On Certain Arithmetic Integer Additive set-indexers of Graphs (original) (raw)

Arithmetic Integer Additive Set-Indexers of Graph Operations

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.

A Study on Arithmetic Integer Additive Set-Indexers of Graphs

arXiv: Combinatorics, 2013

A set-indexer of a graph GGG is an injective set-valued function f:V(G)rightarrow2Xf:V(G) \rightarrow2^{X}f:V(G)rightarrow2X such that the function foplus:E(G)rightarrow2X−emptysetf^{\oplus}:E(G)\rightarrow2^{X}-\{\emptyset\}foplus:E(G)rightarrow2X−emptyset defined by foplus(uv)=f(u)oplusf(v)f^{\oplus}(uv) = f(u){\oplus} f(v)foplus(uv)=f(u)oplusf(v) for every uvinE(G)uv{\in} E(G)uvinE(G) is also injective, where 2X2^{X}2X is the set of all subsets of XXX and oplus\oplusoplus is the symmetric difference of sets. An integer additive set-indexer is defined as an injective function f:V(G)rightarrow2mathbbN_0f:V(G)\rightarrow 2^{\mathbb{N}_0}f:V(G)rightarrow2mathbbN0 such that the induced function f+:E(G)rightarrow2mathbbN0f^+:E(G) \rightarrow 2^{\mathbb{N}_0}f+:E(G)rightarrow2mathbbN_0 defined by f+(uv)=f(u)+f(v)f^+ (uv) = f(u)+ f(v)f+(uv)=f(u)+f(v) is also injective. A graph GGG which admits an IASI is called an IASI graph. An IASI fff is said to be a weak IASI if ∣f+(uv)∣=max(∣f(u)∣,∣f(v)∣)|f^+(uv)|=max(|f(u)|,|f(v)|)∣f+(uv)∣=max(∣f(u)∣,∣f(v)∣) and an IASI fff is said to be a strong IASI if ∣f+(uv)∣=∣f(u)∣∣f(v)∣|f^+(uv)|=|f(u)| |f(v)|∣f+(uv)∣=∣f(u)∣∣f(v)∣ for all u,vinV(G)u,v\in V(G)u,vinV(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...

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

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.

A Study on Semi-arithmetic Integer Additive Set-Indexers of Graphs

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...

On Integer Additive Set-Indexers of Graphs

A set-indexer of a graph GGG is an injective set-valued function f:V(G)rightarrow2Xf:V(G) \rightarrow2^{X}f:V(G)rightarrow2X such that the function foplus:E(G)rightarrow2X−emptysetf^{\oplus}:E(G)\rightarrow2^{X}-\{\emptyset\}foplus:E(G)rightarrow2X−emptyset defined by foplus(uv)=f(u)oplusf(v)f^{\oplus}(uv) = f(u){\oplus} f(v)foplus(uv)=f(u)oplusf(v) for every uvinE(G)uv{\in} E(G)uvinE(G) is also injective, where 2X2^{X}2X is the set of all subsets of XXX and oplus\oplusoplus is the symmetric difference of sets. An integer additive set-indexer is defined as an injective function f:V(G)rightarrow2mathbbN0f:V(G)\rightarrow 2^{\mathbb{N}_0}f:V(G)rightarrow2mathbbN0 such that the induced function gf:E(G)rightarrow2mathbbN0g_f:E(G) \rightarrow 2^{\mathbb{N}_0}gf:E(G)rightarrow2mathbbN0 defined by gf(uv)=f(u)+f(v)g_f (uv) = f(u)+ f(v)gf(uv)=f(u)+f(v) is also injective. A graph GGG which admits an IASI is called an IASI graph. An IASI fff is said to be a {\em weak IASI} if ∣gf(uv)∣=max(∣f(u)∣,∣f(v)∣)|g_f(uv)|=max(|f(u)|,|f(v)|)∣gf(uv)∣=max(∣f(u)∣,∣f(v)∣) and an IASI fff is said to be a {\em strong IASI} if ∣gf(uv)∣=∣f(u)∣∣f(v)∣|g_f(uv)|=|f(u)| |f(v)|∣gf(uv)∣=∣f(u)∣∣f(v)∣ for all u,vinV(G)u,v\in V(G)u,vinV(G). In this paper, we study about inter additive set-indexers.

