Fuzzy Graphs and Fuzzy Hypergraphs

2011 ◽  
pp. 704-709 ◽  
Author(s):  
Leonid S. Bershtein ◽  
Alexander V. Bozhenyuk
Keyword(s):  
Communication Networks ◽  
Independent Set ◽  
Economic Systems ◽  
Graph Models ◽  
Graph Theoretic ◽  
Fuzzy Graphs ◽  
Theoretic Problem ◽  
Fuzzy Hypergraphs ◽  
Vehicle Capacity ◽  
Vehicle Travel Time

Graph theory has numerous application to problems in systems analysis, operations research, economics, and transportation. However, in many cases, some aspects of a graph-theoretic problem may be uncertain. For example, the vehicle travel time or vehicle capacity on a road network may not be known exactly. In such cases, it is natural to deal with the uncertainty using the methods of fuzzy sets and fuzzy logic. Hypergraphs (Berge,1989) are the generalization of graphs in case of set of multiarity relations. It means the expansion of graph models for the modeling complex systems. In case of modelling systems with fuzzy binary and multiarity relations between objects, transition to fuzzy hypergraphs, which combine advantages both fuzzy and graph models, is more natural. It allows to realise formal optimisation and logical procedures. However, using of the fuzzy graphs and hypergraphs as the models of various systems (social, economic systems, communication networks and others) leads to difficulties. The graph isomorphic transformations are reduced to redefinition of vertices and edges. This redefinition doesn’t change properties the graph determined by an adjacent and an incidence of its vertices and edges. Fuzzy independent set, domination fuzzy set, fuzzy chromatic set are invariants concerning the isomorphism transformations of the fuzzy graphs and fuzzy hypergraph and allow make theirs structural analysis.

scholarly journals Degree of a Vertex in Max-Product of Intuitionistic Fuzzy Graph

2019 ◽  
Vol 8 (4) ◽  
pp. 2902-2905 ◽  
Keyword(s):  
Automata Theory ◽  
Fuzzy Graph ◽  
Graph Theoretic ◽  
Fuzzy Models ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Graphs ◽  
Wide Range ◽  
Theoretic Problem ◽  
Intuitionistic Fuzzy Graphs ◽  
Necessary And Sufficient

Graph theory has applications in many areas of computer science, including data mining, image segmentation, clustering and networking. Product on graphs has a wide range of application in networking system, automata theory, game theory and structural mechanics. In many cases, some aspects of a graph-theoretic problem may be uncertain. Intuitionistic fuzzy models provide more compatible to the system compared to the fuzzy models. An intuitionistic fuzzy graph can be derived from two given intuitionistic fuzzy graphs using max-product. In this paper, we studied the degree of vertex in intuitionistic fuzzy graph by the max-product of two given intuitionistic fuzzy graph. Also find the necessary and sufficient condition for max-product of two intuitionistic fuzzy graphs to be regular.

Network Theory and Intergroup Approaches

2018 ◽  
Author(s):  
Cynthia Stohl
Keyword(s):  
Communication Networks ◽  
Network Theory ◽  
Social Contexts ◽  
Central Feature ◽  
Large Set ◽  
Social Unit ◽  
Individual Level ◽  
Graph Theoretic ◽  
Group Members ◽  
Structural Configurations

A social network consists of interactive patterns among individuals and groups that are created by transmitting and exchanging messages through time and space. A central feature of intergroup settings is that group members are embedded in multiple, previously established, as well as emerging, communication networks that vary in their structure, the nature of the relationships, and the diversity of the links. A network perspective extends and complements traditional social scientific approaches to intergroup communication. Rather than focusing on the attributes of individuals, a network perspective focuses on the causes and consequences of relations and connections between and among sets of people and groups. A network approach invigorates intergroup theory by focusing on the dynamic structures of connectedness, treating identity, social categorization, and representativeness as fluid rather than fixed factors within interactions. A basic principle of network theory is that behavior can best be understood socially; every social unit stands at the nexus of a multitude of constraining and enabling alignments. Structural network dynamics include, but are not limited to, density, diversity, clustering, equivalence, and centrality of the network. These structural configurations combined with the strength and multiplexity of specific network linkages strongly influence social identities, values, attitudes, experiences, and behavior. Using graph-theoretic models, network analysts are able to identify specific types of structures that are highly effective in predicting ingroup and intergroup attitudes and behaviors above and beyond individual-level characteristics. Structural dynamics can further amplify intergroup principles through exploring the degree to which ingroup boundaries are loosely or tightly connected and the types and nature of linkages and communication exchanges within and between groups. For example, network theory suggests that the greater ingroup overlap across social contexts, the more likely group members perceive higher status for that particular ingroup than for other social categories to which they belong. It is also more likely the boundary between groups will be linguistically marked. In organizations, intergroup conflict and the capacity for successful adaptation and intergroup cooperation are strongly related to the extent and the alignment of intergroup “weak” ties across traditional communication channels and online. Identifying network structures can help explain a large set of multilevel intergroup outcomes such as linguistic accommodation and stereotyping, group level conflict, organizational productivity and innovation, political attitudes, and community resilience.

