The document defines and discusses global domination sets in intuitionistic fuzzy graphs (IFGs). Some key points:
- A global domination set of an IFG G is a domination set that dominates both G and its complement. The global domination number γg(G) is the minimum cardinality of a global domination set.
- Bounds on γg(G) are established, such as Min{|Vi|+|Vj|} ≤ γg(G) ≤ p, where Vi and Vj are vertices.
- Properties of γg(G) are proved, including γg(G) = γg(Gc) where Gc is the complement of G.
- Special
Some Concepts on Constant Interval Valued Intuitionistic Fuzzy Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Solving Fuzzy Matrix Games Defuzzificated by Trapezoidal Parabolic Fuzzy NumbersIJSRD
This document discusses solving fuzzy matrix games where the payoff elements are fuzzy numbers. It begins with definitions related to fuzzy sets and fuzzy numbers. A two-person zero-sum matrix game model is presented where the payoff matrix contains trapezoidal fuzzy numbers. The fuzzy game is converted to a crisp equivalent game using defuzzification techniques. Different defuzzification methods are applied to a numerical example and the results are compared. The key concepts of mixed strategies, maximin-minimax criteria and saddle points in fuzzy matrix games are also covered.
The document introduces new classes of odd graceful graphs called m-shadow graphs and m-splitting graphs. It proves that m-shadow graphs of paths, complete bipartite graphs, and symmetric products of paths and null graphs are odd graceful. It also proves that m-splitting graphs of paths, stars, and symmetric products of paths and null graphs are odd graceful. Examples are provided to illustrate the theories.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
The document discusses efficient algorithms for performing approximate matching queries on strings that have been grammar-compressed. It introduces the concept of implicit unit-Monge matrices which can represent permutation matrices in a space-efficient way using a range tree data structure. This representation allows dominance counting queries, needed for string comparison, to be performed in O(log2 n) time after an O(n log n) preprocessing step. More advanced data structures can improve these asymptotic time and space bounds further.
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
This document summarizes a research paper that proposes using Milne's predictor-corrector method to solve the dependency problem in fuzzy computations when numerically solving fuzzy differential equations (FDEs). The paper first provides background on fuzzy sets, fuzzy numbers, fuzzy processes, and fuzzy initial value problems (FIVPs). It then describes Milne's predictor-corrector method and illustrates how it can be applied to solve some example FIVPs. The goal is to address issues that arise from dependencies in fuzzy computations by using this numerical method to find solutions to FDEs.
The document summarizes a two-stage image segmentation method and super pixels. It discusses a two-stage convex image segmentation method that transforms an image into a piecewise smooth one that can be segmented based on pixel values. It introduces super pixels, which group pixels into perceptually meaningful atomic regions. The document studied these methods through MATLAB and also explored applications like saliency detection based on super pixel segmentation. It focused on understanding the algorithms through code implementation and improving the efficiency of methods like the SLIC super pixel algorithm.
In this paper, we study the cartesian product of intuitionistic fuzzy soft normal subgroup structure over snorm. By using s-norm of S, we characterize some basic results of intuitionistic S-fuzzy soft subgroup of normal subgroup. Also, we define the relationship between intuitionistic S-fuzzy soft subgroup and intuitionistic S-fuzzy soft normal subgroup. Finally we prove some basic properties
Some Concepts on Constant Interval Valued Intuitionistic Fuzzy Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Solving Fuzzy Matrix Games Defuzzificated by Trapezoidal Parabolic Fuzzy NumbersIJSRD
This document discusses solving fuzzy matrix games where the payoff elements are fuzzy numbers. It begins with definitions related to fuzzy sets and fuzzy numbers. A two-person zero-sum matrix game model is presented where the payoff matrix contains trapezoidal fuzzy numbers. The fuzzy game is converted to a crisp equivalent game using defuzzification techniques. Different defuzzification methods are applied to a numerical example and the results are compared. The key concepts of mixed strategies, maximin-minimax criteria and saddle points in fuzzy matrix games are also covered.
The document introduces new classes of odd graceful graphs called m-shadow graphs and m-splitting graphs. It proves that m-shadow graphs of paths, complete bipartite graphs, and symmetric products of paths and null graphs are odd graceful. It also proves that m-splitting graphs of paths, stars, and symmetric products of paths and null graphs are odd graceful. Examples are provided to illustrate the theories.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
The document discusses efficient algorithms for performing approximate matching queries on strings that have been grammar-compressed. It introduces the concept of implicit unit-Monge matrices which can represent permutation matrices in a space-efficient way using a range tree data structure. This representation allows dominance counting queries, needed for string comparison, to be performed in O(log2 n) time after an O(n log n) preprocessing step. More advanced data structures can improve these asymptotic time and space bounds further.
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
This document summarizes a research paper that proposes using Milne's predictor-corrector method to solve the dependency problem in fuzzy computations when numerically solving fuzzy differential equations (FDEs). The paper first provides background on fuzzy sets, fuzzy numbers, fuzzy processes, and fuzzy initial value problems (FIVPs). It then describes Milne's predictor-corrector method and illustrates how it can be applied to solve some example FIVPs. The goal is to address issues that arise from dependencies in fuzzy computations by using this numerical method to find solutions to FDEs.
The document summarizes a two-stage image segmentation method and super pixels. It discusses a two-stage convex image segmentation method that transforms an image into a piecewise smooth one that can be segmented based on pixel values. It introduces super pixels, which group pixels into perceptually meaningful atomic regions. The document studied these methods through MATLAB and also explored applications like saliency detection based on super pixel segmentation. It focused on understanding the algorithms through code implementation and improving the efficiency of methods like the SLIC super pixel algorithm.
