Abstract
Let \(\mathbb{N}_{0}\) be the set of all non-negative integers and \(\mathcal{P}(\mathbb{N}_{0})\) be its power set. Then, an integer additive set-indexer (IASI) of a given graph \(G\) is defined as an injective function \(f:V(G)\to \mathcal{P}(\mathbb{N}_{0})\) such that the induced edge-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 sumset of \(f(u)\) and \(f(v)\). An IASI \(f\) of \(G\) is said to be a strong IASI of \(G\) if \(|f^+(uv)|=|f(u)|\,|f(v)|\) for all \(uv\in E(G)\). The nourishing number of a graph \(G\) is the minimum order of the maximal complete subgraph of \(G\) so that \(G\) admits a strong IASI. In this paper, we study the characteristics of certain graph classes and graph powers that admit strong integer additive set-indexers and determine their corresponding nourishing numbers.
Highlights
For all terms and definitions, not defined in this paper, we refer to [1,2,3,4]
The following theorem on graph powers is the most relevant result in our present discussion
We have established some results on the nourishing number of certain graphs and graph powers
Summary
For all terms and definitions, not defined in this paper, we refer to [1,2,3,4]. A complete graph Kn admits a strong IASI if and only if the difference sets of the set-labels of all vertices of Kn are pairwise disjoint. [1] The r-th power of a simple graph G is the graph Gr whose vertex set is V , two distinct vertices being adjacent in Gr if and only if their distance in G is at most r.
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