A Characterisation of Strong Integer Additive Set-Indexers of Graphs

Abstract

Let $\mathbb{N}_0$ be the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ 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, where $f(u)+f(v)$ is the sum set of $f(u)$ and $f(v)$. If $f^+(uv)=k~\forall~uv\in E(G)$, then $f$ is said to be a $k$-uniform IASI. An IASI $f$ is said to be a strong IASI if $|f^+(uv)|=|f(u)|.|f(v)|~\forall ~ uv\in E(G)$. In this paper, we study the characteristics of certain graph classes, graph operations and graph products that admit strong integer additive set-indexers.