A Note on the Sparing Number of the Sieve Graphs of Certain 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-indexer (IASI) of a given graph $G$ is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ of a graph $G$ is said to be a weak IASI of $G$ if $|f^+(uv)|=\max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The 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 introduce the notion of $k$-sieve graphs of a given graph and study their sparing numbers.Comment: 9 pages, 3 figures, Publishe

Topological Integer Additive Set-Sequential Graphs

Let $\mathbb{N}_0$ denote the set of all non-negative integers and $X$ be any non-empty subset of $\mathbb{N}_0$. Denote the power set of $X$ by $\mathcal{P}(X)$. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathcal{P}(X)$ such that the induced function $f^+:E(G) \to \mathcal{P}(X)$ is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. If the associated set-valued edge function $f^+$ is also injective, then such an IASL is called an integer additive set-indexer (IASI). An IASL $f$ is said to be a topological IASL (TIASL) if $f(V(G))\cup \{\emptyset\}$ is a topology of the ground set $X$. An IASL is said to be an integer additive set-sequential labeling (IASSL) if $f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}$. An IASL of a given graph $G$ is said to be a topological integer additive set-sequential labeling of $G$, if it is a topological integer additive set-labeling as well as an integer additive set-sequential labeling of $G$. In this paper, we study the conditions required for a graph $G$ to admit this type of IASL and propose some important characteristics of the graphs which admit this type of IASLs.Comment: 10 pages, 2 figures. arXiv admin note: text overlap with arXiv:1506.0124

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

Sumset Valuations of Graphs and Their Applications

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On the Distance Pattern Distinguishing Number of a Graph

Let G=(V,E) be a connected simple graph and let M be a nonempty subset of V. The M-distance pattern of a vertex u in G is the set of all distances from u to the vertices in M. If the distance patterns of all vertices in V are distinct, then the set M is a distance pattern distinguishing set of G. A graph G with a distance pattern distinguishing set is called a distance pattern distinguishing graph. Minimum number of vertices in a distance pattern distinguishing set is called distance pattern distinguishing number of a graph. This paper initiates a study on the problem of finding distance pattern distinguishing number of a graph and gives bounds for distance pattern distinguishing number. Further, this paper provides an algorithm to determine whether a graph is a distance pattern distinguishing graph or not and hence to determine the distance pattern distinguishing number of that graph

Topological Integer Additive Set-Sequential Graphs

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Some new results on integer additive set-valued signed graphs

Let X denotes a set of non-negative integers and P(X) be its power set. An integer additive set-labeling (IASL) of a graph G is an injective set-valued function f : V (G) → P(X) − {∅} such that the induced function f+ : E(G) → P(X) − {∅} is defined by f+(uv) = f(u) + f(v); ∀ uv ∈ E(G), where f(u) + f(v) is the sumset of f(u) and f(v). An IASL of a signed graph is an IASL of its underlying graph G together with the signature σ defined by σ(uv) = (−1)|f+(uv)|; ∀ uv ∈ E(Σ). In this paper, we discuss certain characteristics of the signed graphs which admits certain types of integer additive set-labelings.Publisher's Versio