249 results on '"Germina, K. A."'
Search Results
2. Mycielskian of Signed Graphs
- Author
-
Mathew, Albin and A., Germina K.
- Subjects
Mathematics - Combinatorics - Abstract
In this paper, we define the Mycielskian of a signed graph and discuss the properties of balance and switching in the Mycielskian of a given signed graph. We provide a condition for ensuring the Mycielskian of a balanced signed graph remains balanced, leading to the construction of a balanced Mycielskian. We establish a relation between the chromatic numbers of a signed graph and its Mycielskian. We also study the structure of different matrices related to the Mycielskian of a signed graph.
- Published
- 2023
3. Vector Valued Switching in Signed Graphs
- Author
-
K, Shahul Hameed, Mathew, Albin, A, Germina K, and Zaslavsky, Thomas
- Subjects
Mathematics - Combinatorics ,05C22 - Abstract
A signed graph is a graph with edges marked positive and negative; it is unbalanced if some cycle has negative sign product. We introduce the concept of vector valued switching function in signed graphs, which extends the concept of switching to higher dimensions. Using this concept, we define balancing dimension and strong balancing dimension for a signed graph, which can be used for a new classification of degree of imbalance of unbalanced signed graphs. We provide bounds for the balancing and strong balancing dimensions, and calculate these dimensions for some classes of signed graphs., Comment: 15 pp., 5 figs. V2 12 pp., 4 figs., more data, slightly shorter. V3 typos fixed
- Published
- 2022
4. Signed Distance Laplacian Matrices for Signed Graphs
- Author
-
Roy, Roshni T, Germina, K A, Hameed, K Shahul, and Zaslavsky, Thomas
- Subjects
Mathematics - Combinatorics ,05C12 (Primary), 05C22, 05C50, 05C75 (Secondary) - Abstract
A signed graph is a graph whose edges are labeled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize balance in signed graphs using these matrices and find signed distance laplacian spectra of some classes of unbalanced signed graphs., Comment: 12 pp
- Published
- 2020
5. On the Powers of Signed Graphs
- Author
-
T V, Shijin, A, Germina K, and K, Shahul Hameed
- Subjects
Mathematics - Combinatorics ,05C12, 05C22, 05C50 - Abstract
A signed graph is an ordered pair $\Sigma=(G,\sigma),$ where $G=(V,E)$ is the underlying graph of $\Sigma$ with a signature function $\sigma:E\rightarrow \{1,-1\}$. In this article, we define $n^{th}$ power of a signed graph and discuss some properties of these powers of signed graphs. As we can define two types of signed graphs as the power of a signed graph, necessary and sufficient conditions are given for an $n^{th}$ power of a signed graph to be unique. Also, we characterize balanced power signed graphs., Comment: 10 pages, 1 figure
- Published
- 2020
6. On Two Laplacian Matrices for Skew Gain Graphs
- Author
-
Roy, Roshni T, K, Shahul Hameed, and A, Germina K
- Subjects
Mathematics - Combinatorics ,05C22, 05C50, 05C76 - Abstract
Let $G=(V,\overrightarrow{E})$ be a graph with some prescribed orientation for the edges and $\Gamma$ be an arbitrary group. If $f\in \mathrm{Inv}(\Gamma)$ be an anti-involution then the skew gain graph $\Phi_f=(G,\Gamma,\varphi,f)$ is such that the skew gain function $\varphi:\overrightarrow{E}\rightarrow \Gamma$ satisfies $\varphi(\overrightarrow{vu})=f(\varphi(\overrightarrow{uv}))$. In this paper, we study two different types, Laplacian and $g$-Laplacian matrices for a skew gain graph where the skew gains are taken from the multiplicative group $F^\times$ of a field $F$ of characteristic zero. Defining incidence matrix, we also prove the matrix tree theorem for skew gain graphs in the case of the $g$-Laplacian matrix., Comment: 15 pages
- Published
- 2020
7. On Signed Distance in Product of Signed Graphs
- Author
-
Shijin, T. V., Soorya, P., Hameed, K. Shahul, and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C12, 05C22, 05C50, 05C76 - Abstract
