavaseelan Levels Assigning labels Range i=1 (/ 4 ) " 2 5 %1 7%1) 1 k 2 i i=2 (/ 4 ) " 2 5 %1 7%3 1 k 2 i i=3 (/ 4 ) " 2 5 %1 7%7 1 k 2 i i=4 (/ 4 ) " 2 5 %1 7%15) 1 k 2 i i=5 (/ 4 ) " 2 5 %1 7%31) 1 k 2 i i=m (/ 4 ) " 2 5 %1 (7%(2 & 1)) 1 k 2 i Thus we have e * (0) " e * 1) " 2 ; 1. Thus we get e * 0) e * 1) 1. Hence T is an even sum cordial graph. xample 3.6: Binary tree with 2 levels. Proposition 3.7: Triangular snake T n is an even sum cordial graph. Proof: Let G be the graph T. Let u 1,u 2,.u n be the vertices of the ven sum cordial path =@ABCD P.Let =(> ) "? N> " NA O/ &P BCD =@AB F G! & " ' I %1 ' $J, $ 'K LMM and F G! & " ' %1 ' $J, $ 'K!$! &,! & / &IP BCD I %1 ' $Q, VG = 2n-1, G = 3(n-1). First we construct even sum cordial path P by Proposition 2.13.Then join the edges by above axioms. Thus, we get Hence T n is an even sum cordial graph. xample 3.7: Triangular snake T 4 and T 5. Proposition 3.8: The complete bipartite graph K 1,n is an even sum cordial graph. Proof: Let G be the graph K 1,n. VG = n+1 and G = n. Define f: V(G) {1,2,3,.,n+1} as follows. (/) " 1; / & " '%1 ; 1 ' $ and construct the edges as follows NR " O/! & ; 1 ' $Q. Thus we obtained the complete bipartite graph K 1,n, with e * (0) " e * 1), when n is even. e * 0) " e * 1)% 1 when n is odd and we get e * 0) e * 1) 1. Hence G is an even sum cordial graph editor@iaeme.com

6 ven Sum Cordial Labeling for Some New Graphs xample 3.8: Complete bipartite graph K 1,7 Proposition 3.9: The complete bipartite graph K 2,n is an even sum cordial graph. Proof: Let G be the graph K 2,n.Then VG = n+2, G = 2n. Define f: V(G) {1,2,3,.,n+2} as follows. (!) " 1; / " 2;! & " '%2; 1 ' $ and construct the edges as follows NR " O/! &,!! & ; 1 ' $Q. Thus, we obtained the complete bipartite graph K 1,n with e * (0) " e * 1) and we get e * 0) e * 1) 1. Hence K 2,n is an even sum cordial graph. Similarly, we can prove the above result for K n,n graph. xample 3.9: Complete bipartite graph K 2,5 Proposition 3.10: A book with triangular pages is an even sum cordial graph. Proof: Let B be a book with triangular pages. Let! and! be the vertices of the common edge of the triangular pages and let v 2, v 3,..., v n 1 be the vertices of other ends of the triangle. Here B has 2n 3 edges. Now assign the label the vertices as follows (! ) " 1 ;! " $ and! & " ' ; 2 ' $ 1. Then construct the edges as follows O!! &,!! &,Y$M!! :2 ' $ 1 Q. Thus we have e * 0) " n1, e * 1) "n2 and 0) 1) 1. Hence B is even sum cordial graph. xample 3.10: Triangular book with 10 vertices editor@iaeme.com

7 S. Abhirami, R. Vikramaprasad and R. Dhavaseelan 4. CONCLUSION: In this paper, we discussed the subdivision graph K,, Crown graph, Comb graph, Shell graph, Binary tree graph and Triangular snake graph are even sum cordial. RFRNCS [1] M. M. Andar, Samina Boxwala and N. B. Limaye, On the cordiality of corona graphs, Ars combinatoria 78(2006), [2] I.Cahit. Cordial graphs: A weaker version of graceful and harmonious graph. Ars combinatorial l23(1987), [3] I.Cahit. On Cordial And 3-quitable Labeling of Graph. Utilitas Math, 370(1990), [4] R.Dhavaseelan, R.Vikramaprasad and S.Abhirami.A New Notions of Cordial Labeling Graphs, Global Journal Of Pure And Applied Mathematics, 11 (4) (2015), [5] J. A. Gallian, A dynamic survey of graph labeling, lectronic Journal of Combinatorics, 16(2009), DS6. [6] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, [7] R. Ponraj and S. Sathish Narayanan, Difference Cordial Labeling of Graphs Obtained from Triangular Snakes. Applications and Applied Mathematics: An International Journal Vol. 9, Issue 2 (December 2014), pp [8] A.Rosa. On Certain Valuations of the Vertices of A Graph, Theory of Graphs, International Symposium Rome. (1996), [9] M. Sundaram, R. Ponraj and S. Somasundaram, Some results on product cordial labelling, Pure and Applied Mathematica Sciences, Vol, N01-2, (March 2006), [10] R. Varatharajan, S. Navaneethakrishnan, K. Nagarajan, Special classes of divisor cordial graphs, International Mathematical forum, Vol.7, 2012, no. 35, editor@iaeme.com