A Study on Prime Arithmetic Integer Additive Set-Indexers of Graphs

TL;DR

This paper investigates prime arithmetic integer additive set-indexers (IASIs) of graphs, focusing on their properties and conditions for graphs to admit such labelings, extending the concept of arithmetic IASIs with prime-related constraints.

Contribution

It introduces and studies prime arithmetic IASIs, a novel class of IASIs where set-labels follow prime-related arithmetic progressions, expanding the theory of graph labelings.

Findings

Characterization of graphs admitting prime arithmetic IASIs

Conditions under which prime arithmetic IASIs exist

Extension of arithmetic IASI concepts with prime number constraints

Abstract

Let $\mathbb{N}_0$ be the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-indexer (IASI) is defined as 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 a particular type of arithmetic IASI called prime arithmetic IASI.