Associated Graphs of Certain Arithmetic IASI Graphs

An integer additive set-indexer is defined as an injective function f:V(G)rightarrow2mathbbN_0f:V(G)\rightarrow 2^{\mathbb{N}_0}f:V(G)rightarrow2mathbbN0 such that the induced function f+:E(G)rightarrow2mathbbN0f^+:E(G) \rightarrow 2^{\mathbb{N}_0}f+:E(G)rightarrow2mathbbN_0 defined by f+(uv)=f(u)+f(v)f^+ (uv) = f(u)+ f(v)f+(uv)=f(u)+f(v) is also injective. A graph GGG which admits an IASI is called an IASI graph. An arithmetic integer additive set-indexer is an integer additive set-indexer fff, under which the set-labels of all elements of a given graph GGG are arithmetic progressions. In this paper, we discuss about admissibility of arithmetic integer additive set-indexers by certain associated graphs of the given graph GGG, like line graph, total graph, etc.

On Weak Integer Additive Set-Indexers of Certain Graph Classes

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.

Associated Graphs of Arithmetic IASI Graphs

An integer additive set-indexer is defined as an injective function f:V(G)rightarrow2mathbbN0f:V(G)\rightarrow 2^{\mathbb{N}_0}f:V(G)rightarrow2mathbbN0 such that the induced function gf:E(G)rightarrow2mathbbN0g_f:E(G) \rightarrow 2^{\mathbb{N}_0}gf:E(G)rightarrow2mathbbN0 defined by gf(uv)=f(u)+f(v)g_f (uv) = f(u)+ f(v)gf(uv)=f(u)+f(v) is also injective. A graph GGG which admits an IASI is called an IASI graph. An arithmetic integer additive set-indexer is an integer additive set-indexer fff, under which the set-labels of all elements of a given graph GGG are arithmetic progressions. In this paper, we discuss about admissibility of arithmetic integer additive set-indexers by certain associated graphs of the given graph GGG, like line graph, total graph, etc.

On Integer Additive set-Sequential Graphs

arXiv: Combinatorics, 2014

A set-labeling of a graph GGG is an injective function f:V(G)tomathcalP(X)f:V(G)\to \mathcal{P}(X)f:V(G)tomathcalP(X), where XXX is a finite set of non-negative integers and a set-indexer of GGG is a set-labeling such that the induced function foplus:E(G)rightarrowmathcalP(X)−emptysetf^{\oplus}:E(G)\rightarrow \mathcal{P}(X)-\{\emptyset\}foplus:E(G)rightarrowmathcalP(X)−emptyset defined by foplus(uv)=f(u)oplusf(v)f^{\oplus}(uv) = f(u){\oplus}f(v)foplus(uv)=f(u)oplusf(v) for every uvinE(G)uv{\in} E(G)uvinE(G) is also injective. A set-indexer f:V(G)tomathcalP(X)f:V(G)\to \mathcal{P}(X)f:V(G)tomathcalP(X) is called a set-sequential labeling of GGG if foplus(V(G)cupE(G))=mathcalP(X)−emptysetf^{\oplus}(V(G)\cup E(G))=\mathcal{P}(X)-\{\emptyset\}foplus(V(G)cupE(G))=mathcalP(X)−emptyset. A graph GGG which admits a set-sequential labeling is called a set-sequential graph. An integer additive set-labeling is an injective function f:V(G)rightarrowmathcalP(mathbbN_0)f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)f:V(G)rightarrowmathcalP(mathbbN0), mathbbN0\mathbb{N}_0mathbbN0 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)rightarrowmathcalP(mathbbN0)f^+:E(G) \rightarrow \mathcal{P}(\mathbb{N}_0)f+:E(G)rightarrowmathcalP(mathbbN_0) defined by f+(uv)=f(u)+f(v)f^+ (uv) = f(u)+ f(v)f+(uv)=f(u)+f(v) is also injective. In this paper, we extend the concepts of set...