A Study on the Nourishing Number of Graphs and Graph Powers

TL;DR

This paper investigates the properties of specific graph classes and their powers that support strong integer additive set-indexers, a type of graph labeling with injective sumset functions.

Contribution

It introduces the concept of strong integer additive set-indexers and explores which graphs and their powers admit such labelings, expanding understanding of graph labelings.

Findings

Characterization of graphs admitting strong integer additive set-indexers

Conditions under which graph powers support these labelings

Identification of graph classes with this property

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-indexer. An integer additive set-indexer $f$ is said to be a strong integer additive set-indexer if $|g_f(uv)|=|f(u)|.|f(v)|~\forall ~ uv\in E(G)$. In this paper, we study the characteristics of certain graph classes and graph powers that admit strong integer additive set-indexers.