Q(a)- Balance Edge Magic of Sun family Graphs

Transcription

1 Volume-6, Issue-3, May-June 2016 International Journal of Engineering and Management Research Page Number: Q(a)- Balance Edge Magic of Sun family Graphs S.Vimala 1, R.Prabavathi 2 1 Assistant Professor, Department of Mathematics, Mother Teresa Women's University, Kodaikanal, Tamilnadu, INDIA 2 Research Scholar, Department of Mathematics, Mother Teresa Women's University, Kodaikanal, Tamilnadu, INDIA ABSTRACT Let G be (p,q)-graph in which the edges are labeled 1,2,3,4, q so that the vertex sums are constant, mod p, then G is called an Edge magic graph. A (p,q)-graph G in which the edges are labeled by Q(a) so that vertex sums mod p is constant, is called Q(a) Balance Edge-Magic Graph(BEM), which is introduced about Q(a)-BEM[12]. This article extends and discussed Q(a) Balance Edge Magic graphs of Sun family graphs, Haar graph, cubical graph, flower graph. Keywords cubical graph, flower graph, haar graph, labeling, sun graph, raising sun graph, Q(a)- BEM. I. INTRODUCTION A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. Labeled graphs are becoming an increasingly useful family of Mathematical Models from abroad range of applications. In 1967 Alex Rosa introduced labeling. An enormous body of literature has grown around labeling in about more than 1300 papers. Labelling includes lots of verities such as magic, antimagic, bimagic, vertex magic, edge magic, total magic, graceful, Harmonious, felicitous, elegant, cordial, prime labeling, etc. All labeling graphs using in varies places like networking, neural networks, stochastic models, etc., II. METHODOLOGY A useful survey to know about the numerous graph labeling methods is the one by one in J.A. Gallian[5]. All graphs considered here are finite simple and undirected. Magic labeling were initially developed by Sedl aˇcek in 1963 and after many people worked.[1,7,8] Kotzig and Rosa introduced and defined magic labeling of finite graphs and complete graphs etc., [2,9,10,11] improved edge- magic results. In 2007, introduced Q(a) balance edge magic graphs and Q(a) balance super edge magic graphs by Sin- Min Lee and Thomas Wong, Sheng Ping Bill Lo[12]. Proved many results in Q(a)-Balance super Edge-magic Graphs and Q(a) Balance edge-magic Graphs for wheel graph, Fan graphs, Complete graphs, Book graph, Ladder Graph. Here after extend results in [9,14]. In this paper we extends research work to sun graph, raising sun graph, Haar graph, cubical graph, flower graph. III. PRIOR APPROACH All graphs in this paper are connected (multi-) graphs without lops. Definition: A finite graph G = [V(G), E(G)] without loops, multiple edges or isolated vertices. If there exists a mapping f from the set of edges E(G) into positive real numbers such that (i) f(ei) f(ej) for all ei ej : ei, ej є E(G), (ii) Σe єe(g) ƞ(v, e)f(e) = r for all v є E(G) where ƞ(v, e) is 1 when vertex v and edge e are incident and 0 in the opposite case. Then the graph G is called magic. The mapping f is called a labeling of G and the value r is the index of the label f. Definition: A graph G is called edge-magic if there exists a bijective function φ:v(g) ᴜ E(G) {1,2,3,, V(G) ᴜ E(G) } such that φ(x) + φ(xy) + φ(y) is a constant c(φ) for every edge xy є E(G): here c(φ) is called the valence of φ. Unless, A Graph (p,q) graph G such that the edges are labeled 1,2,3,,p so that the vertex sums are constant, mod p, is called edge-magic. Definition: A (p,q) graph G in which the edges are labeled by Q(a) so that the vertex sums mod p is a constant is called Q(a)- balance edge-magic(bem). For a 1, we denote Q(a) = {±a,, ±(a 1 +q/2)}, if q is even, Q(a) = 143 Copyright Vandana Publications. All Rights Reserved.

