The Exquisite Integer Additive Set-Labeling of Graphs

Let $\mathbb{N}_0$ denote the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. In this paper, we introduce the notion of a particular type of integer additive set-indexers called integer additive set-filtered labeling of given graphs and study their characteristics.

Journal of Fuzzy Set Valued Analysis, 2014

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, k-uniform integer additive set-indexers, weak and strong integer additive set-indexers, weakly and strongly uniform integer additive set-indexers, completely uniform integer additive set-indexers and arbitrarily uniform integer additive set-indexers based on the cardinality of the labeling sets. In this paper, we study further characteristics of the graphs which admit weak integer additive set-indexers as extensions to our earlier studies and provide some useful results on these types of set-indexers.

Discrete Mathematics, Algorithms and Applications, 2014

Let N 0 be the set of all non-negative integers. An integer additive setindexer of a graph G is an injective function f :

Journal of Discrete Mathematical Sciences and Cryptography, 2015

Let N 0 denote the set of all non-negative integers and P(N 0) be its power set. An integer additive set-indexer (IASI) of a graph G is 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. An IASI f is said to be a weak IASI if |f + (uv)| = max(|f (u)|, |f (v)|) for all adjacent vertices u, v ∈ V (G). The sparing number of a weak IASI graph G is the minimum number of edges in G with singleton set-labels. In this paper, we study the admissibility of weak integer additive set-indexers by certain graph classes and associated graphs of given graphs.

Journal of Advanced Research in Pure Mathematics

Let N 0 be the set of non-negative integers. An integer additive set-indexer (IASI) of a graph G is an injective function f : V (G) → 2 N0 such that the induced function f + : E(G) → 2 N0 defined by f + (uv) = f (u) + f (v); uv ∈ E(G), is also injective, where f (u) + f (v) is the sum set of the sets f (u) and f (v). A graph G which admits an IASI is called an IASI graph. An arithmetic IASI is an IASI f , under which the elements of the set-labels of all vertices and edges of a given graph G are in arithmetic progressions. In this paper, we discuss about admissibility of arithmetic IASIs by certain operations and products of graphs.

arXiv: Combinatorics, 2014

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 $f^{\oplus}:E(G)\rightarrow \mathcal{P}(X)-\{\emptyset\}$ 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 $f^{\oplus}(V(G)\cup E(G))=\mathcal{P}(X)-\{\emptyset\}$. 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 $f^+:E(G) \rightarrow \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of set...

Proyecciones (Antofagasta)

Let 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.

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 sum set of $f(u)$ and $f(v)$ and $\mathbb{N}_0$ is the set of all non-negative integers. 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. In this paper, we study the admissibility of weak integer additive set-indexer by certain finite graph operations.

2013

We have the notion of set-indexers, integer additive set-indexers and k-uniform integer additive set-indexers of graphs. In this paper, we initiate a study of the graphs which admit k-uniform integer additive set-indexers and introduce the notion of weakly uniform integer additive set-indexers and arbitrarily uniform integer additive set-indexers and provide some useful results on these types of set-indexers.

International Journal of Scientific and Engineering Research

For a non-empty ground set $X$, finite or infinite, the {\em set-valuation} or {\em set-labeling} of a given graph $G$ is an injective function $f:V(G) \to \mathcal{P}(X)$, where $\mathcal{P}(X)$ is the power set of the set $X$. A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \to \mathcal{P}(X)$ such that the function $f^{\ast}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\ast}(uv) = f(u){\ast} f(v)$ for every $uv{\in} E(G)$ is also injective, where $\ast$ is a binary operation on sets. An integer additive set-indexer 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. In this paper, we critically and creatively review the concepts and properties of integer additive set-valued graphs.