Graph Colorings Applied in Scale-Free Networks

2013 ◽  
Vol 760-762 ◽  
pp. 2199-2204 ◽  
Author(s):  
Chao Yang ◽  
Bing Yao ◽  
Hong Yu Wang ◽  
Xiang'en Chen ◽  
Si Hua Yang
Keyword(s):  
Communication Networks ◽  
Information Networks ◽  
Chromatic Numbers ◽  
Graph Colorings ◽  
Graph Models ◽  
Important Method ◽  
Scale Free ◽  
Scale Free Networks ◽  
As Graph

Building up graph models to simulate scale-free networks is an important method since graphs have been used in researching scale-free networks and communication networks, such as graph colorings can be used for distinguishing objects of communication and information networks. In this paper we determine the avdtc chromatic numbers of some models related with researching networks.

scholarly journals Identification Conditions for the Solvability of NP-complete Problems for the Class of Pre-fractal Graphs

2021 ◽  
Vol 28 (2) ◽  
pp. 126-135
Author(s):  
Aleksandr Vasil'evich Tymoshenko ◽  
Rasul Ahmatovich Kochkarov ◽  
Azret Ahmatovich Kochkarov
Keyword(s):  
Communication Networks ◽  
Hamiltonian Cycle ◽  
Optimization Problems ◽  
Independent Set ◽  
Network Systems ◽  
Logistics Networks ◽  
Complete Problems ◽  
Discrete Problems ◽  
Np Complete ◽  
Discrete Optimization Problems

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, cryptocurrency networks) are distinguished by their multi-element nature and the dynamics of connections between its elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems leads to an "additional" complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamical graphs, which are used to model the structures of network systems, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes pre-fractal graphs. The article investigates NP-complete problems on pre-fractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, an independent set. The conditions under which for some problems it is possible to obtain an answer about the existence and to construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions are identified.

SPACETIME DISCOUNTED VALUE OF NETWORK CONNECTIVITY

2018 ◽  
Vol 21 (08) ◽  
pp. 1850018 ◽  
Author(s):  
ARNAUD Z. DRAGICEVIC
Keyword(s):  
Network Connectivity ◽  
Threshold Level ◽  
Opportunity Costs ◽  
Dynamic Graph ◽  
Consensus Protocol ◽  
Graph Theoretic ◽  
Leaders And Followers ◽  
Distance Weighted ◽  
Theoretic Problem ◽  
Control Dynamic

In order to unveil the value of network connectivity, discounted both in space and time, we formalize the construction of networks as an optimal control dynamic graph-theoretic problem. The network is based on a set of leaders and followers linked through edges. The node dynamics, built upon the consensus protocol, form a time evolutive Mahalanobis distance weighted by the opportunity costs. The results show that the network equilibrium depends on the influence of leader nodes, while the network connectivity depends on the cohesiveness among followers. Through numerical simulations, we find that — past a threshold level of opportunity costs — the values of shadow prices become stationary. Likewise, the model outputs show that, at a fixed level of foregone gains, agents value the safeguard of connections less in time than in space.

scholarly journals Generalized Fuzzy Graph Connectivity Parameters with Application to Human Trafficking