In this paper, we study the cartesian product of intuitionistic fuzzy soft normal subgroup structure over snorm. By using s-norm of S, we characterize some basic results of intuitionistic S-fuzzy soft subgroup of normal subgroup. Also, we define the relationship between intuitionistic S-fuzzy soft subgroup and intuitionistic S-fuzzy soft normal subgroup. Finally we prove some basic properties
Abstract: An edge dominating set D of a fuzzy graph G= (σ, µ) is a non-split edge dominating set if the induced fuzzy sub graph H= (<e-d>, σ¢, µ¢) is connected. The split edge domination number γ¢ns(G)or γ¢ns is the minimum fuzzy cardinality of a non-split edge dominating set. In this paper we study a non-split edge dominating set of fuzzy graphs and investigate the relationship of γ¢ns(G)with other known parameter of G. Keywords: Fuzzy graphs, fuzzy domination, fuzzy edge domination, fuzzy non split edge domination number.
Title: Non Split Edge Domination in Fuzzy Graphs
Author: C.Y. Ponnappan, S. Basheer Ahamed, P. Surulinathan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
Research on information and communication technologies have been developed rapidly since it can be applied easily to several areas like computer science, medical science, economics, environments, engineering, among other. Applications of soft set theory, especially in information systems have been found paramount importance. Recently, Mukherjee and Das defined some new operations in fuzzy soft multi set theory and show that the De-Morgan’s type of results hold in fuzzy soft multi set theory with respect to these newly defined operations. In this paper, we extend their work and study some more basic properties of their defined operations. Also, we define some basic supporting tools in information system also application of fuzzy soft multi sets in information system are presented and discussed. Here we define the notion of fuzzy multi-valued information system in fuzzy soft multi set theory and show that every fuzzy soft multi set is a fuzzy multi valued information system.
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
This document summarizes key concepts from a PhD dissertation on uncertainty in deep learning:
1) There are two types of uncertainties - epistemic uncertainty from lack of knowledge that decreases with more data, and aleatoric uncertainty from inherent noise that cannot be reduced. Deep learning models need to estimate both to provide predictive uncertainty.
2) Variational inference allows approximating intractable Bayesian posteriors by minimizing the KL divergence between an approximating distribution and the true posterior. Dropout can be seen as a Bayesian approximation where weights follow a Bernoulli distribution.
3) With dropout as a variational distribution, predictive uncertainty in regression is estimated from multiple stochastic forward passes, with aleatoric uncertainty from noise and epistem
This document discusses a fusion of soft expert set and matrix models. It begins by introducing soft sets, soft expert sets, fuzzy soft sets, and intuitionistic fuzzy soft sets. It then defines various types of matrices in the context of soft expert sets, including soft expert matrices, soft expert equal matrices, soft expert complement matrices, and operations on soft expert matrices like addition, subtraction, and multiplication. An example is provided to illustrate a soft expert matrix model for a manufacturing firm choosing a location based on expert opinions. The document aims to provide a new dimension to soft expert sets through the use of matrices to solve decision making problems.
COMPARISON OF DIFFERENT APPROXIMATIONS OF FUZZY NUMBERSijfls
The notions of interval approximations of fuzzy numbers and trapezoidal approximations of fuzzy numbers have been discussed. Comparisons have been made between the close-interval approximation, valueambiguity
interval approximation and distinct approximation with the corresponding crisp and trapezoidal fuzzy numbers. A numerical example is included to justify the above mentioned notions.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document discusses simplifying complex rational expressions, which have numerators or denominators containing fractions. It provides two methods for simplification:
1) Multiplying the numerator and denominator by the lowest common denominator to clear fractions. Examples and steps are shown.
2) Dividing the numerator by the denominator after collecting like terms. An example problem is worked through to demonstrate the process. Objectives and learning outcomes are stated to guide readers.
Multilayerity within multilayerity? On multilayer assortativity in social net...Moses Boudourides
This document discusses assortative mixing and partitions in multilayer networks. It defines graph partitions, ordering of partitions, self-similarity of partitions, and how partitions can be viewed as enumerative attribute assignments. It then defines the assortativity coefficient of a partition to measure how strongly vertices in the same group of a partition are connected. Finally, it discusses how to measure assortativity between two different partitions of a graph by focusing on their intersection partition.
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be “entangled” such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique, which was provided previously. For polyadic semigroups and groups we introduce two external products: (1) the iterated direct product, which is componentwise but can have an arity that is different from the multipliers and (2) the hetero product (power), which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. We show in which cases the product of polyadic groups can itself be a polyadic group. In the same way, the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), in which all multipliers are zeroless fields. Many illustrative concrete examples are presented. Thу proposed construction can lead to a new category of polyadic fields.
A fusion of soft expert set and matrix modelseSAT Journals
Abstract
The purpose of this paper is to define different types of matrices in the light of soft expert sets. We then propose a decision making
model based on soft expert set.
Keywords: Soft set, soft expert set, Soft Expert matrix.
The document discusses matroids and parameterized algorithms. It begins with an overview of Kruskal's greedy algorithm for finding a minimum spanning tree in a graph. It then introduces matroids and defines them as structures where greedy algorithms find optimal solutions. Several examples of matroids are provided, including uniform matroids, partition matroids, graphic matroids, and gammoids. The document also presents an alternate definition of matroids using basis families.