A signed graph is a graph in which each edge has a positive or negative sign. In this article, first we characterize the distance compatibility in the case of a connected signed graph and discussed the distance compatibility criterion for the cartesian product, lexicographic product and tensor product of signed graphs. We also deal with the distance matrix of the cartesian product, lexicographic product and tensor product of signed graphs in terms of the distance matrix of the factor graphs., Comment: 21 pages
- Published
- 2020
8. On the Characteristic Polynomial of Skew Gain Graphs
- Author
-
Hameed, K. Shahul, Roy, Roshni T, Soorya, P., and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C22, 05C50, 05C76 - Abstract
Gain graphs are graphs where the edges are given some orientation and labeled with the elements (called gains) from a group so that gains are inverted when we reverse the direction of the edges. Generalizing the notion of gain graphs, skew gain graphs have the property that the gain of a reversed edge is the image of edge gain under an anti-involution. In this paper, we deal with the adjacency matrix of skew gain graphs with involutive automorphism on a field of characteristic zero and their charactersitic polynomials. Spectra of some particular skew gain graphs are also discussed. Meanwhile it is interesting to note that weighted graphs are particular cases of skew gain graphs., Comment: 13 pages
- Published
- 2020
9. Signed Distance in Signed Graphs
- Author
-
K, Shahul Hameed, T V, Shijin, P, Soorya, A, Germina K, and Zaslavsky, Thomas
- Subjects
Mathematics - Combinatorics ,Primary 05C12, Secondary 05C22, 05C50, 05C75 - Abstract
Signed graphs have their edges labeled either as positive or negative. Here we introduce two types of signed distance matrix for signed graphs. We characterize balance in signed graphs using these matrices and we obtain explicit formulae for the distance spectrum of some unbalanced signed graphs. We also introduce the notion of distance-compatible signed graphs and partially characterize it., Comment: v1: 11 pp. v2: 16 pp with more eigenvalue results and 2 figures
- Published
- 2020
10. On Certain Colouring Parameters of Mycielski Graphs of Some Graphs
- Author
-
Chithra, K. P., Sudev, N. K., Satheesh, S., Germina, K. A., and Kok, Johan
- Subjects
Mathematics - General Mathematics ,05C15, 05C75, 62A01 - Abstract
Colouring the vertices of a graph $G$ according to certain conditions can be considered as a random experiment and a discrete random variable $X$ can be defined as the number of vertices having a particular colour in the proper colouring of $G$. The concepts of mean and variance, two important statistical measures, have also been introduced to the theory of graph colouring and determined the values of these parameters for a number of standard graphs. In this paper, we discuss the colouring parameters of the Mycielskian of certain standard graphs., Comment: 18 Pages, 2 Figures, Discrete Mathematics Algorithms and Applications, 2017
- Published
- 2017
11. Vector valued switching in the products of signed graphs.
- Author
-
Mathew, Albin and Germina, K. A.
- Subjects
- *
GRAPH theory , *LEXICOGRAPHY , *EDGES (Geometry) , *MATHEMATICAL bounds , *MATHEMATICAL models - Abstract
A signed graph is a graph whose edges are labeled either as positive or negative. The concepts of vector valued switching and balancing dimension of signed graphs were introduced by S. Hameed et al. In this paper, we deal with the balancing dimension of various products of signed graphs, namely the Cartesian product, the lexicographic product, the tensor product, and the strong product. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
12. Friendship-Like Graphs and It’s Classiffication
- Author
-
Nageswara Rao, K., Shaini, P., Germina, K. A., Deo, Naokant, editor, Gupta, Vijay, editor, Acu, Ana Maria, editor, and Agrawal, P. N., editor
- Published
- 2020
- Full Text
- View/download PDF
13. On α-Vertex Choosability of Graphs
- Author
-
Soorya, P., Germina, K. Augusthy, and Sudev, Naduvath
- Published
- 2021
- Full Text
- View/download PDF
14. A Study on Set-Valuations of Signed Graphs
- Author
-
Ashraf, P. K., Germina, K. A., and Sudev, N. K.
