A Note on the Sparing Number 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 IASI $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph $G$, a vertex or an edge, is the cardinality of its set-labels. The sparing number of a graph $G$ is the minimum number of edges with singleton set-labels, required for a graph $G$ to admit a weak IASI. In this paper, we study the sparing number of certain graphs and the relation of sparing number with some other parameters like matching number, chromatic number, covering number, independence number etc.Comment: 10 pages, 10 figures, submitte

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

A study on prime arithmetic integer additive set-indexers of graphs

International audienceLet N 0 be the set of all non-negative integers and P(N 0) be its power set. An integer additive set-indexer (IASI) is defined as an injective function f : V (G) → P(N 0) such that the induced function f + : E(G) → P(N 0) defined by f + (uv) = f (u)+f (v) is also injective, where N 0 is the set of all non-negative integers. A graph G which admits an IASI is called an IASI graph. An IASI of a graph G is said to be an arithmetic IASI if the elements of the set-labels of all vertices and edges of G are in arithmetic progressions. In this paper, we discuss about a particular type of arithmetic IASI called prime arithmetic IASI

A Study on the Nourishing Number of Graphs and Graph Powers

International audienceLet N 0 be the set of all non-negative integers and P(N 0) be its power set. Then, an integer additive set-indexer

Topological Integer Additive Set-Valued Graphs: A Review

International audienceLet í µí±‹ denote a set of non-negative integers and í µí±ƒ 0 (í µí±‹) be the collection of all non-empty subsets of í µí±‹. An integer additive set-labeling (IASL) of a graph í µí°º is an injective set-valued function í µí±“: í µí±‰(í µí°º) → í µí±ƒ 0 (í µí±‹) where induced function í µí±“ + : í µí°¸(í µí°º) → í µí±ƒ 0 (í µí±‹) is defined byí µí±“ + (í µí±¢í µí±£) = í µí±“(í µí±¢) + í µí±“(í µí±£), where í µí±“(í µí±¢) + í µí±“(í µí±£) is the sumset of í µí±“(í µí±¢) and í µí±“(í µí±£). A set-labeling í µí±“: í µí±‰(í µí°º) → í µí±ƒ 0 (í µí±‹) is said to be a topological set-labeling if í µí±“(í µí±‰(í µí°º)) ∪ {∅} is a topology on the ground set í µí±‹ and a set-labeling í µí±“: í µí±‰(í µí°º) → í µí±ƒ 0 (í µí±‹) is said to be a topogenic set-labeling if í µí±“(í µí±‰(í µí°º)) ∪ í µí±“ + (í µí°¸(í µí°º)) ∪ {∅} is a topology on í µí±‹. In this article, we critically review some interesting studies on the properties and characteristics of different topological and topogenic integer additive set-labeling of certain graphs