2 {0, ±a,, ±(a 1 +(q-1)/2)}, if q is odd then Q(a)- Balance Edge Magic(BEM). Lemma1 The complete bipartite graph, Fan graph, book graph, complete graph(k n, n < 5) and other multi graphs were Q(a)- BEM[9,12]. Definition A Haar graph H(n) is a bipartite regular vertextransitive graph, indexed by a positive integer and obtained by a simple binary encoding of cyclically adjacent vertices, Haar graphs may be connected or disconnected. There are 2k-1 Haar graphs on 2k vertices, so the vertex count of H(n). Definition The cubical graph is the platonic graph corresponding to the connectivity of the cube. Definition The flower graph Fln is the graph obtained from a helm Hn joins each pendant vertex to the apex of helm. The helm Hn is the graph obtained from a wheel Wn by attaching a pendant edge to each rim vertex. IV. OUR APPROACH Theorem 2 If the sun graph Sn is strong Q(a)-BEM for n 6. Let n = 6, It suffices to show that S6 is Q(a)-Balanced Edge Magic for 1,2,3,4,5,6 For a 1, we denote {±a,,±(a 1 +q/2)}, if q is even, Q(a) = { {0, ±a,, ±(a 1 +(q-1)/2)}, if q is odd. Here q is even, Fig 1 shows that s6 is strong Q(a)-BEM Fig 2 A Q (2)-BEM labeling for S 8 :{ -2.-4,-5, 2, 3,-3, 5,7,-8, -7,8,4} A Q (3)-BEM labeling for S 8 :{ -6.-4,-5, 6, 3,-3, 5,7,-7, -8,8,-9,-9} A Q (4)-BEM labeling for S 8 :{ -6.-4,-5, 6, 10,-10, 5,7,-7, -8,8,-9,-9} A Q (5)-BEM labeling for S 8 :{10,-9,-10, -6,-5, 6, 5,7,-7, -8,11,8,-11,9} A Q (6)-BEM labeling for S 8 :{-6,-9,-10, 10,11, 6, -11,7, -7,-8,8,12,-12,9} A Q (7)-BEM labeling for S 8 :{-13,-9,-10, 10,11, 13, -11,7,-7,-8,8,12,-12,9} A Q (8)-BEM labeling for S 8 :{9,-9,-10, 10,11, 13, -11, -13-8,8,12,-12,9,14,-14} Theorem 4 If the Raising sun graph (p, q) n 7 is strong Q(a)-BEM : Generalized Raising sun graph is 2n-n If n = 7,q = 12 It suffices to show that Raising sun graph is Q(a)-Balanced edge magic for 1,2,3,4,5,6,7 Here q is even,fig 3,shows that Raising sun graph Q(a)- BEM Fig 1 A Q (2)-BEM labeling for S6 : { 0,-2.-4,-5, 2, 3,-3, 5} A Q (3)-BEM labeling for S6 : { 0,-5.-4,-5, 6, 3,-3,-6} A Q (4)-BEM labeling for S6 : {-4, 0, 4,-5,-7,6,7,-6} A Q (5)-BEM labeling for S6 : {-7, 7,-5, 0, 5,8,-6,-8,6} A Q (6)-BEM labeling for S6 : {-7,7,9,0,-9,8,-6,-8,6} Theorem 3 If the sun graph S8 satisfies the Q(a)-BEM conditions. n = 8, q = 14 It is suffices the following conditions, Here n is even. Fig 2,shows that the S 8 is Q(a)-BEM Fig 3 A Q (2)-BEM labeling for Raising sun graph : {-2,-6, -2,3,4,-4,3,6,-5, 7,-7,5} A Q (3)-BEM labeling for Raising sun graph : {-3,-6,8, -8,4,-4,3,6,-5, 7,-7,5} A Q (4)-BEM labeling for Raising sun graph : {4,-6,-8,4, 144 Copyright Vandana Publications. All Rights Reserved.

