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

# Topological Integer Additive Set-Sequential Graphs

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Department of Mathematics, Vidya Academy of Science & Technology, Thrissur 680501, India
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PG & Research Department of Mathematics, Mary Matha Arts & Science College, Mananthavady 670645, India
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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
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MDPI and ACS Style

Naduvath, S.; Augustine, G.; Sudev, C. Topological Integer Additive Set-Sequential Graphs. Mathematics 2015, 3, 604-614.

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