On Integer Additive set-Sequential Graphs

A set-labeling of a graph $G$ is an injective function $f:V(G)\to $\mathcal{P}$(X)$,where$X$ is a finite set of non-negative integers and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus }:E(G)\to \mathcal{P}(X)-\{\mathrm{\varnothing }\}$ defined by $f^{\oplus }(uv)=f(u)\oplus f(v)$ for every $uv\in E(G)$ is also injective. A set-indexer $f:V(G)\to \mathcal{P}(X)$ is called a set-sequential labeling of $G$ if $f^{\oplus }(V(G)\cup E(G))=\mathcal{P}(X)-\{\mathrm{\varnothing }\}$. A graph $G$ which admits a set-sequential labeling is called a set-sequential graph. An integer additive set-labeling is an injective function $f:V(G)\rightarrow $\mathcal{P}$($\mathbb{N}$_0)$,$$\mathbb{N}$_0$ is the set of all non-negative integers and an integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:E(G) \rightarrow $\mathcal{P}$($\mathbb{N}$_0)$definedby$f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of set-sequential labeling to integer additive set-labelings of graphs and provide some results on them.