3 -4,8,6,-5, 7,-7,5,9,-9} A Q (5)-BEM labeling for Raising sun graph : {10,-6,-10, -9,9-8,6,-5,8, 7,-7,5} A Q (6)-BEM labeling for Raising sun graph : {10,-6,-10, -9,9-8,6,-11,8, 7,-7,11} A Q (7)-BEM labeling for Raising sun graph : {12,-12,10, -10,-9,9-8,-11,8, 7,-7,11 Theorem 5 If the Haar graph H(7) is strong Q(a)-BEM. If p = 7,q = 12.It suffices to show that Haar graph H(7) is strong s Q(a)-Balanced edge magic for 1,2,3,4,5,6 Here q is even, Fig 4, shows that Haar graph H(7) is strong Q(a)-BEM. Fig 4, Q(a)-BEM A Q (2)-BEM labeling for Haar graph H(7) : {-2,-6,2,-3,4, -4,3,6,-5, 7-7,5 A Q (3)-BEM labeling for Haar graph H(7) : {3,-3,4,-4,5, -5,6,-6,-7,7,8,-8} A Q (4)-BEM labeling for Haar graph H(7) : {4,-4,5,-5,6, -6,-7,7,8,-8,9,-9} A Q (5)-BEM labeling for Haar graph H(7) : {5,-5,6,-6, -7,7,8,-8,9,-9,10,-10} A Q (6)-BEM labeling for Haar graph H(7) : {6,-6,-7,7,8, -8,9,-9,10,-10,11,-11} A Q (7)-BEM labeling for Haar graph H(7) : {-7,7,8,-8,9, -9,10,-10,11,-11,12,-12} Theorem 6 If the Haar graph H(13) is strong Q(a)-BEM. If p = 8,q = 12.It suffices to show that Haar graph H(13) is strong s Q(a)-Balanced edge magic for 1,2,3,4,5,6,7,8 Here q is even,fig 5, shows that Haar graph H(13) is strong Q(a)-BEM. Fig 5 A Q (2)-BEM labeling for Haar graph H(13) : {-2,-6,2, -3,4,-4,3,6,-5, 7-7,5} A Q (3)-BEM labeling for Haar graph H(13) : {-6,-3,4, -4,3,6,-5, 7-7,5,8,-8} A Q (4)-BEM labeling for Haar graph H(13) : {-6,-9,4, -4,9,6,-5, 7-7,5,8,-8} A Q (5)-BEM labeling for Haar graph H(13) : {-6,-9,10, -10,9,6,-5, 7-7,5,8,-8} A Q (6)-BEM labeling for Haar graph H(13) : {-11,-9,10, -10,9,11,-6, 7-7,6,8,-8} A Q (7)-BEM labeling for Haar graph H(13) : {-11,-9,10, -10,9,11,-7, 12,-12,7,8,-8} A Q (8)-BEM labeling for Haar graph H(13) : {-11,-9,10, -10,9,11,-13,12,-12,13,8,-8} A Q (9)-BEM labeling for Haar graph H(13) : {-11,-9,10, -10,9,11,-13,12,-12,13,14,-14} A Q (10)-BEM labeling for Haar graph H(13) : {-11, -15,10,-10,15,11,-13,12,-12,13,14,-14} Theorem 7 If the Haar graph H(15) is strong Q(a)-BEM. If p = 8,q = 16 It suffices to show that Haar graph H(13) is strong s Q(a)- Balanced edge magic for 1,2,3,4,5,6,7,8 Here q is even, fig 6, shows that Haar graph H(15) is strong Q(a)-BEM. Fig 6 A Q (2)-BEM labeling for Haar graph H(13) : {-2,-6,2, -3,4,-4,3,6,-5, 7-7,5,8,-8,9,-9} A Q (3)-BEM labeling for Haar graph H(13) : {-6,,4,-4, -3,3,6,-5, 7-7,5,8,-8,9,-9,10,-10} A Q (4)-BEM labeling for Haar graph H(13) : {-6,4,-4,6, -5, 7-7,5,8,-8,9,-9,10,-10,11,-11} A Q (5)-BEM labeling for Haar graph H(13) : {-6,6,-5, 7, -7,5,8, A Q (6)-BEM labeling for Haar graph H(13) : {13,-13, -6,6,7,-7,8, A Q (7)-BEM labeling for Haar graph H(13) : {7,-7,13, -13,8,-8,9,-9,10,-10,11,-11,12,-12,14,-14} 145 Copyright Vandana Publications. All Rights Reserved.

