Some New Results on Integer Additive Set-Valued Signed Graphs

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.