# A Study on Topological Integer Additive Set-Labeling of Graphs Source: [A Study on Topological Integer Additive Set-Labeling of Graphs]() Collection index: [llms.txt]() Data updated: 2025-10-27T04:57:47.618Z Authors: N. K. Sudev, K. A. Germina Published: 2014-07-17T01:17:31.000Z ## 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 $f^{\\oplus}:E(G)\\to \\mathcal{P}(X)-\\{\\emptyset\\}$ 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()