A Study on Topological Integer Additive Set-Labeling of Graphs

Abstract

A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus }:E(G)\to \mathcal{P}(X)-\{\varnothing \}$ defined by $f^{\oplus }(uv)=f(u)\oplus f(v)$ for every $uv\in E(G)$ is also injective. Let $G$ be a graph and let $X$ be a non-empty set. A set-indexer $f:V(G)\to \mathcal{P}(X)$ is called a topological set-labeling of $G$ if $f(V(G))$ is a topology of $X$. An integer additive set-labeling is an injective function $f:V(G)\to \mathcal{P}({\mathbb{N}}_0)$, whose associated function $f^{+}:E(G)\to \mathcal{P}({\mathbb{N}}_0)$ is defined by $f(uv)=f(u)+f(v),uv\in E(G)$, where ${\mathbb{N}}_0$ is the set of all non-negative integers and $\mathcal{P}({\mathbb{N}}_0)$ is its power set. An integer additive set-indexer is an integer additive set-labeling such that the induced function $f^{+}:E(G)\to \mathcal{P}({\mathbb{N}}_0)$ defined by $f^{+}(uv)=f(u)+f(v)$ is also injective. In this paper, we extend the concepts of topological set-labeling of graphs to topological integer additive set-labeling of graphs.

16 pages, 7 figures, Accepted for publication. arXiv admin note: text overlap with arXiv:1403.3984

References in corpus (1)

• On Integer Additive Set-Indexers of Graphs