International journal of applied sciences and innovation vol 2015 - no 2 - ...sophiabelthome
This document discusses domatic subdivision stable and just excellent graphs. It begins by introducing key concepts related to domination in graphs such as domatic number, dominating sets, and private neighborhood. It then defines domatic subdivision stable graphs as graphs where the domatic number is unchanged after any edge subdivision. Just excellent graphs are defined as graphs where every vertex is contained in a unique minimum dominating set. The document proves several results about the relationships between these graph properties, including that just excellent graphs are not domatic subdivision stable, and that the domatic number decreases to 2 after any edge addition between vertices in the same dominating set of a just excellent graph with domatic number 3.
IRJET- Optimization of 1-Bit ALU using Ternary LogicIRJET Journal
This document summarizes a research paper that proposes a novel approach to implementing a 1-bit arithmetic logic unit (ALU) using ternary logic. Ternary logic offers potential advantages over binary logic, including reduced transistor count and hardware. The authors designed a 1-bit ALU using ternary logic gates (T-gates) for ternary arithmetic and logic operations. Simulation results showed the ternary logic ALU design achieved a 25% reduction in transistor usage compared to an equivalent binary logic ALU design. The ternary logic ALU design approach could potentially be extended to multi-bit ALUs for applications where reduced transistor count is important.
The document discusses the Expectation-Maximization (EM) algorithm for estimating the parameters (φ, μ, Σ) of a mixture of Gaussian distributions model. The EM algorithm iteratively estimates the latent variables z that indicate which Gaussian each data point was drawn from (E-step), and updates the model parameters based on these estimates (M-step). This process repeats until convergence. The EM algorithm provides a way to perform maximum likelihood estimation for the mixture of Gaussians model when the latent variables z are unknown.
This document summarizes key aspects of Chapter 5 from the book "Periodic Differential Operators". It discusses how:
1) The spectral bands of a periodic Sturm-Liouville operator remain intervals of purely absolutely continuous spectrum when a perturbation is added, if the perturbation decays sufficiently at infinity.
2) If the perturbation tends to 0 at infinity, each compact interval within an instability interval contains finitely many eigenvalues and no other spectrum, so instability intervals remain spectral gaps.
3) In the limit of slow perturbation variation, it derives asymptotics for the distribution of eigenvalues introduced into the gaps.
The International Journal of Engineering and Sciencetheijes
This document summarizes a research paper that proposes a block cipher involving a key matrix and key bunch matrix supplemented with permutation. The cipher encrypts a plaintext matrix using modular multiplication with the key matrices over 256. It adds a permutation function that circularly rotates and swaps bits in the plaintext matrix in each round. Cryptanalysis showed the cipher cannot be broken by general attacks. The decryption uses the inverse key matrix and multiplicative inverse of the encryption key bunch matrix.
This document provides definitions and theorems related to domination and strong domination of graphs. It begins with introductions to graph theory concepts like vertex degree. It then defines different types of domination like dominating sets, connected dominating sets, and k-dominating sets. Further definitions include total domination, strong domination, and dominating cycles. Theorems are provided that relate strong domination number to independence number and domination number. The document concludes by discussing applications of domination in fields like communication networks and distributing computer resources.
This document defines and describes concepts related to fuzzy graphs and fuzzy digraphs. Key points include:
- A fuzzy graph is defined by two functions that assign membership values to vertices and edges.
- A fuzzy subgraph has lower or equal membership values for vertices and edges compared to the original graph.
- Effective edges and effective paths only include edges/paths where the membership equals the minimum vertex membership.
- Various graph measures are generalized to fuzzy graphs, such as vertex degree, order, size, and domination number.
- Fuzzy digraphs are defined similarly but with directed edges. Concepts like paths, independence, and domination are extended to fuzzy digraphs.
This document discusses operations on interval-valued fuzzy graphs. It begins with an introduction to fuzzy graphs and definitions related to fuzzy graph theory. It then presents the main results which prove properties of the union and join operations on interval-valued fuzzy graphs. Specifically, it proves that the union of two interval-valued fuzzy graphs G1 and G2 is isomorphic to their join, and vice versa. It also discusses subgraphs, complements, and neighbors in fuzzy graph theory.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Abstract: An edge dominating set D of a fuzzy graph G= (σ, µ) is a non-split edge dominating set if the induced fuzzy sub graph H= (<e-d>, σ¢, µ¢) is connected. The split edge domination number γ¢ns(G)or γ¢ns is the minimum fuzzy cardinality of a non-split edge dominating set. In this paper we study a non-split edge dominating set of fuzzy graphs and investigate the relationship of γ¢ns(G)with other known parameter of G. Keywords: Fuzzy graphs, fuzzy domination, fuzzy edge domination, fuzzy non split edge domination number.
Title: Non Split Edge Domination in Fuzzy Graphs
Author: C.Y. Ponnappan, S. Basheer Ahamed, P. Surulinathan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
Research on information and communication technologies have been developed rapidly since it can be applied easily to several areas like computer science, medical science, economics, environments, engineering, among other. Applications of soft set theory, especially in information systems have been found paramount importance. Recently, Mukherjee and Das defined some new operations in fuzzy soft multi set theory and show that the De-Morgan’s type of results hold in fuzzy soft multi set theory with respect to these newly defined operations. In this paper, we extend their work and study some more basic properties of their defined operations. Also, we define some basic supporting tools in information system also application of fuzzy soft multi sets in information system are presented and discussed. Here we define the notion of fuzzy multi-valued information system in fuzzy soft multi set theory and show that every fuzzy soft multi set is a fuzzy multi valued information system.
