A Study on Topological Integer Additive Set-Labeling of Graphs

A set-labeling of a graph $G$ is an injective function $f:V(G)\to $\mathcal{P}$(X)$,where$X$isafinitesetandaset-indexerof$G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\to $\mathcal{P}$(X)-{\emptyset}$definedby$f^{\oplus}(uv) = f(u){\oplus}f(v)$forevery$uv{\in} E(G)$isalsoinjective\mathrm{.}Let$G$beagraphandlet$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.