- Subjects
Mathematics - General Mathematics ,05C78, 05C22 - Abstract
Let $X$ be a non-empty ground set and $\mathcal{P}(X)$ be its power set. A set-labeling (or a set-valuation) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathcal{P}(X)$ such that the induced function $f^\oplus:E(G) \to \mathcal{P}(X)$ is defined by $f^\oplus(uv) = f(u)\oplus f(v)$, where $f(u)\oplus f(v)$ is the symmetric difference of the sets $f(u)$ and $f(v)$. A graph which admits a set-labeling is known to be a set-labeled graph. A set-labeling $f$ of a graph $G$ is said to be a set-indexer of $G$ if the associated function $f^\oplus$ is also injective. In this paper, we define the notion of set-valuations of signed graphs and discuss certain properties of signed graphs which admits certain types of set-valuations., Comment: 8 pages, submitted to Carpathian Mathematical Publications. arXiv admin note: text overlap with arXiv:1609.00295
- Published
- 2016
15. Some New Results on Integer Additive Set-Valued Signed Graphs
- Author
-
Sudev, N. K., Ashraf, P. K., and Germina, K. A.
- Subjects
Mathematics - General Mathematics ,05C78, 05C22 - Abstract
Let $X$ denotes a set of non-negative integers and $\mathscr{P}(X)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathscr{P}(X)-\{\emptyset\}$ such that the induced function $f^+:E(G) \to \mathscr{P}(X)-\{\emptyset\}$ is defined by $f^+(uv)=f(u)+f(v);\ \forall\, uv\in E(G)$, 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 $\sigma$ defined by $\sigma(uv)=(-1)^{|f^+(uv)|};\ \forall\, uv\in E(\Sigma)$. In this paper, we discuss certain characteristics of the signed graphs which admits certain types of integer additive set-labelings., Comment: 9 pages, Submitted to European J Pure. Appl. Math. arXiv admin note: text overlap with arXiv:1511.00678
- Published
- 2016
16. On the Vertex In-Degrees of Certain Jaco-Type Graphs
- Author
-
Kok, Johan, Sudev, N. K., Chithra, K. P., Germina, K. A., and Mary, U.
- Subjects
Mathematics - General Mathematics ,05C07, 05C38, 05C75, 05C85 - Abstract
The concepts of linear Jaco graphs and Jaco-type graphs have been introduced as certain types of directed graphs with specifically defined adjacency conditions. The distinct difference between a pure Jaco graph and a Jaco-type graph is that for a pure Jaco graph, the total vertex degree $d(v)$ is well-defined, while for a Jaco-type graph the vertex out-degree $d^+(v)$ is well-defined. Hence, in the case of pure Jaco graphs a challenge is to determine $d^-(v)$ and $d^+(v)$ respectively and for Jaco-type graphs a challenge is to determine $d^-(v)$. In this paper, the vertex in-degrees for Fibonaccian and modular Jaco-type graphs are determined., Comment: 11 Pages, 2 Figures, Southeast Asian Bulletin of Mathematics, 2016
- Published
- 2016
17. A Note on Sparing Number Algorithm of Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - General Mathematics ,05C78 - Abstract
Let $X$ denote a set of all non-negative integers and $\sP(X)$ be its power set. A weak integer additive set-labeling (WIASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \sP(X)-\{\emptyset\}$ where induced function $f^+:E(G) \to \sP(X)-\{\emptyset\}$ is defined by $f^+ (uv) = f(u)+ f(v)$ such that either $|f^+ (uv)|=|f(u)|$ or $|f^+ (uv)|=|f(v)|$ , where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. The sparing number of a WIASL-graph $G$ is the minimum required number of edges in $G$ having singleton set-labels. In this paper, we discuss an algorithm for finding the sparing number of arbitrary graphs., Comment: 6 pages, 2 figures, submitted
- Published
- 2015
18. A Study on Integer Additive Set-Valuations of Signed Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - General Mathematics ,05C78, 05C22 - Abstract
Let $\N$ denote the set of all non-negative integers and $\cP(\N)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \cP(\N)-\{\emptyset\}$ such that the induced function $f^+:E(G) \to \cP(\N)-\{\emptyset\}$ 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 admits 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., Comment: 12 pages, Carpathian Mathematical Publications, Vol. 8, Issue 2, 2015, 12 pages
- Published
- 2015
19. On Integer Additive Set-Filtered Graphs
- Author
-
Sudev, N. K., Chithra, K. P., and Germina, K. A.
- Subjects
Mathematics - General Mathematics ,05C78 - Abstract
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., Comment: 12 Pages, 4 figures, Submitted
- Published
- 2015
20. Topological Integer Additive Set-Graceful Graphs
- Author
-
Sudev, N. K., Germina, K. A., and Chithra, K. P.