Entity Linking via Graph-Distance MinimizationRoi Blanco
Entity-linking is a natural-language--processing task that consists in identifying strings of text that refer to a particular
item in some reference knowledge base.
One instance of entity-linking can be formalized as an optimization problem on the underlying concept graph, where the quantity to be optimized is the average distance between chosen items.
Inspired by this application, we define a new graph problem which is a natural variant of the Maximum Capacity Representative Set. We prove that our problem is NP-hard for general graphs; nonetheless, it turns out to be solvable in linear time under some more restrictive assumptions. For the general case, we propose several heuristics: one of these tries to enforce the above assumptions while the others try to optimize similar easier objective functions; we show experimentally how these approaches perform with respect to some baselines on a real-world dataset.
This document summarizes key concepts from a PhD dissertation on uncertainty in deep learning:
1) There are two types of uncertainties - epistemic uncertainty from lack of knowledge that decreases with more data, and aleatoric uncertainty from inherent noise that cannot be reduced. Deep learning models need to estimate both to provide predictive uncertainty.
2) Variational inference allows approximating intractable Bayesian posteriors by minimizing the KL divergence between an approximating distribution and the true posterior. Dropout can be seen as a Bayesian approximation where weights follow a Bernoulli distribution.
3) With dropout as a variational distribution, predictive uncertainty in regression is estimated from multiple stochastic forward passes, with aleatoric uncertainty from noise and epistem
This document discusses a fusion of soft expert set and matrix models. It begins by introducing soft sets, soft expert sets, fuzzy soft sets, and intuitionistic fuzzy soft sets. It then defines various types of matrices in the context of soft expert sets, including soft expert matrices, soft expert equal matrices, soft expert complement matrices, and operations on soft expert matrices like addition, subtraction, and multiplication. An example is provided to illustrate a soft expert matrix model for a manufacturing firm choosing a location based on expert opinions. The document aims to provide a new dimension to soft expert sets through the use of matrices to solve decision making problems.
COMPARISON OF DIFFERENT APPROXIMATIONS OF FUZZY NUMBERSijfls
The notions of interval approximations of fuzzy numbers and trapezoidal approximations of fuzzy numbers have been discussed. Comparisons have been made between the close-interval approximation, valueambiguity
interval approximation and distinct approximation with the corresponding crisp and trapezoidal fuzzy numbers. A numerical example is included to justify the above mentioned notions.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document discusses simplifying complex rational expressions, which have numerators or denominators containing fractions. It provides two methods for simplification:
1) Multiplying the numerator and denominator by the lowest common denominator to clear fractions. Examples and steps are shown.
2) Dividing the numerator by the denominator after collecting like terms. An example problem is worked through to demonstrate the process. Objectives and learning outcomes are stated to guide readers.
Multilayerity within multilayerity? On multilayer assortativity in social net...Moses Boudourides
This document discusses assortative mixing and partitions in multilayer networks. It defines graph partitions, ordering of partitions, self-similarity of partitions, and how partitions can be viewed as enumerative attribute assignments. It then defines the assortativity coefficient of a partition to measure how strongly vertices in the same group of a partition are connected. Finally, it discusses how to measure assortativity between two different partitions of a graph by focusing on their intersection partition.
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be “entangled” such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique, which was provided previously. For polyadic semigroups and groups we introduce two external products: (1) the iterated direct product, which is componentwise but can have an arity that is different from the multipliers and (2) the hetero product (power), which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. We show in which cases the product of polyadic groups can itself be a polyadic group. In the same way, the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), in which all multipliers are zeroless fields. Many illustrative concrete examples are presented. Thу proposed construction can lead to a new category of polyadic fields.
A fusion of soft expert set and matrix modelseSAT Journals
Abstract
The purpose of this paper is to define different types of matrices in the light of soft expert sets. We then propose a decision making
model based on soft expert set.
Keywords: Soft set, soft expert set, Soft Expert matrix.
The document discusses matroids and parameterized algorithms. It begins with an overview of Kruskal's greedy algorithm for finding a minimum spanning tree in a graph. It then introduces matroids and defines them as structures where greedy algorithms find optimal solutions. Several examples of matroids are provided, including uniform matroids, partition matroids, graphic matroids, and gammoids. The document also presents an alternate definition of matroids using basis families.
International journal of applied sciences and innovation vol 2015 - no 2 - ...sophiabelthome
This document discusses domatic subdivision stable and just excellent graphs. It begins by introducing key concepts related to domination in graphs such as domatic number, dominating sets, and private neighborhood. It then defines domatic subdivision stable graphs as graphs where the domatic number is unchanged after any edge subdivision. Just excellent graphs are defined as graphs where every vertex is contained in a unique minimum dominating set. The document proves several results about the relationships between these graph properties, including that just excellent graphs are not domatic subdivision stable, and that the domatic number decreases to 2 after any edge addition between vertices in the same dominating set of a just excellent graph with domatic number 3.
IRJET- Optimization of 1-Bit ALU using Ternary LogicIRJET Journal
This document summarizes a research paper that proposes a novel approach to implementing a 1-bit arithmetic logic unit (ALU) using ternary logic. Ternary logic offers potential advantages over binary logic, including reduced transistor count and hardware. The authors designed a 1-bit ALU using ternary logic gates (T-gates) for ternary arithmetic and logic operations. Simulation results showed the ternary logic ALU design achieved a 25% reduction in transistor usage compared to an equivalent binary logic ALU design. The ternary logic ALU design approach could potentially be extended to multi-bit ALUs for applications where reduced transistor count is important.
