Further Studies on the Sparing Number of Graphs

TL;DR

This paper investigates the properties and conditions for weak integer additive set-indexers in graphs, focusing on their admissibility under various graph structures and operations.

Contribution

It introduces the concept of weak integer additive set-indexers and explores their applicability to different graphs and operations, expanding understanding of graph labelings.

Findings

Characterization of graphs admitting weak integer additive set-indexers

Conditions for the existence of weak integer additive set-indexers under graph operations

Identification of classes of graphs with specific set-indexer properties

Abstract

Let $\mathbb{N}_0$ denote the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-indexer is 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 integer additive set-indexer. An integer additive set-indexer $f$ is said to be a weak integer additive set-indexer if $|f^+(uv)|=\max(|f(u)|,|f(v)|)~\forall ~ uv\in E(G)$. In this paper, we study the admissibility of weak integer additive set-indexer by certain graphs and graph operations.