Some New Results on Integer Additive Set-Valued Signed Graphs

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