A Study on Integer Additive Set-Valuations of Signed Graphs

TL;DR

This paper explores integer additive set-labelings of signed graphs, introducing new definitions and analyzing properties of graphs that admit such labelings, expanding the understanding of set-labeling in signed graph theory.

Contribution

It defines integer additive set-labelings for signed graphs and investigates properties of graphs that admit these labelings, providing new theoretical insights.

Findings

Characterization of signed graphs with integer additive set-labelings

Conditions for the existence of integer additive set-indexers in signed graphs

Properties of signed graphs admitting specific set-labelings

Abstract

Let $\N$ denote the set of all non-negative integers and $\cP(\N)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \cP(\N)-\{\emptyset\}$ such that the induced function $f^+:E(G) \to \cP(\N)-\{\emptyset\}$ is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. A graph which admits an IASL is usually called an IASL-graph. An IASL $f$ of a graph $G$ is said to be an integer additive set-indexer (IASI) of $G$ if the associated function $f^+$ is also injective. In this paper, we define the notion of integer additive set-labeling of signed graphs and discuss certain properties of signed graphs which admits certain types of integer additive set-labelings.