- Subjects
Mathematics - General Mathematics ,05C78 - Abstract
Let $\mathbb{N}_0$ denote the set of all non-negative integers and $X$ be any subset of $X$. Also denote the power set of $X$ by $\mathcal{P}(X)$. An integer additive set-labeling (IASL) of a graph $G$ is an injective 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 $f(v)$. An IASL $f$ is said to be a topological IASL (Top-IASL) if $f(V(G))\cup \{\emptyset\}$ is a topology of the ground set $X$. An IASL is said to be an integer additive set-graceful labeling (IASGL) if for the induced edge-function $f^+$, $f^+(E(G))= \mathcal{P}(X)-\{\emptyset, \{0\}\}$. In this paper, we study certain types of IASL of a given graph $G$, which is a topological integer additive set-labeling as well as an integer additive set-graceful labeling of $G$., Comment: 9 pages, 2 figure, submitted
- Published
- 2015
- Full Text
- View/download PDF
21. On the Sparing Number of the Edge-Corona of Graphs
- Author
-
Chithra, K. P., Germina, K. A., and Sudev, N. K.
- Subjects
Mathematics - General Mathematics ,05C78 - Abstract
Let $\mathbb{N}_0$ be the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its the power set. An integer additive set-indexer (IASI) of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective, where $f(u)+f(v)$ is the sum set of $f(u)$ and $f(v)$. An integer additive set-indexer $f$ is said to be a weak integer additive set-indexer (weak IASI) if $|f^+(uv)|=\max(|f(u)|,|f(v)|)~\forall ~ uv\in E(G)$. The minimum number of singleton set-labeled edges required for the graph $G$ to admit an IASI is called the sparing number of the graph. In this paper, we discuss the admissibility of weak IASI by a particular type of graph product called the edge corona of two given graphs and determine the sparing number of the edge corona of certain graphs., Comment: 10 pages, 1 figure, published. arXiv admin note: text overlap with arXiv:1407.5092
- Published
- 2015
- Full Text
- View/download PDF
22. Strong Integer Additive Set-valued Graphs: A Creative Review
- Author
-
Sudev, N. K., Germina, K. A., and Chithra, K. P.
- Subjects
Mathematics - General Mathematics ,05C78 - Abstract
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 $g_f:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective, where $\mathbb{N}_0$ is the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ is its power set. An IASI $f$ is said to be a strong IASI if $|f^+(uv)|=|f(u)|\,|f(v)|$ for every pair of adjacent vertices $u,v$ in $G$. In this paper, we critically and creatively review the concepts and properties of strong integer additive set-valued graphs., Comment: 13 pages, Published. arXiv admin note: text overlap with arXiv:1407.4677, arXiv:1405.4788, arXiv:1310.6266
- Published
- 2015
- Full Text
- View/download PDF
23. Weak Set-Labeling Number of Certain IASL-Graphs
- Author
-
Sudev, N. K., Chithra, K. P., and Germina, K. A.
- Subjects
Mathematics - General Mathematics ,05C78 - Abstract
Let $\mathbb{N}_0$ be the set of all non-negative integers, let $X\subset \mathbb{N}_0$ and $\mathcal{P}(X)$ be the the power set of $X$. An integer additive set-labeling (IASL) of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ 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 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 $|f^+(uv)|=\max(|f(u)|,|f(v)|)~\forall ~ uv\in E(G)$. 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., Comment: 8 figures, Published
- Published
- 2015
- Full Text
- View/download PDF
24. Further Studies on the Sparing Number of Graphs
- Author
-
Sudev, N K and Germina, K A
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 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, where $f(u)+f(v)$ is the sum set of $f(u)$ and $f(v)$. If $f^+(uv)=k~\forall~uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexer. An integer additive set-indexer $f$ is said to be a weak integer additive set-indexer if $|f^+(uv)|=\max(|f(u)|,|f(v)|)~\forall ~ uv\in E(G)$. In this paper, we study the admissibility of weak integer additive set-indexer by certain graphs and graph operations., Comment: 10 Pages, Submitted. arXiv admin note: substantial text overlap with arXiv:1310.6091
- Published
- 2014
25. A Creative Review on Integer Additive Set-Valued Graphs
- Author
-
Sudev, N. K., Germina, K. A., and Chithra, K. P.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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., Comment: 14 pages, submitted. arXiv admin note: text overlap with arXiv:1312.7672, arXiv:1312.7674
- Published
- 2014
26. The Sparing Number of the Cartesian Products of Certain Graphs
- Author
-
Chithra, K. P., Germina, K. A., and Sudev, N. K.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
