Integer Additive Set-Indexers Of Graphs Research Articles

A Study on Arithmetic Integer Additive Set-Indexers of Graphs

Let $\N$ be the set of all non-negative integers and $\cP(\N)$ be its power set. An integer additive set-indexer (IASI) of a graph $G$ is an injective function $f:V(G)\to \cP(\N)$ such that the induced function $f^+:E(G) \to \cP(\N)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which admits an IASI is called an IASI-graph. An IASI $f$ is said to be a {\em weak IASI} if $|f^+(uv)|=\max(|f(u)|,|f(v)|)$ and an IASI $f$ is said to be a {\em strong IASI} if $|f^+(uv)|=|f(u)|\,|f(v)|$ for all $uv\in E(G)$. In this paper, we introduce the notion of arithmetic integer additive set-indexers of a given graph $G$ as an IASI with respect to which all elements of $G$ have arithmetic progressions as their set-labels and study the characteristics of this type of IASIs.

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

Let $\mathbb{N}_0$ be the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+(uv) = f(u)+ f(v)$ is also injective, where $\mathbb{N}_0$ 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.

On certain arithmetic integer additive set-indexers of graphs

Let ℕ0 denote the set of all non-negative integers and 𝒫(ℕ0) be its power set. An integer additive set-indexer (IASI) of a graph G is an injective function f : V(G) → 𝒫(ℕ0) such that the induced function f+ : E(G) → 𝒫(ℕ0) defined by f+ (uv) = f(u) + f(v) is also injective. A graph G which admits an IASI is called an integer additive set-indexed graph (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 two special types of arithmetic IASIs.

Some new results on strong integer additive set-indexers of graphs

Let ℕ0 be the set of all non-negative integers. An integer additive set-indexer (IASI) of a graph G is an injective function f : V(G) → 2ℕ0 such that the induced function gf : E(G) → 2ℕ0 defined by f+(uv) = f(u) + f(v) is also injective. An IASI is said to be k-uniform if |f+(e)| = k for all e ∈ E(G). In this paper, we introduce the notions of strong IASIs and initiate a study of the graphs which admit strong IASIs.

A Study on Arithmetic Integer Additive Set-Indexers of Graphs

A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \rightarrow2^{X}$ such that the function $f^{\oplus}:E(G)\rightarrow2^{X}-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus} f(v)$ for every $uv{\in} E(G)$ is also injective, where $2^{X}$ is the set of all subsets of $X$ and $\oplus$ is the symmetric difference of sets. 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 IASI $f$ is said to be a weak IASI if $|f^+(uv)|=max(|f(u)|,|f(v)|)$ and an IASI $f$ is said to be a strong IASI if $|f^+(uv)|=|f(u)| |f(v)|$ for all $u,v\in V(G)$. In this paper, we discuss about a special type of integer additive set-indexers called arithmetic integer additive set-indexer and establish some results on this type of integer additive set-indexers. We also check the admissibility of arithmetic integer additive set-indexer by certain graphs associated with a given graph.

Erratum to A Characterisation of Weak Integer Additive Set-Indexers of Graphs [Journal of Fuzzy Set Valued Analysis 2014 (2014) 1-7

In the recently published article titled A Characterisation of Weak Integer Additive Set-Indexers of Graphs (article id:jfsva-00189) an inadvertent error in the final simplification has been occured.

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.

On weakly uniform integer additive set-indexers of graphs

Acharya introduced the notion of set-valuations of graphs as a set analogue of the number valuations of graphs. Also we have the notion of set-indexers, integer additive set-indexers and k-uniform integer additive set-indexers. In this paper, we initiate a study of the graphs which admit k-uniform integer additive set-indexers as a generalisation to the earlier studies of 1-uniform and 2-uniform integer additive set-indexers. We introduce the notion of weakly uniform integer additive set-indexers and arbitrarily uniform integer additive set-indexers based on the cardinality of the labeling sets and provide some useful results on these types of set-indexers.