Some New Results on Integer Additive Set-Valued Signed Graphs

TL;DR

This paper explores properties of signed graphs that admit specific types of integer additive set-labelings, focusing on the mathematical structure and characteristics of such labelings.

Contribution

It introduces new results on integer additive set-labelings for signed graphs, expanding understanding of their structural properties and labeling conditions.

Findings

Characterization of signed graphs with certain IASL properties

Conditions for the existence of specific IASL types in signed graphs

New theoretical insights into set-labeling structures

Abstract

Let $X$ denotes a set of non-negative integers and $\mathscr{P}(X)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathscr{P}(X)-\{\emptyset\}$ such that the induced function $f^+:E(G) \to \mathscr{P}(X)-\{\emptyset\}$ is defined by $f^+(uv)=f(u)+f(v);\ \forall\, uv\in E(G)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. An IASL of a signed graph is an IASL of its underlying graph $G$ together with the signature $\sigma$ defined by $\sigma(uv)=(-1)^{|f^+(uv)|};\ \forall\, uv\in E(\Sigma)$. In this paper, we discuss certain characteristics of the signed graphs which admits certain types of integer additive set-labelings.