Strong Integer Additive Set-valued Graphs: A Creative Review

For a non-empty ground set $X$, finite or infinite, the {\em set-valuation} or {\em set-labeling} of a given graph $G$ is an injective function $f:V(G) \to $\mathcal{P}$(X)$,where$$\mathcal{P}$(X)$isthepowersetoftheset$X$. A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \to $\mathcal{P}$(X)$suchthatthefunction$f^{\ast}:E(G)\to $\mathcal{P}$(X)-{\emptyset}$definedby$f^{\ast}(uv) = f(u){\ast} f(v)$ for every $uv\in E(G)$ is also injective., where $\ast$ is a binary operation on sets. An integer additive set-indexer is defined as an injective function $f:V(G)\to \mathcal{P}({\mathbb{N}}_0)$ such that the induced function $g_f:E(G)\to \mathcal{P}({\mathbb{N}}_0)$ defined by $g_f(uv)=f(u)+f(v)$ is also injective, where ${\mathbb{N}}_0$ is the set of all non-negative integers and $\mathcal{P}({\mathbb{N}}_0)$ is its power set. An IASI $f$ is said to be a strong IASI if $\mathrm{\mid }{f}^{+}(uv)\mathrm{\mid }=\mathrm{\mid }f(u)\mathrm{\mid }\mathrm{\mid }f(v)\mathrm{\mid }$ for every pair of adjacent vertices $u,v$ in $G$. In this paper, we critically and creatively review the concepts and properties of strong integer additive set-valued graphs.