A Characterisation of Strong Integer Additive Set-Indexers of Graphs

TL;DR

This paper explores the properties of specific graph classes, operations, and products that allow for strong integer additive set-indexers, extending previous characterizations of such graphs.

Contribution

It provides new insights into which graph classes and operations admit strong integer additive set-indexers, expanding the understanding of these graph labelings.

Findings

Characterizes graph classes admitting strong integer additive set-indexers

Analyzes the effect of graph operations on the existence of such indexers

Identifies graph products that preserve strong integer additive set-indexers

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, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. If $g_f(uv)=k~\forall~uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexers. An integer additive set-indexer $f$ is said to be a strong integer additive set-indexer if $|g_f(uv)|=|f(u)|.|f(v)|~\forall ~ uv\in E(G)$. We already have some characteristics of the graphs which admit strong integer additive set-indexers. In this paper, we study the characteristics of certain graph classes, graph operations and graph products that admit strong integer additive set-indexers.