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 424
Author(s):  
Arya Sebastian ◽  
John N Mordeson ◽  
Sunil Mathew
Keyword(s):  
Human Trafficking ◽  
Network Theory ◽  
Graph Connectivity ◽  
Fuzzy Graph ◽  
Graph Models ◽  
Edge Connectivity ◽  
Clustering Techniques ◽  
New Class ◽  
Large Size ◽  
Fuzzy Graphs

Graph models are fundamental in network theory. But normalization of weights are necessary to deal with large size networks like internet. Most of the research works available in the literature have been restricted to an algorithmic perspective alone. Not much have been studied theoretically on connectivity of normalized networks. Fuzzy graph theory answers to most of the problems in this area. Although the concept of connectivity in fuzzy graphs has been widely studied, one cannot find proper generalizations of connectivity parameters of unweighted graphs. Generalizations for some of the existing vertex and edge connectivity parameters in graphs are attempted in this article. New parameters are compared with the old ones and generalized values are calculated for some of the major classes like cycles and trees in fuzzy graphs. The existence of super fuzzy graphs with higher connectivity values are established for both old and new parameters. The new edge connectivity values for some wider classes of fuzzy graphs are also obtained. The generalizations bring substantial improvements in fuzzy graph clustering techniques and allow a smooth theoretical alignment. Apart from these, a new class of fuzzy graphs called generalized t-connected fuzzy graphs are studied. An algorithm for clustering the vertices of a fuzzy graph and an application related to human trafficking are also proposed.

scholarly journals Algorithms for Finding Diameter Cycles of Biconnected Graphs

2021 ◽  
Vol 28 (4) ◽  
pp. 225-240
Author(s):  
Mehmet Hakan Karaata
Keyword(s):  
Space Complexity ◽  
Brute Force ◽  
Expansion Process ◽  
Time And Space ◽  
Graph Theoretic ◽  
Theoretic Problem ◽  
Biconnected Graph ◽  
Force Mechanism ◽  
Time And Space Complexity ◽  
Initial Cycle

In this paper, we first coin a new graph theoretic problem called the diameter cycle problem with numerous applications. A longest cycle in a graph G = (V, E) is referred to as a diameter cycle of G iff the distance in G of every vertex on the cycle to the rest of the on-cycle vertices is maximal. We then present two algorithms for finding a diameter cycle of a biconnected graph. The first algorithm is an abstract intuitive algorithm that utilizes a brute-force mechanism for expanding an initial cycle by repeatedly replacing paths on the cycle with longer paths. The second algorithm is a concrete algorithm that uses fundamental cycles in the expansion process and has the time and space complexity of O(n^6) and O(n^2), respectively. To the best of our knowledge, this problem was neither defined nor addressed in the literature. The diameter cycle problem distinguishes itself from other cycle finding problems by identifying cycles that are maximally long while maximizing the distances between vertices in the cycle. Existing cycle finding algorithms such as fundamental and longest cycle algorithms do not discover cycles where the distances between vertices are maximized while also maximizing the length of the cycle.

scholarly journals Certain Properties of Domination in Product Vague Graphs With an Application in Medicine

2021 ◽  
Vol 9 ◽  
Author(s):  
Xiaolong Shi ◽  
Saeed Kosari
Keyword(s):  
Graph Theory ◽  
Real World ◽  
Dominating Set ◽  
Independent Set ◽  
Fuzzy Graph ◽  
Medical Sciences ◽  
Total Dominating Set ◽  
Fuzzy Graphs ◽  
Vague Graph ◽  
Real World Problems

The product vague graph (PVG) is one of the most significant issues in fuzzy graph theory, which has many applications in the medical sciences today. The PVG can manage the uncertainty, connected to the unpredictable and unspecified data of all real-world problems, in which fuzzy graphs (FGs) will not conceivably ensue into generating adequate results. The limitations of previous definitions in FGs have led us to present new definitions in PVGs. Domination is one of the highly remarkable areas in fuzzy graph theory that have many applications in medical and computer sciences. Therefore, in this study, we introduce distinctive concepts and properties related to domination in product vague graphs such as the edge dominating set, total dominating set, perfect dominating set, global dominating set, and edge independent set, with some examples. Finally, we propose an implementation of the concept of a dominating set in medicine that is related to the COVID-19 pandemic.

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