4 A Q (8)-BEM labeling for Haar graph H(13) : {13,-13,8, -8,9,-9,10,-10,11,-11,12,-12,14,-14,15, -15} Theorem 8 If the Haar graph H(23) is strong Q(a)-BEM. If p = 10,q = 24 It suffices to show that Haar graph H(23) is strong s Q(a)- Balanced edge magic for 1,2,3,4,5,6,7,8,9,10 Here q is even, Fig 7,shows that Haar graph H(23) is strong Q(a)-BEM. -14,16,-17,-19,-20,-21} Theorem 9 If the cubical graph is strong Q(a)-BEM. Here q is even, fig 8, shows that strong Q(1)-balanced Fig 8 Fig 7 A Q(2)-BEM labeling for Haar graph H(23) : {-2,-6,2, -3,4,-4,3,6,-5, 7-7,5,8,-8,9,-9,10,-10,11, -11,12,-12,13,-13} A Q(3)-BEM labeling for Haar graph H(23) :{-6,-3,4, -4,3,6,-5,7,-7,5,8,-8,9,-9,10,-10,11,-11,12,-12,13,-13,14, -14} A Q(4)-BEM labeling for Haar graph H(23) : {-6,4,-4,6, -5,7,-7,5,8,-8,9,-9,10,-10,11,-11,15,-15,12,-12,13,-13,14, -14} A Q(5)- BEM labeling for Haar graph H(23) : {-6,6,-5,7-7,5,8,-8,9,-9,10,-10,11,-11,15,-15,16,12,-12,13,-13,14, -14,16} A Q(6)-BEM labeling for Haar graph H(23) : {-6,6,-7-7,8, -8,9,-9,10,-10,11,-11,15,-15,16,17,12,-12,13,-13,14, -14,16,-17} A Q(7)-BEM labeling for Haar graph H(23) : {-7-7,8,-8,9, -9,10,-10,11,-11,15,-15,16,17,18,-18,12,-12,13,-13,14,- 14,16,-17} A Q(8)-BEM labeling for Haar graph H(23) : {8,-8,9, -9,10,-10,11,-11,15,-15,16,17,18,-18,19,12,-12,13,-13,14, -14,16,-17,-19} A Q(9)-BEM labeling for Haar graph H(23) : {9,-9,10, -10,11,-11,15,-15,16,17,18,-18,19,20,12,-12,13,-13,14, -14,16,-17,-19,-20} A Q(10)-BEM labeling for Haar graph H(23) : {10, -10,11,-11,15,-15,16,17,18,-18,19,20,21,12,-12,13,-13,14, A strong Q(2)-BEM labeling for cubical graph : {2,-2,3, A strong Q(3)-BEM labeling for cubical graph : {3,-3,4, A strong Q(4)-BEM labeling for cubical graph : {4,-4,5, Preposition 10 If the following cubical graph is strong Q(a)-BEM. Solution Here q is even, fig 9, shows that strong Q(1)-balanced Fig Copyright Vandana Publications. All Rights Reserved.

5 Preposition 11 Is the given graph is strong Q(a)-BEM. Solution Given graph is cubical n = 8, q = 12 Here q is even, fig 10, shows that strong Q(1)-balanced Fig 10 Theorem 12 If the cubical graph is strong Q(a)-BEM. Here q is even, fig 11, shows that strong Q(1)-balanced Fig 11 Theorem 13 If the cubical graph is strong Q(a)-BEM. Here q is even, fig 12, shows that strong Q(1)-balanced Fig Copyright Vandana Publications. All Rights Reserved.

6 Theorem 14 If the Flower graph Fl 7 is weak Q(a)-BEM. Given n = 7,q = 8 Flower graph Fl 7 is satisfies the Q(a)-BEM for a = 1,2,3,4,5,6,7 Here q is even, fig 13, shows that Q(1)-balanced edge magic for Fl7. Fig 13 A Q (2)-BEM labeling for Flower graph Fl7 : {-4, -3,5,3,2,4,-2,-5} A Q (3)-BEM labeling for Flower graph Fl7 : {-4,-3,5,3,6,4,-6,-5} A Q (4)-BEM labeling for Flower graph Fl7 : {7,-7, -4,5,6,4,-6,-5} A Q (5)-BEM labeling for Flower graph Fl7 : {7,-7,8, -8,5,6,-6,-5} A Q (6)-BEM labeling for Flower graph Fl7 : {7,-7,8, -8,9,6,-6,-9} A Q (7)-BEM labeling for Flower graph Fl7 : {10,-10,7, -7,8,-8,9,-9} Theorem 15 If the Flower graph Fl 10 is weak Q(a)-BEM Given n = 10,q = 12 Flower graph Fl 10 is satisfies the weak Q(a)-BEM for a = 1,2,3,4,5,6,7,8,9,10 Here q is even, fig 14, shows that Q(1)-balanced edge