The document discusses the Expectation-Maximization (EM) algorithm for estimating the parameters (φ, μ, Σ) of a mixture of Gaussian distributions model. The EM algorithm iteratively estimates the latent variables z that indicate which Gaussian each data point was drawn from (E-step), and updates the model parameters based on these estimates (M-step). This process repeats until convergence. The EM algorithm provides a way to perform maximum likelihood estimation for the mixture of Gaussians model when the latent variables z are unknown.
This document summarizes key aspects of Chapter 5 from the book "Periodic Differential Operators". It discusses how:
1) The spectral bands of a periodic Sturm-Liouville operator remain intervals of purely absolutely continuous spectrum when a perturbation is added, if the perturbation decays sufficiently at infinity.
2) If the perturbation tends to 0 at infinity, each compact interval within an instability interval contains finitely many eigenvalues and no other spectrum, so instability intervals remain spectral gaps.
3) In the limit of slow perturbation variation, it derives asymptotics for the distribution of eigenvalues introduced into the gaps.
The International Journal of Engineering and Sciencetheijes
This document summarizes a research paper that proposes a block cipher involving a key matrix and key bunch matrix supplemented with permutation. The cipher encrypts a plaintext matrix using modular multiplication with the key matrices over 256. It adds a permutation function that circularly rotates and swaps bits in the plaintext matrix in each round. Cryptanalysis showed the cipher cannot be broken by general attacks. The decryption uses the inverse key matrix and multiplicative inverse of the encryption key bunch matrix.
This document provides definitions and theorems related to domination and strong domination of graphs. It begins with introductions to graph theory concepts like vertex degree. It then defines different types of domination like dominating sets, connected dominating sets, and k-dominating sets. Further definitions include total domination, strong domination, and dominating cycles. Theorems are provided that relate strong domination number to independence number and domination number. The document concludes by discussing applications of domination in fields like communication networks and distributing computer resources.
This document defines and describes concepts related to fuzzy graphs and fuzzy digraphs. Key points include:
- A fuzzy graph is defined by two functions that assign membership values to vertices and edges.
- A fuzzy subgraph has lower or equal membership values for vertices and edges compared to the original graph.
- Effective edges and effective paths only include edges/paths where the membership equals the minimum vertex membership.
- Various graph measures are generalized to fuzzy graphs, such as vertex degree, order, size, and domination number.
- Fuzzy digraphs are defined similarly but with directed edges. Concepts like paths, independence, and domination are extended to fuzzy digraphs.
This document discusses operations on interval-valued fuzzy graphs. It begins with an introduction to fuzzy graphs and definitions related to fuzzy graph theory. It then presents the main results which prove properties of the union and join operations on interval-valued fuzzy graphs. Specifically, it proves that the union of two interval-valued fuzzy graphs G1 and G2 is isomorphic to their join, and vice versa. It also discusses subgraphs, complements, and neighbors in fuzzy graph theory.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
To find a non-split strong dominating set of an interval graph using an algor...IOSR Journals
In graph theory, a connected component of an undirected graph is a sub graph in which any two
vertices are connected to each other by paths. For a graph G, if the subgraph of G itself is a connected
component then the graph is called connected, else the graph G is called disconnected and each connected
component sub graph is called it’s components. A dominating set Dst of graph G=(V,E) is a non-split strong
dominating set if the induced sub graph < V-Dst > is connected. The non-split strong domination number of G is
the minimum cardinality of a non-split strong dominating set . In this paper constructed a verification method
algorithm for finding a non-split strong dominating set of an interval graph.
This document introduces the concept of weak triple connected domination number (γwtc) of a graph. A subset S of vertices is a weak triple connected dominating set if S is a weak dominating set and the induced subgraph <S> is triple connected. The γwtc is defined as the minimum cardinality of such a set. Some standard graphs are used to illustrate the concept and determine this number. Bounds on γwtc are obtained for general graphs, and its relationship to other graph parameters are investigated. The paper aims to develop this new graph invariant and establish basic results about weak triple connected domination.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found
applications in optimization, navigation, network theory, image processing, pattern recognition etc.
Several other authors have studied metric dimension of various standard graphs. In this paper we
introduce a real valued function called generalized metric G X × X × X ® R+ d : where X = r(v /W) =
{(d(v,v1),d(v,v2 ),...,d(v,v ) / v V (G))} k Î , denoted d G and is used to study metric dimension of graphs. It
has been proved that metric dimension of any connected finite simple graph remains constant if d G
numbers of pendant edges are added to the non-basis vertices.
AN APPLICATION OF Gd -METRIC SPACES AND METRIC DIMENSION OF GRAPHSFransiskeran
The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found
applications in optimization, navigation, network theory, image processing, pattern recognition etc.
Several other authors have studied metric dimension of various standard graphs. In this paper we
introduce a real valued function called generalized metric → + Gd
: X × X × X R where X = r(v /W) =
{(d(v,v1
),d(v,v2
),...,d(v,vk
/) v∈V (G))}, denoted Gd
and is used to study metric dimension of graphs. It
has been proved that metric dimension of any connected finite simple graph remains constant if Gd
numbers of pendant edges are added to the non-basis vertices.
