The sparing number of certain graph powers

Abstract

Let ℕ0 be the set of all non-negative integers and 𝒫(ℕ0) be its power set. Then, an integer additive set-indexer (IASI) of a given graph G is an injective function f : V(G) → P(ℕ0) such that the induced function f+ : E(G) → 𝒫(ℕ0) defined by f+(uv) = f(u) + f(v) is also injective. An IASI f is said to be a weak IASI if |f+(uv)| = max(|f(u)|, |f(v)|) for all u, v ∈ 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 corresponding sparing numbers.

Keywords

• graph powers
• integer additive set-indexers
• weak integer additive set-indexers
• mono-indexed elements of a graph
• sparing number of a graph
• 05c78