# 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

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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