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Global Domination Set in Intuitionistic Fuzzy Graph
1. ISSN (e): 2250 – 3005 || Vol, 04 || Issue, 9 || September – 2014 ||
International Journal of Computational Engineering Research (IJCER)
www.ijceronline.com Open Access Journal Page 55
Global Domination Set in Intuitionistic Fuzzy Graph R. JahirHussain 1 ,S. Yahya Mohamed 2 1P.G and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli-620 020, India 2P.G and Research Department of Mathematics, Govt. Arts College, Trichy-22, India
I. INTRODUCTION
Atanassov [1] introduced the concept of intuitionistic fuzzy (IF) relations and intuitionistic fuzzy graphs (IFGs). Research on the theory of intuitionistic fuzzy sets (IFSs) has been witnessing an exponential growth in Mathematics and its applications. R. Parvathy and M.G.Karunambigai’s paper [7] introduced the concept of IFG and analyzed its components. Nagoor Gani, A and Sajitha Begum, S [5] defined degree, Order and Size in intuitionistic fuzzy graphs and extend the properties. The concept of Domination in fuzzy graphs is introduced by A. Somasundaram and S. Somasundaram [8] in the year 1998. Parvathi and Thamizhendhi[6] introduced the concepts of domination number in Intuitionistic fuzzy graphs. Study on domination concepts in Intuitionistic fuzzy graphs are more convenient than fuzzy graphs, which is useful in the traffic density and telecommunication systems. The Global domination number of a Graph was discussed by E. Sampathkumar [10] in 1989.In this paper, We define global Intuitionistic fuzzy domination set of IFG and discuss the situation of this concept used in network. Also some thereoms and bounds of global Intuitionistic fuzzy domination number of IFGs are established. II. PRELIMINARIES Definition 2.1: An Intuitionistic fuzzy graph is of the form G = (V, E ) where (i) V={v1,v2,….,vn} such that μ1: V[0,1]and γ1: V [0,1] denote the degree of membership and non-membership of the element vi V, respectively, and 0 ≤ μ1 (vi) + γ1 (vi) ≤ 1 for every vi V, (i = 1,2, ……. n), (ii) E V x V where μ2: VxV [0,1] and γ2: VxV [0,1] are such that μ2 (vi , vj) ≤ min [μ1(vi), μ1(vj)] and γ2 (vi , vj) ≤ max [γ1(vi), γ1(vj) ] and 0 ≤ μ2 (vi, vj) + γ2 (vi, vj) ≤ 1 for every (vi ,vj) E, ( i, j = 1,2, ……. n) Definition 2.2 An IFG H = < V’, E’ > is said to be an Intuitionistic fuzzy subgraph (IFSG) of the IFG, G = < V, E > if V’ V and E’ ⊆ E. In other words, if μ1i’ ≤ μ1i ; γ1i’ ≥ γ1i and μ2ij’ ≤ μ2ij ; γ2ij’ ≥ γ2ij for every i, j = 1,2………n. Definition 2.3: Let G = (V,E) be a IFG. Then the cardinality of G is defined as |G| = Definition 2.4: The vertex cardinality of IFG G is defined by |V| = = p and The edge cardinality of IFG G is defined by |E| = = q. The vertex cardinality of IFG is called the order of G and denoted by O(G). The cardinality of G is called the size of G, denoted by S(G).
ABSTRACT
In this paper, We define global Intuitionistic fuzzy domination set and its number of IFGs. Also connected Intuitionistic fuzzy domination number of IFGs are discussed. Some results and bounds of global Intutionistic fuzzy domination number of IFGs are established.
KEYWORDS: Intuitionistic fuzzy graph, connected Intuitionistic fuzzy dominating set, global Intuitionistic fuzzy dominating set, effective degree.
2010Mathematics Subject Classification: 05C69, 03F55, 05C72, 03E72.
2. Global Domination Set In …
www.ijceronline.com Open Access Journal Page 56
Definition 2.5: An edge e = (x, y) of an IFG G = ( V, E ) is called an effective edge if μ2(x, y) = μ1(x) Λ μ1 (y) and γ2(x, y) = γ1(x) V γ1(y). Definition 2.6: An Intuitionistic fuzzy graph is complete if μ2ij = min ( μ1i, μ1j ) and γ2ij = max (γ2i ,γ2j) for all ( vi , vj ) V. Definition 2.7: An Intuitionistic fuzzy graph G is said to be strong IFG if μ2(x, y) = μ1(x) Λ μ1 (y) and γ2(x, y) = γ1(x) V γ1(y) for all ( vi , vj ) E. That is every edge is effective edge. Definition 2.8 : The complement of an IFG G = < V, E > is denoted by = ( , ) and is defined as i) = μ1(v) and = γ1(v) ii) (u,v) = μ1 (u) Λ μ1 (v) - μ2 (u,v) and (u,v) = γ1(u) V γ1(v) - γ2(u,v) for u,v in V Definition 2.9: Let G = (V,E) be an IFG. The neighbourhood of any vertex v is defined as N(v) = (Nμ(v) , Nγ(v)), Where Nμ(v) = and Nγ(v) = . N[v] = N (v) {v} is called the closed neighbourhood of v. Definition 2.10: The neighbourhood degree of a vertex is defined as dN(v) = (dNμ(v),dNγ(v)) where dNμ(v) = and dNγ(v) = . The minimum neighbourhood degree is defined as N(G) = (Nμ(v),Nγ(v)), where Nμ(v) = { dNμ(v): vV} and Nγ(v) = { dNγ(v): vV}. Definition 2.11: The effective degree of a vertex v in a IFG. G = (V, E) is defined to be sum of the effective edges incident at v, and denoted by dE(v). The minimum effective degree of G is E(G) = Λ {dE (v)/v V} Definition 2.12: Let G= (V, E) be an IFG. Let u, v V, we say that u dominated v in G if there exist a strong arc between them. A Intuitionistic fuzzy subset D V is said to be dominating set in G if for every v V-D, there exist u in D such that u dominated v. The minimum scalar cardinality taken over all Intuitionisitc fuzzy dominating sets is called Intuitionistic fuzzy domination