# 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

# 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

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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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