Let $\mathbb{N}_0$ be the set of all non-negative integers. An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)$ 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, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$ and $\mathcal{P}(\mathbb{N}_0)$ is the power set of $\mathbb{N}_0$. If $f^+(uv)=k \forall ~ uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexer. An integer additive set-indexer $f$ is said to be a weak integer additive set-indexer if $|f^+(uv)|=max(|f(u)|,|f(v)|) \forall ~ uv\in E(G)$. In this paper, we study about the sparing number of the cartesian product of two graphs., Comment: 8 pages, published. arXiv admin note: substantial text overlap with arXiv:1311.0858
- Published
- 2014
27. A Study on the Sparing Number of the Corona of Certain Graphs
- Author
-
Chithra, K. P., Germina, K. A., and Sudev, N. K.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)$ 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, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$ and $\mathcal{P}(\mathbb{N}_0)$ is the power set of $\mathbb{N}_0$. If $f^+(uv)=k \forall uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexer. An integer additive set-indexer $f$ is said to be a weak integer additive set-indexer if $|f^+(uv)|=max(|f(u)|,|f(v)|) \forall ~ uv\in E(G)$. We have some characteristics of the graphs which admit weak integer additive set-indexers. In this paper, we study about the sparing number of the corona of two graphs., Comment: 12 pages, submitted. arXiv admin note: substantial text overlap with arXiv:1311.0858, arXiv:1407.4869, arXiv:1310.6091
- Published
- 2014
28. A Study on Topological Integer Additive Set-Labeling of Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. Let $G$ be a graph and let $X$ be a non-empty set. A set-indexer $f:V(G)\to \mathcal{P}(X)$ 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 $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$, whose associated function $f^+:E(G)\to \mathcal{P}(\mathbb{N}_0)$ is defined by $f(uv)=f(u)+f(v), uv\in E(G)$, where $\mathbb{N}_0$ is the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ is its power set. An integer additive set-indexer is an integer additive set-labeling 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. In this paper, we extend the concepts of topological set-labeling of graphs to topological integer additive set-labeling of graphs., Comment: 16 pages, 7 figures, Accepted for publication. arXiv admin note: text overlap with arXiv:1403.3984
- Published
- 2014
29. A Study on Prime Arithmetic Integer Additive Set-Indexers of Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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., Comment: 8 pages, submitted. arXiv admin note: text overlap with arXiv:1312.7674, arXiv:1405.6617, arXiv:1403.6435
- Published
- 2014
30. Weak Integer Additive Set-Indexed Graphs: A Creative Review
- Author
-
Chithra, K P, Germina, K A, and Sudev, N K
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 (IASI) is defined as an injective function $f:V(G)\to \mathcal{P}({\mathbb{N}_0})$ such that the induced function $g_f:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective, where $\mathbb{N}_0$ is the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ is its power set. A weak IASI is an IASI $f$ such that $|f^+(uv)|= \text{max}(f(u),f(v))$. In this paper, we critically and creatively review the concepts and properties of weak integer additive set-valued graphs., Comment: 18 pages, review paper, submitted
- Published
- 2014
- Full Text
- View/download PDF
31. On Integer Additive set-Sequential Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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-sequential labeling to integer additive set-labelings of graphs and provide some results on them., Comment: 11 pages, 2 figures, submitted. arXiv admin note: substantial text overlap with arXiv:1403.3984, arXiv:1407.4533
- Published
- 2014
32. Arithmetic Intger Additive Set-Idexers of Graph Operations
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
An integer additive set-indexer is an injective function $f:V(G)\to 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \to 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which admits an IASI is called an IASI graph. An arithmetic integer additive set-indexer is an integer additive set-indexer $f$, under which the set-labels of all elements of a given graph $G$ are arithmetic progressions. In this paper, we discuss about admissibility of arithmetic integer additive set-indexers by certain graph operations and certain products of graphs., Comment: 10 pages, submitted. arXiv admin note: substantial text overlap with arXiv:1401.6040, arXiv:1405.6617, arXiv:1312.7674
- Published
- 2014
- Full Text
- View/download PDF
33. On Certain Arithmetic Integer Additive set-indexers of Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 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, 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 two special types of arithmetic IASIs., Comment: 14 pages, submitted. arXiv admin note: substantial text overlap with arXiv:1312.7674, arXiv:1403.6435