number and is denoted by γ. The maximum scalar cardinality of a minimal domination set is called upper Intuitionistic fuzzy domination number and is denoted by the symbol Γ . Definition 2.13: A Intuitionistic fuzzy dominating set D⊆V of IFG G is said to be a Intutitionisitc fuzzy connected dominating set of G if the subgraph <D> induced by D is connected. The minimum cardinality taken over all minimal Intuitionistic fuzzy connected dominating sets is called Intuitionistic fuzzy domination number of G and it is denoted by γc(G). Definition 2.14: An independent set of an Intuitionistic fuzzy graph G = (V, E) is a subset S of V such that no two vertices of S are adjacent in G. Definition 2.15: A Bipartite IFG, G= (V,E) is said to be complete Bipartite IFG, if μ2(vi, vj) = μ1(vi) Λ μ1 (vj) and γ2(vi, vj) = γ1(vi) V γ1(vj) for all vi V1 and vj V2. It is denoted by . III. GLOBAL INTUITIONISTIC FUZZY DOMINATION SET IN IFG Definition 3.1: Let G = (V, E) be an IFG. A Intuitionistic fuzzy dominating set S ⊆V is said to be global Intuitionistic fuzzy dominating set of G if S is also a Intuitionistic fuzzy dominating set of . The minimum cardinality of global Intuitionistic fuzzy dominating sets is global Intuitionistic fuzzy domination number and is denoted by γg(G). Example: In case of transportation and road networks, the travel time is mostly used as weight. The travel time is a function of the traffic density on the road and/or the length of the road. The length of a road is a crisp quantity but the traffic density is fuzzy. In a road network, we represent crossings as nodes and roads as edges. The traffic density is mostly calculated on the road between adjacent crossings. These numbers can be represented as intuitionistic fuzzy numbers. Road network represented as an intuitionistic fuzzy graph R* = (C, L), where C is an intuitionistic fuzzy set of crossings at which the traffic density is calculated and L is an intuitionistic fuzzy set of roads between two crossings. The degrees of membership, μL(xy), and non membership, νL(xy), are calculated as μL(xy) = min (μC(x), μC(y)), γL(xy) = max (γC(x), γC(y)).
3. Global Domination Set In …
www.ijceronline.com Open Access Journal Page 57
Some essential goods are being supplied to some crossings from supplying stations located some other crossings. It may happen that the roads(edges of G) may be closed for some reason or the other. So, we have to think of maintaining the supply of goods to various crossing uninterrupted through secret links ( i.e. edges of the complement of network). We have to find minimum number of supplying stations(crossings) needed which is called global Intuitionistic fuzzy domination number. Example 3.2: Let G = (V,E) be IFG be defined as follows
v1(0.2, 0.4) (0.2, 0.4) v2(0.3,0.3) (0.2,0.6) (0.3,0.6) (0.1,0.5) v4(0.4,0.6) (0.1, 0.6) V3(0.1,0.5) Fig- 1: Intuitionistic fuzzy graph(G) Here |v1| = 0.4 , |v2| = 0.5, |v3| = 0.3, |v4| = 0.4 and minimum γg-set is { v2, v3, v4} and therefore γg(G) = 1.2. Observations 3.3:
(i) γg(Kn) = γg() = p
(ii) γg() = Min {|vi|} + Min {|vj|}, where vi V1 and vj V2.
Proposition 3.4: The global Intuitionistic fuzzy dominating set is not singleton. Proof: Since gifd-set contain dominating set for both G and Gsc then at least two vertices are in the set. i.e) The gifd-set containing at least two vertices. Theorem 3.5: For any IFG G= (V, E) with effective edges, Min{|Vi|+|Vj|} ≤ γg(G) ≤ p, i ≠ j Proof: We know that global Intuitonistic fuzzy dominating set has at least two vertices. Let {vi, vj} are the vertices, then Min{|Vi|+|Vj|} = γg(G) If the set contains other than {vi, vj} then Min{|Vi|+|Vj|} < γsg(G), i ≠ j If the given G is complete IFG then gifd-set contains all the vertices of the G, that is γg(G) ≤ O(G) = p i.e.) We get, Min{|Vi|+|Vj|} ≤ γg(G) ≤ p. Theorem 3.6: Let G = (V, E) be the IFG and the Intuitionistic fuzzy dominating set S of G is global Intuitionistic fuzzy dominating set if and only if, for each v V-S, there exists a u S such that u is not adjacent to v. Proof: Let S is global dominating set and also dominating set. Suppose u is adjacent to v then we get S is not a dominating set. Which is contradiction. That is u is not adjacent to v. Conversely, for each v V-S and u is not adjacent to v then the set S is dominating both G and . That is S is global Intuitionistic fuzzy dominating set. Theorem 3.7: Let G = (V, E) be an IFG then (i) γg(G) = γg() (ii) γ (G) ≤ γg(G) Proof: G is connected IFG and γg-set dominating vertices of G and . Clearly γg(G) = γg() . Suppose D is the γ-set of G then the number of vertices in the dominating set is less than or some time equal to γg-set. That is γ (G) ≤ γg(G). Theorem 3.8: Let S be the minimum Intuitionisitc fuzzy dominating set of IFG G containing t vertices. If there exist a vertex v V-S adjacent to only vertices in S then γg-set contain atmost t+1 vertices. Proof: Since S is the γ-set and v V-S adjacent to only vertices in S then we get S{v} is a global Intuitionistics fuzzy dominating set. That is γg-set contain atmost t+1 vertices. Theorem 3.9: Let G = (V, E) be strongly connected IFG then, at least one of the following holds.
