On the Sumset Labeling of Graphs

Let N 0 denote the set of all non-negative integers and P(N 0) be its power set. An integer additive set-labeling (IASL) 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) is defined by f + (uv) = f (u)+f (v), where f (u)+f (v) is the sumset of f (u) and f (v). An IASL f is said to be an integer additive set-indexer (IASI) if the associated edge-function f + is also injective. An IASL f of a given graph G is said to be a weak integer additive set-labeling (WIASL) of G if the cardinality of the set-label of every edge of G is equal to the cardinality of the set-label of at least one end vertex of it. In this paper, we study the admissibility of weak integer additive set-labeling by different graphs.

Discrete Mathematics, Algorithms and Applications, 2015

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.

Carpathian Mathematical Publications, 2015

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 set-valued function $f:V(G)\to\mathcal{P}(\mathbb{N}_0)\setminus\{\emptyset\}$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)\setminus \{\emptyset\}$ is defined by $f^+(uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. A graph which has an IASL is usually called an IASL-graph. An IASL $f$ of a graph $G$ is said to be an integer additive set-indexer (IASI) of $G$ if the associated function $f^+$ is also injective. In this paper, we define the notion of integer additive set-labeling of signed graphs and discuss certain properties of signed graphs which admits certain types of integer additive set-labelings.

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.

International Journal of Computer Applications

Let N 0 be the set of all non-negative integers, let X ⊂ N 0 and P(X) be the the power set of X. An integer additive set-labeling (IASL) 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) is defined by f + (uv) = f (u)+f (v), where f (u)+f (v) is the sum set of f (u) and f (v). An IASL f is said to be an integer additive set-indexer (IASI) of a graph G if the induced edge function f + is also injective. An integer additive set-labeling f is said to be a weak integer additive set-labeling (WIASL) if |f + (uv)| = max(|f (u)|, |f (v)|) ∀ uv ∈ E(G). The minimum cardinality of the ground set X required for a given graph G to admit an IASL is called the set-labeling number of the graph. In this paper, we introduce the notion of the weak set-labeling number of a graph G as the minimum cardinality of X so that G admits a WIASL with respect to the ground set X and discuss the weak set-labeling number of certain graphs. Keywords: Intege...

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...

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.

Electronic Journal of Graph Theory and Applications, 2015

A set-labeling of a graph G is an injective function f : V (G) → P(X) such that the induced function f ⊕ : E(G) → P(X) − {∅} defined by f ⊕ (uv) = f (u)⊕f (v) for every uv∈E(G), where X is a non-empty finite set and P(X) be its power set. A set-indexer of G is a set-labeling such that the induced function f ⊕ is also injective. A set-indexer f : V (G) → P(X) of a given graph G is called a topological set-labeling of G if f (V (G)) is a topology of X. An integer additive set-labeling is an injective function f : V (G) → P(N 0 ), whose associated function

2016

An integer additive set-indexer is defined as an injective function f: V (G) → 2N0 such that the induced function gf: E(G) → 2N0 defined by gf (uv) = f(u) + f(v) is also injective. An integer additive set-indexer f is said to be an arithmetic integer additive set-indexer if every element of G are labeled by non-empty sets of non negative integers, which are in arithmetic progressions. An integer additive set-indexer f is said to be a semi-arithmetic integer additive set-indexer if vertices of G are labeled by non-empty sets of non negative inte-gers, which are in arithmetic progressions, but edges are not labeled by non-empty sets of non negative integers, which are in arithmetic progressions. In this paper, we discuss about semi-arithmetic integer additive set-indexer and establish some results on this type of integer additive set-indexers. Key words: Integer additive set-indexers, uniform integer additive set-indexers, arithmetic integer additive set-indexers, semi-arithmetic inte...

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.