- Published
- 2014
- Full Text
- View/download PDF
34. A Study on the Nourishing Number of Graphs and Graph Powers
- Author
-
Sudev, N K and Germina, K A
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 sumset of $f(u)$ and $f(v)$. If $g_f(uv)=k~\forall~uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexer. An integer additive set-indexer $f$ is said to be a strong integer additive set-indexer if $|g_f(uv)|=|f(u)|.|f(v)|~\forall ~ uv\in E(G)$. In this paper, we study the characteristics of certain graph classes and graph powers that admit strong integer additive set-indexers., Comment: 10 pages, submitted
- Published
- 2014
- Full Text
- View/download PDF
35. The Sparing Number of Certain Graph Powers
- Author
-
Sudev, N. K., Chithra, K. P., and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 IASI $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph $G$, a vertex or an edge, is the cardinality of its set-labels. 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 study the admissibility of weak IASI by certain graph powers and their sparing numbers., Comment: 14 pages, 6 figures, submitted
- Published
- 2014
36. Interference in Graphs
- Author
-
Acharya, B. D., A., Germina K., Kurian, Rency, Paul, Viji, and Zaslavsky, Thomas
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
Given a graph $I=(V, E),$ $\emptyset \ne D \subseteq V,$ and an arbitrary nonempty set $X,$ an injective function $f: V\to 2^X \setminus \{\emptyset\}$ is an interference of $D$ with respect to $I,$ if for every vertex $u\in V\setminus D$ there exists a neighbor $v\in D$ such that $f(u)\cap f(v) \ne \emptyset.$ We initiate a study of interference in graphs. We study special cases of the difficult problem of finding a smallest possible set $X,$ and we decide when, given a graph $G=(V,E(G))$ (resp., its line graph $L(G)$) the open neighborhood function $N_G: V \to 2^V$ (resp., $N_{L(G)}: E \to 2^E$) or its complementary function is an interference with respect to the complete graph $I=K_n.$, Comment: 17 pp
- Published
- 2014
37. A Study on Semi-arithmetic Integer Additive Set-Indexers of Graphs
- Author
-
Sudev, N K and Germina, K A
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 $f$ 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 integers, 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., Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1312.7674, arXiv:1312.7672
- Published
- 2014
38. A Study on Integer Additive Set-Graceful Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set 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. 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-graceful labeling to integer additive set-labelings of graphs and provide some results on them., Comment: 11 pages, submitted to JARPM
- Published
- 2014
39. On Weak Integer Additive Set-Indexers of Certain Graph Classes
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 $|g_f(uv)|=k$ for all $u,v\in V(G)$. An integer additive set-indexer $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. The sparing number of a graph $G$ is the minimum number of edges in $G$ with singleton set-labels, so that $G$ admits a weak integer additive set-indexer. In this paper, we study the admissibility of weak integer additive set-indexers by certain graph classes and certain associated graphs of given graphs., Comment: 12 pages, submitted, Journal of Discrete Mathematical Sciences & Cryptography, 2014
- Published
- 2014
- Full Text
- View/download PDF
40. A Note on the Sparing Number of Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 IASI $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph $G$, a vertex or an edge, is the cardinality of its set-labels. 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 study the sparing number of certain graphs and the relation of sparing number with some other parameters like matching number, chromatic number, covering number, independence number etc., Comment: 10 pages, 10 figures, submitted
- Published
- 2014
41. A Characterisation of Strong Integer Additive Set-Indexers of Graphs
- Author
-
Sudev, N K and Germina, K A
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 sumset of $f(u)$ and $f(v)$. 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 strong integer additive set-indexer if $|g_f(uv)|=|f(u)|.|f(v)|~\forall ~ uv\in E(G)$. We already have some characteristics of the graphs which admit strong integer additive set-indexers. In this paper, we study the characteristics of certain graph classes, graph operations and graph products that admit strong integer additive set-indexers., Comment: 10 pages
- Published
- 2014
