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Mathematics 2015, 3(3), 604-614; doi:10.3390/math3030604

Topological Integer Additive Set-Sequential Graphs

1
Department of Mathematics, Vidya Academy of Science & Technology, Thrissur 680501, India
2
PG & Research Department of Mathematics, Mary Matha Arts & Science College, Mananthavady 670645, India
3
Naduvath Mana, Nandikkara, Thrissur 680301, India
*
Author to whom correspondence should be addressed.
Academic Editor: Michael Falk
Received: 7 April 2015 / Revised: 24 June 2015 / Accepted: 26 June 2015 / Published: 3 July 2015
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Abstract

Let \(\mathbb{N}_0\) denote the set of all non-negative integers and \(X\) be any non-empty subset of \(\mathbb{N}_0\). Denote the power set of \(X\) by \(\mathcal{P}(X)\). An integer additive set-labeling (IASL) of a graph \(G\) is an injective function \(f : V (G) \to P(X)\) such that the image of the induced function \(f^+: E(G) \to \mathcal{P}(\mathbb{N}_0)\), defined by \(f^+(uv)=f(u)+f(v)\), is contained in \(\mathcal{P}(X)\), where \(f(u) + f(v)\) is the sumset of \(f(u)\) and \(f(v)\). If the associated set-valued edge function \(f^+\) is also injective, then such an IASL is called an integer additive set-indexer (IASI). An IASL \(f\) is said to be a topological IASL (TIASL) if \(f(V(G))\cup \{\emptyset\}\) is a topology of the ground set \(X\). An IASL is said to be an integer additive set-sequential labeling (IASSL) if \(f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}\). An IASL of a given graph \(G\) is said to be a topological integer additive set-sequential labeling of \(G\), if it is a topological integer additive set-labeling as well as an integer additive set-sequential labeling of \(G\). In this paper, we study the conditions required for a graph \(G\) to admit this type of IASL and propose some important characteristics of the graphs which admit this type of IASLs. View Full-Text
Keywords: integer additive set-labeling; integer additive set-sequential labeling; topological integer additive set-labeling; topological integer additive set-sequential labeling integer additive set-labeling; integer additive set-sequential labeling; topological integer additive set-labeling; topological integer additive set-sequential labeling
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Naduvath, S.; Augustine, G.; Sudev, C. Topological Integer Additive Set-Sequential Graphs. Mathematics 2015, 3, 604-614.

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