(i) γc(G) ≤ γg(G). (ii) (G) ≤ γg(G).
Proof: Since γc-set also dominating set and induced Intuitionistic fuzzy subgraph is connected then the Gc may be disconnected and it is less than or equal to γg-set. That is γc(G) ≤ γg(G). Similarly, we have (G) ≤ γg(G).
4. Global Domination Set In …
www.ijceronline.com Open Access Journal Page 58
Definition 3.10: G = (V, E ) be a connected IFG with effective edges which is said to be semi complete IFG, if every pair of vertices have a common neighbor in G . The IFG G is said to be purely semi complete IFG if G is semi complete IFG but not complete IFG. Example 3.11:
v1(0.2, 0.3) (0.2, 0.4) v2(0.3,0.4) (0.2,0.5) (0.2,0.6) (0.3,0.6) v4(0.4,0.5) (0.3, 0.6) v3(0.3,0.6) Fig-2: Purely Semi complete IFG Theorem 3.12: Let G =(V, E) be the purely semi complete IFG. Then γg-set contains at least three vertices. Proof: Since G is purely semi complete IFG then it contains triangles with common vertex. Let v be the common vertex then, In Gc , the vertex v is isolated vertex. Also global Intuitionistic dominating set contains Intuitionistic fuzzy dominating vertices of G and Gc. Suppose gifd-set contains less than three vertices We know that gifd-set not a singleton. i.e) gifd-set contains at least two vertices Let D = { v1, v2} be a gifd-set in G. Case 1: <D> is connected in G Then v1v2 is an effective edge in G. By the definition of semi complete IFG, there is a v3 in G such that <v1v2v3> is triangle in G, i.e) D is not a Intuitonistic fuzzy domination set in Gc. Which is contradiction to D is a gifd-set in G. Case 2: <D> is disconnected in G. i.e.) There is no effective edge between v1 and v2 . Since G is semi complete IFG, there is v3 in G such that v1v3and v3v2 are the effective edges in G Therefore, In Gc, v3 is not dominated by a vertex in D. Which implies, D is not a gifd-set in G Which is contradiction to our assumption. That is γg-set contains at least three vertices Theorem 3.13: Let G = (V, E) be the IFG with effective edges . γg(G) = Min{|Vi|+|Vj|} i ≠j if and only if there is an effective edge uv in G such that each vertex in V – {u, v} is adjacent to u or v but not both. Proof: Suppose γg(G) = Min{|Vi|+|Vj|} i≠j, We assume D = {u, v} be the gifd-set in G Let <D> is connected in G, then uv is an effective edge in G. If any vertex w in V-{u, v} is adjacent to both u and v. Which implies D is not a dominating set for Gc. which is contradiction to our assumption. i.e) effective edge uv in G such that each vertex in V – {u, v} is adjacent to u or v but not both. Conversely, each vertex in V – {u, v} is adjacent to u or v but not both, then we get γg(G) = Min{|Vi|+|Vj|} i≠j. IV. CONCLUSION Here ,We defined global Intuitionistic fuzzy domination set of IFG and discussed the situation of this concept used in network. Also some theorems and bounds of global Intuitionistic fuzzy domination number of IFGs are established. Further we going to establish more results and bounds on this gifd number with other domination parameters. REFERENCES
[1]. Atanassov. KT. Intuitionistic fuzzy sets: theory and applications. Physica, New York, 1999.
[2]. Bhattacharya, P. Some Remarks on fuzzy graphs, Pattern Recognition Letter 5: 297-302,1987.
[3]. Harary,F., Graph Theory, Addition Wesley, Third Printing, October 1972.
[4]. Kulli,V.R., Theory of domination in graph, Vishwa International Publications, 2010.math. Phys. Sci., 13 (1979), 607-613.
[5]. Nagoor Gani. A and Shajitha Begum.S, Degree, Order and Size in Intuitionistic Fuzzy Graphs, International Journal of Algorithms, Computing and Mathematics, (3) 3 (2010).
[6]. Parvathi,R., and Thamizhendhi, G. Domination in Intuitionistic fuzzy graphs, Fourteenth Int. conf. on IFGs, Sofia, NIFS Vol.16, 2, 39- 49, 15-16 May 2010.
[7]. Parvathi, R. and Karunambigai, M.G., Intuitionistic Fuzzy Graphs, Computational Intelligence,Theory and applications,International Conference in Germany, Sept 18 -20, 2006.
[8]. Somasundaram, A and Somasundaram, S., Domination in Fuzzy graph-I, Patter Recognition Letter 19(9), 1998, 787-791.
[9]. Sampathkumar.E, and Walikar.H.B., The connected domination number of a graph, J. math. Phys. Sci., 13 (1979), 607-613.
[10]. Sampathkumar. E., The Global Domination number of a Graph, Jour. Math. Phy. Sc., vol. 23, No. 5, pp. 377-385.
[11]. Tamizh Chelvam. T and Robinson Chelladuari. S, Complementary nil domination number of a graph, Tamkang Journal of mathematics, Vol.40, No.2 (2009),165-172.