magic for Fl 10 Fig 14 A weak Q (2)-BEM labeling for Flower graph Fl10 : {7, -7,2,-2,3,-3,4,-4,5,-5,6,-6} A weak Q (3)-BEM labeling for Flower graph Fl10 : {8, -8,7,-7,3,-3,4,-4,5,-5,6,-6} A weak Q (4)-BEM labeling for Flower graph Fl10 : {4, -4,5, A weak Q (5)-BEM labeling for Flower graph Fl10 : {5, -5,6, A weak Q (6)-BEM labeling for Flower graph Fl10 : {6, -6,7, A weak Q (7)-BEM labeling for Flower graph Fl10 : {7, -7,8, A weak Q (8)-BEM labeling for Flower graph Fl10 : {8, -8,9, A weak Q (9)-BEM labeling for Flower graph Fl10 : {9, -9,10,-10,11,-11,12,-12,13,-13,14,-14} A weak Q (10)-BEM labeling for Flower graph Fl 10 : {10, -10,11,-11,12,-12,13,-13,14,-14,15, -15} Theorem 16 If the Flower graph Fl 13 is weak Q(a)-BEM for n 13. Given n = 13,q = 16 Flower graph Fl16 is satisfies the Q(a)-BEM for a = 1,2,3,4,5,6,7,8,9,10,11,12,13 Here q is even, fig 15, shows that weak Q(1)-balanced edge magic for Fl 13 Fig 15 A weak Q (2)-BEM labeling for Flower graph Fl13 : {2, -2,3,-3,4,-4,5, A weak Q (3)-BEM labeling for Flower graph Fl13 : {10, -10,3,-3,4,-4,5, A weak Q (4)-BEM labeling for Flower graph Fl13 : {10, -10,11,-11,4,-4,5, A weak Q (5)-BEM labeling for Flower graph Fl13 : {10, -10,11,-11,12,-12,5,-5,6,-6,7,-7,8,-8,9, -9} A weak Q (6)-BEM labeling for Flower graph Fl13 : {10, -10,11,-11,12,-12,13,-13,6,-6,7,-7,8, -8,9,-9} A weak Q (7)-BEM labeling for Flower graph Fl13 : {10, -10,11,-11,12,-12,13,-13,14,-14,7,-7,8, -8,9,-9} A weak Q (8)-BEM labeling for Flower graph Fl13 : {10, -10,11,-11,12,-12,13,-13,14,-14,15, -15,8,-8,9,-9} A weak Q (9)-BEM labeling for Flower graph Fl 13 : {10, -10,11,-11,12,-12,13,-13,14,-14,15, -15,16,-16,9,-9} A weak Q (10)-BEM labeling for Flower graph Fl 13 : {10, -10,11,-11,12,-12,13,-13,14,-14,15,-15,16,-16,10,-10} A weak Q (11)-BEM labeling for Flower graph Fl 13 : {17, -17,11,-11,12,-12,13,-13,14,-14,15, -15,16,-16,10,-10} A weak Q (12)-BEM labeling for Flower graph Fl 13 : {17, -17,18,-18,12,-12,13,-13,14,-14,15, -15,16,-16,10,-10} A weak Q (13)-BEM labeling for Flower graph Fl 13 : {17, -17,18,-18,19,-19,13,-13,14,-14,15, -15,16,-16,10,-10} Theorem 17 If the Flower graph Fln is Q(a)-BEM for n 16. Given n = 16,q = 20 Flower graph Fl16 is satisfies the weak Q(a)-BEM for 148 Copyright Vandana Publications. All Rights Reserved.

7 a = 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 Here q is even, fig 16, shows that Q(1)-balanced edge magic for Fl10-27,-18,18,27,21,-25,25,22,24, 23, -23,-24,-22,26,19, A Q (19)-BEM labeling for Flower graph Fl16 : {-21,-26, -27,-28,28,27,21,-25,25,22,24, 23,-23,-24,-22,26,19, A Q (20)-BEM labeling for Flower graph Fl16 : {-21,-26, -27,-28,28,27,21,-25,25,22,24, 23,-23,-24,-22,26,20,-20} Fig 16 A Q(2)-BEM labeling for Flower graph Fl16 : {-10,-6,-9, -7,-8,8,9,7,10,6,-5,11,5,-11,4,2,3,-3, -4,-2} A Q(3)-BEM labeling for Flower graph Fl16 : {-10,-6,-9, -7,-8,8,9,7,10,6,-5,-11,5,11,4,12,3,-3, -4,-12} A Q(4)-BEM labeling for Flower graph Fl16 : {-10,-6,-9, -7,-8,8,9,7,10,6,-5,-11,5,11,4,12,13, -13,-4,-12} A Q(5)-BEM labeling for Flower graph Fl16 : {-10,-6,-9, -7,-8,8,9,7,10,6,-5,-11,5,11,14,12,13,-13,-14,-12} A Q(6)-BEM labeling for Flower graph Fl16 : {-10,-6,-9, -7,-8,8,9,7,10,6,-15,-15,11,14,12,13, -13,-14,-12,-11} A Q(7)-BEM labeling for Flower graph Fl16 : {-10,-16, -9,-7,-8,8,9,7,10,16,-15,15,11,14,12,13, -13,-14,-12,-11} A Q(8)-BEM labeling for Flower graph Fl16 : {-10,-16, -9,-17,-8,8,9,17,10,-15,15,11,14,12,13, -13,-14,-12,-11,16} A Q(9)-BEM labeling for Flower graph Fl16 : {-10,-16, -9,-17,-18,18,9,17,10,15,11,14,12,13, -13,-14,-12,-11,16, -15} A Q(10)-BEM labeling for Flower graph Fl16 :{-10,-16, -17,-18,18,17,10,-15,15,11,14,12,13,-13,-14,-12,-1,16,19, -19} A Q(11)-BEM labeling for Flower graph Fl16 :{-20,-16, -17,-18,18,17,20,-15,15,11,14,12,13,-13,-14,-12,-1,16,19, -19} A Q(12)-BEM labeling for Flower graph Fl16 : {-21,-16, -17,-18,18,17,21,-15,15,20,14,12,13,-13,-14,-12,-0,16,19, -19} A Q(13)-BEM labeling for Flower graph Fl16 : {19, -19,20,-20,-21,-16,-17,-18,18,17,21, -15,15,22,14, 13,-13, -14,-22,16} A Q(14)-BEM labeling for Flower graph Fl16 : {19, -19,20,-20,-21,-16,-17,-18,18,17,21,-15,15,22,14,23,-23, -14,-22,16} A Q(15)-BEM labeling for Flower graph Fl16 : {-21,-16, -17,-18,18,17,21,-15,15,22,24,23,-23,-24,-22,16,19, A Q(16)-BEM labeling for Flower graph Fl16 : {-21,-16, -17,-18,18,17,21,-25,25,22,24, 23, -23,-24,-12,-22,16,19, A Q(17)-BEM labeling for Flower graph Fl16 : {-21,-26, -17,-18,18,17,21,-25,25,22,24,23,-23,-24,-22,2619,-19,20, -20} A Q(18)-BEM labeling for Flower graph Fl16 : {-21,-26, V. CONCLUSION This paper discussed Q(a)- Balance Edge Magic of sunfamily, Haar graph etc., The author working on Q(a)- Balance Edge Magic(BEM) of prism graph and antiprism graph and other graphs. REFERENCES [1] Alison M. Marr W.D. Wallis, Magic Graphs, 2 nd edn., Springer New York Heidelberg Dordrecht London,2013. [2] Dharam Chopra, H.Kwong and Sin-Min Lee, On the Edge magic (p,3p-1)- Graphs, Congressus Numerantium 179, 49-63,2006. [3] F.Harary, Graph theory, Addision Wesley, New Delhi, [4] H. Enomoto, K. Masuda and T. Nakamigawa, Induced graph theorem on magic valuations, Ars Combinatoria 56 (2000), [5] J.A Gallian, A dynamic survey of graph labeling, the Electronic J of Combin, # DS6, 1-180,2007 [6] Jiří Sedláček, On magic graphs, Mathematica Slovaca, Vol. 26 (1976), No. 4, [7] Kotzig and A. Rosa, Magic valuations of finite graphs, Canada Math. Bull. 13 (1970), [8] Kotzig and A. Rosa, Magic valuations of complete graphs, Publications du Centre de Recherches Mathematiques Universite de Montreal 175 (1972). [9] Ping-Tsai Chung, Sin-Min Lee, On Computing edge magic graphs and Q(a) Balance Edge Magic Graphs, Congressus Numerantium 199(2009),pg: [10] Seah.E, Lee. S.M and Tan. S.K, On Edge-Magic Graphs, Congress Num.,86, , [11] Sin-Min Lee, Eric Seah and S.K. Tan, On edge-magic graphs, Congressus Numberantium, 86, , [12] Sin-Min Lee and Thomas Wong and Sheng Ping Bill Lo, On the Q(a) Balance Edge-magic Graphs and Q(a) Balance Super Edge-magic Graphs, Congressus Numberantium 188, 33-57, [13] S.M. Hegde and S. Shetty, On magic graphs, Australasian Journal of Combinatorics, 27 (2003), [14] S.Vimala, Some New Results On Q(a)-Balance Edge-Magic Graphs, The Research Journal of Science & IT Management( RJSITM), Volume: 05, Number: 01, November-2015, pg: Copyright Vandana Publications. All Rights Reserved.