Dr. P. Lawrence Rozario Raj SIGNED PRODUCT CORDIAL LABELING IN THE CONTEXT OF ARBITRARY SUPERSUBDIVISION OF SOME SPECIAL GRAPHS Dr. P. Lawrence Rozario Raj Assistant Professor, Department of Mathematics, St. Joseph’s College Trichirappalli – 620 002, Tamil Nadu, India. lawraj2006@yahoo.co.in

The fear of the LORD is the beginning of wisdom … Pro 9:10 Praise the Lord The fear of the LORD is the beginning of wisdom … Pro 9:10

All graphs in this paper are finite, simple and undirected All graphs in this paper are finite, simple and undirected. We follow the basic notation and terminology of graph theory as in Harary and of graph labelling as in Gallian. Cahit defines cordial labeling and cordial graph. The concept of signed product cordial labeling was introduced by Baskar Babujee. Baskar Babujee et al. proved that the path graph, cycle graphs, star, and bistar are signed product cordial and some general results on signed product cordial labeling are also studied.

Definition 1.1 The assignment of values subject to certain conditions to the vertices of a graph is known as graph labeling. Definition 1.2 Let G = (V, E) be a graph. A mapping f : V(G) →{0,1} is called binary vertex labeling of G and f(v) is called the label of the vertex v of G under f. For an edge e = uv, the induced edge labeling f*:E(G)→{0,1} is given by f*(e) = |f(u) − f(v)|. Let vf(0), vf(1) be the number of vertices of G having labels 0 and 1 respectively under f and let ef(0), ef(1) be the number of edges having labels 0 and 1 respectively under f*. Definition 1.3 A binary vertex labeling of a graph G is called a cordial labeling if | vf(0) − vf(1) |  1 and | ef(0) − ef(1) |  1. A graph G is cordial if it admits cordial labeling.

Definition 1.4: A vertex labeling of graph G, f : V(G) → {–1,1} with induced edge labeling f∗ : E(G) → {–1,1} defined by f∗(uv) = f(u)f(v) is called a signed product cordial labeling if |vf(–1)–vf(1)|  1 and | ef(–1)–ef(1) |  1, where vf(–1)is the number of vertices labeled with –1, vf(1) is the number of vertices labeled with 1, ef(–1) is the number of edges labeled with –1 and ef(1) is the number of edges labeled with 1. A graph G is signed product cordial if it admits signed product cordial labeling.

Sethuraman et al. introduced a new method of construction called supersubdivision of graph. Sethuraman et al. proved that arbitrary supersubdivision of any path and cycle are graceful. Kathiresan et al. proved that arbitrary supersubdivision of any star are graceful. Vaidya et al. proved that arbitrary supersubdivision of any path, star and cycle Cn except when n and all mi are simultaneously odd numbers are cordial. Vaidya et al. proved that arbitrary supersubdivision of any tree, grid graph, complete bipartite graph star and CnPm except when n and all mi are simultaneously odd numbers are cordial.

Definition 1.5 Let G be a graph with q edges. A graph H is called a supersubdivision of G if H is obtained from G by replacing every edge ei of G by a complete bipartite graph for some mi, 1  i  q in such a way that the end vertices of each ei are identified with the two vertices of 2-vertices part of after removing the edge ei from graph G. If mi is varying arbitrarily for each edge ei then supersubdivision is called arbitrary supersubdivision of G.

m1=1, m2=2, m3=2, m4=3

Definition 1.6 Tadpole Tn,l is a graph in which path Pl is attached by an edge to any one vertex of cycle Cn. Tn,l has n+l vertices and edges. T5,4

Main Results The graphs obtained by arbitrary supersubdivision of (Pn×Pm)Ps is signed product cordial. The tadpole Tn,l is signed product cordial except mi (1  i  n) are odd, mi (n+1  i  n+l) are even and n is odd CnPm is signed product cordial except mi (1  i  n) are odd, mi (n+1  i  nm) are even and n is odd.

If both u and v are same sign then the value of the edges obtained by supersubdivision are balanced If both u and v are different sign then the value of the edges obtained by supersubdivision are doubled -1 1 -1 1 -1 1 -1 u v 1 -1 -1 u v u v -1 -1 -1 -1 -1 1 w1 w2 w3 w1 w2 1 -1 1 -1 -1 1 -1

Conclusion Labeled graph is the topic of current interest for many researchers as it has diversified applications. I discuss here totally magic cordial labeling of some special graphs. This approach of these graphs is novel and contributes to new graphs to the theory of signed product cordial graphs. The results reported here are new and will add new dimension in the theory of totally magic cordial graphs.

References J. Baskar Babujee, Shobana Loganathan, On Signed Product Cordial Labeling, Applied Mathematics, 2, pp. 1525-1530, 2011. I. Cahit, Cordial Graphs: A weaker version of graceful and harmonious Graphs, Ars Combinatoria 23, pp. 201-207, 1987. F. Harary, Graph theory, Addison Wesley, Reading, Massachusetts, 1972. K.M. Kathiresan, S. Amutha, Arbitrary supersubdivisions of stars are graceful, Indian J. pure appl. Math. 35(1), pp. 81-84, 2004. P. Lawrence Rozario Raj, R. Lawrence Joseph Manoharan, Signed Product Cordialness of Arbitrary Supersubdivisions of Graphs, International Journal of Mathematics and Computational Methods in Science & Technology, Vol.2(2), pp.1-6, 2012 G. Sethuraman, P. Selvaraju, Gracefulness of arbitrary supersubdivisions of graphs, Indian J. pure appl. Math. 32(7), pp. 1059-1064, 2001. S.K. Vaidya, N.A. Dani, Cordial labeling and arbitrary super subdivision of some graphs, International Journal of Information Science and Computer Mathematics, vol. 2(1), pp. 51–60, 2010. S.K. Vaidya, K.K. Kanani, Some new results on cordial labeling in the context of arbitrary supersubdivision of graph, Appl. Math. Sci., vol. 4, no. 45-48, pp. 2323-2329, 2010. S.K. Vaidya, K.K. Kanani, Some strongly multiplicative graphs in the context of arbitrary supersubdivision, International Journal of Applied Mathematics and Computation, vol. 3(1), pp. 60-64, 2011.

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