On Certain Arithmetic Integer Additive set-indexers of Graphs

TL;DR

This paper explores specific types of arithmetic integer additive set-indexers (IASIs) in graphs, focusing on their properties and classifications, with potential implications for graph labeling and combinatorial structures.

Contribution

It introduces and analyzes two special types of arithmetic IASIs, expanding the understanding of set-labeling schemes in graph theory.

Findings

Characterization of arithmetic IASIs in graphs

Conditions for the existence of special arithmetic IASIs

Classification of graphs admitting these IASIs

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 (IASI) of a graph $G$ 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 $\mathbb{N}_0$ is the set of all non-negative integers. A graph $G$ which admits an IASI is called an IASI graph. An IASI of a graph $G$ is said to be an arithmetic IASI if the elements of the set-labels of all vertices and edges of $G$ are in arithmetic progressions. In this paper, we discuss about two special types of arithmetic IASIs.