The Sparing Number of Certain Graph Powers

TL;DR

This paper investigates the conditions under which certain graph powers admit weak integer additive set-indexers and determines the minimum number of singleton-labeled edges needed, focusing on the sparing number.

Contribution

It introduces the concept of weak IASI for graph powers and analyzes their sparing numbers, providing new insights into set-labeling properties of graph powers.

Findings

Determined the sparing numbers for specific graph powers.

Established conditions for the existence of weak IASI in graph powers.

Provided bounds and exact values for sparing numbers in various cases.

Abstract

An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ 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.