Associated Graphs of Certain Arithmetic IASI Graphs

An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $f^+:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which admits an IASI is called an IASI graph. An arithmetic integer additive set-indexer is an integer additive set-indexer $f$, under which the set-labels of all elements of a given graph $G$ are arithmetic progressions. In this paper, we discuss about admissibility of arithmetic integer additive set-indexers by certain associated graphs of the given graph $G$, like line graph, total graph, etc.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1312.7674, arXiv:1312.767

Weak Integer Additive Set-Indexers of Certain Graph Products

An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. If $g_f(uv)=k \forall uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexers. An integer additive set-indexer $f$ is said to be a weak integer additive set-indexer if $|g_f(uv)|=max(|f(u)|,|f(v)|) \forall uv\in E(G)$. We have some characteristics of the graphs which admit weak integer additive set-indexers. We already have some results on the admissibility of weak integer additive set-indexer by certain graphs and finite graph operations. In this paper, we study further characteristics of certain graph products like cartesian product and corona of two weak IASI graphs and their admissibility of weak integer additive set-indexers and provide some useful results on these types of set-indexers.Comment: 7 pages, arXiv admin note: text overlap with arXiv:1310.6091, arXiv:1311.0345, submitte

A Study on Integer Additive Set-Graceful 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 and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\rightarrow \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. 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 $f^+:E(G) \rightarrow \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of set-graceful labeling to integer additive set-labelings of graphs and provide some results on them.Comment: 11 pages, submitted to JARP

A Characterisation of Weak Integer Additive Set-Indexers of Graphs

An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. An integer additive set-indexer is said to be $k$-uniform if $|g_f(e)| = k$ for all $e\in E(G)$. An integer additive set-indexer $f$ is said to be a weak integer additive set-indexer if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. In this paper, we study the characteristics of certain graphs and graph classes which admit weak integer additive set-indexers.Comment: 12pages, 4 figures, arXiv admin note: text overlap with arXiv:1311.085