Abstract

Let N0 denote the set of all non-negative integers and X be any subset of X. Also denote the power set of X by P(X). An integer additive set-labeling (IASL) of a graph G is an injective function f : V (G) ! P(X) such that the induced function f+ : E(G) ! P(X) is defined by f+(uv) = f(u) + f(v), where f(u) + f(v) is the sumset of f(u) and f(v). An IASL f is said to be a topological IASL (Top-IASL) if f(V (G)) [ f;g is a topology of the ground set X. An IASL is said to be an integer additive set-graceful labeling (IASGL) if for the induced edgefunction f+, f+(E(G)) = P(X)??f;; f0gg. In this paper, we study certain types of IASL of a given graph G, which is a topological integer additive set-labeling as well as an integer additive set-graceful labeling of G.

References

• B. D. Acharya, Set-Valuations and Their Applications, MRI Lecture notes in Applied Mathematics, No.2, The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, 1983.
• B. D. Acharya, Set-Indexers of a Graph and Set-Graceful Graphs, Bulletin of Allahabad Mathematical Society, 16(2001), 1-23.
• B. D. Acharya, K. A. Germina, K. L. Princy and S. B. Rao, Topologically Set-Graceful Graphs, Journal of Combinatorics, Information and System Sciences, 37(2-4)(2012), 299-318.
• J. A. Bondy and U. S. R. Murty, Graph theory with applications, Macmillan Press, London, 1976.
• A. Brandst¨adt, V. B. Le and J. P. Spinrad, Graph Classes: A Survey, SIAM, Philadelphia, 1987.
• J. A. Gallian, A Dynamic Survey of Graph Labelling, The Electronic Journal of Combinatorics, DS-6, 2013.
• K. A. Germina and T. M. K. Anandavally, Integer Additive Set-Indexers of a Graph:Sum Square Graphs, Journal of Combinatorics, Information and System Sciences, 37(2-4)(2012), 345-358.
• K. A. Germina and N. K. Sudev, On Weakly Uniform Integer Additive Set-Indexers of Graphs, International Mathematical Forum, 8(37)(2013), 1827-1834. DOI: 10.12988/imf.2013.310188.
• K. D. Joshy, Introduction to General Topology, New Age International, New Delhi, 1983.
• V. Krishnamoorthy, On the Number of Topologies of Finite Sets, The Amer. Math. Monthly, 73(2)(1966), 154-157.
• F. Harary, Graph Theory, Addison-Wesley Publishing Company Inc., Philippines, 1969.
• W. Imrich, S. Klavzar, Product Graphs: Structure and Recognition, Wiley, 2000.
• K. D. Joshi, Applied Discrete Structures, New Age International, New Delhi, 2003.
• J. R. Munkers, Topology, Prentice Hall, Vol.2., 2000.
• A. Rosa, On Certain Valuation of the Vertices of a Graph, in Theory of Graphs, Gordon and Breach, 1967, 349-355.
• N. K. Sudev and K. A. Germina, On Integer Additive Set-Indexers of Graphs, International Journal of Matematical Sciences & Engineering Applications, 8(2)(2014),11-22.
• N. K. Sudev and K. A. Germina, Some New Results on Strong Integer Additive Set-Indexers of Graphs, Discrete Mathematics, Algorithms & Applications, 7(1)(2015),1-11., DOI: DOI:10.1142/S1793830914500657.
• N. K. Sudev, K. A. Germina and K. P. Chithra, A Creative Review on Integer Additive Set-Labeled Graphs, Asian-European Journal of Mathematics, to appear., DOI:10.1142/S1793557115500527.
• N. K. Sudev and K. A. Germina, The exquisite Integer Additive Set-Labeling of Graphs, International Journal of Science and Research, 4(3)(2015), 2858-2862.
• N. K. Sudev and K. A. Germina, A Study on Topological Integer Additive Set-Labeling of Graphs, Electronic Journal of Graph Theory and Applications, 3(1)(2015), 70-84.DOI:10.5614/ejgta.2015.3.1.8.
• N. K. Sudev and K. A. Germina, A Study on Integer Additive Set-Graceful Graphs, Journal of Pure and Applied Mathematics, to appear.
• D. B. West, Introduction to Graph Theory, Pearson Education Inc., 2001.