The Sparing Number of Certain Graph Powers

An integer additive set-indexer is defined as an injective function $f:V(G)\to 2^{{\mathbb{N}}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{$\mathbb{N}$_0}$definedby$g_f (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ is said to be a weak IASI if $\mathrm{\mid }{g}_f(uv)\mathrm{\mid }=max(\mathrm{\mid }f(u)\mathrm{\mid },\mathrm{\mid }f(v)\mathrm{\mid })$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph $G$, a vertex or an edge, is the cardinality of its set-labels. The sparing number of a graph $G$ is the minimum number of edges with singleton set-labels, required for a graph $G$ to admit a weak IASI. In this paper, we study the admissibility of weak IASI by certain graph powers and their sparing numbers.