42. Associated Graphs of Certain Arithmetic IASI Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 arithmetic integer additive set-indexer is an integer additive set-indexer $f$, under which the set-labels of all elements of a given graph $G$ are arithmetic progressions. In this paper, we discuss about admissibility of arithmetic integer additive set-indexers by certain associated graphs of the given graph $G$, like line graph, total graph, etc., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1312.7674, arXiv:1312.7672
- Published
- 2014
43. On Integer Additive Set-Indexers of Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ 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 $|g_f(uv)|=max(|f(u)|,|f(v)|)$ and an IASI $f$ is said to be a {\em strong IASI} if $|g_f(uv)|=|f(u)| |f(v)|$ for all $u,v\in V(G)$. In this paper, we study about certain characteristics of inter additive set-indexers., Comment: 12 pages. arXiv admin note: text overlap with arXiv:1312.7674 To Appear in Int. J. Math. Sci.& Engg. Appl. in March 2014
- Published
- 2013
44. A Study on Arithmetic Integer Additive Set-Indexers of Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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., Comment: 12 pages. arXiv admin note: text overlap with arXiv:1312.7672
- Published
- 2013
45. Weak Integer Additive Set-Indexers of Certain Graph Products
- Author
-
Sudev, N K and Germina, K A
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 sumset of $f(u)$ and $f(v)$. 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. We already have some results on the admissibility of weak integer additive set-indexer by certain graphs and finite graph operations. In this paper, we study further characteristics of certain graph products like cartesian product and corona of two weak IASI graphs and their admissibility of weak integer additive set-indexers and provide some useful results on these types of set-indexers., Comment: 7 pages, arXiv admin note: text overlap with arXiv:1310.6091, arXiv:1311.0345, submitted
- Published
- 2013
46. On the Sparing Number of Certain Graph Structures
- Author
-
Sudev, N K and Germina, K A
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 IASI $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph $G$, a vertex or an edge, is the cardinality of its set-labels. A mono-indexed element of a graph is an element of $G$ which has the set-indexing number $1$. The Sparing number of a graph $G$ is the minimum number of mono-indexed edges required for a graph $G$ to admit a weak IASI. In this paper, we introduce the notion of conjoined graphs, entwined graphs and floral graphs and study further about the sparing number of various finite graph operations as extensions to our earlier studies and provide some useful results on these types of graph structures., Comment: 12 pages, 5 figures. arXiv admin note: text overlap with arXiv:1310.6091
- Published
- 2013
47. Some New Results on Strong Integer Additive Set-Indexers of Graphs
- Author
-
Sudev, N. K. and Germina, K. A.
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
Let $\mathbb{N}_0$ be the set of all non-negative integers. An integer additive set-indexer of a graph $G$ is an injective function $f:V(G)\to 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $f^+(uv) = f(u)+ f(v)$ is also injective. An IASI is said to be {\em $k$-uniform} if $|f^+(e)| = k$ for all $e\in E(G)$. In this paper, we introduce the notions of strong integer additive set-indexers and initiate a study of the graphs which admit strong integer additive set-indexers., Comment: 11 pages, 2 figures
- Published
- 2013
- Full Text
- View/download PDF
48. Weak Integer Additive Set-Indexers of Certain Graph Operations
- Author
-
Sudev, N K and Germina, K A
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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 $|g_f(uv)|=max(|f(u)|,|f(v)|) \forall uv\in E(G)$. A weak integer additive set-indexer $f$ is called a weakly $k$-uniform integer additive set-indexer if $g_f(e)=k \forall e\in E(G)$. We have some characteristics of the graphs which admit weak and weakly uniform integer additive set-indexers. In this paper, we study the admissibility of weak integer additive set-indexer by certain graphs and finite graph operations., Comment: 10 pages, submitted, arXiv admin note: text overlap with arXiv:1310.5779
- Published
- 2013
49. On Weakly Uniform Integer Additive Set-Indexers of Graphs
- Author
-
Germina, K A. and Sudev, N K.
- Subjects
Mathematics - Combinatorics ,O5C78 - Abstract
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., Comment: 8 pages, published in Int. Math. Forum
- Published
- 2013
- Full Text
- View/download PDF
50. A Characterisation of Weak Integer Additive Set-Indexers of Graphs
- Author
-
Sudev, N K and Germina, K A
- Subjects
Mathematics - Combinatorics ,05C78 - Abstract
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., Comment: 12pages, 4 figures, arXiv admin note: text overlap with arXiv:1311.0858
- Published
- 2013
- Full Text
- View/download PDF
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.