Fuzzy Graphs and Fuzzy Hypergraphs

2020 ◽  
pp. 437-468
Author(s):  
Michael G. Voskoglou ◽  
Tarasankar Pramanik
Keyword(s):  
Graph Theory ◽  
Complex Networks ◽  
Shortest Path ◽  
Building Block ◽  
Shortest Path Problem ◽  
Fuzzy Graph ◽  
The Core ◽  
Fuzzy Graphs ◽  
Fuzzy Hypergraphs ◽  
Interval Valued

Relationship is the core building block of a network, and today's world advances through the complex networks. Graph theory deals with such problems more efficiently. But whenever vagueness or imprecision arises in such relationships, fuzzy graph theory helps. However, fuzzy hypergraphs are more advanced generalization of fuzzy graphs. Whenever there is a need to define multiary relationship rather than binary relationship, one can use fuzzy hypergraphs. In this chapter, interval-valued fuzzy hypergraph is discussed which is a generalization of fuzzy hypergraph. Several approaches to find shortest path between two given nodes in an interval-valued fuzzy graphs is described here. Many researchers have focused on fuzzy shortest path problem in a network due to its importance to many applications such as communications, routing, transportation, etc.

scholarly journals On the Shortest Path Problem of Uncertain Random Digraphs

2021 ◽  
Author(s):  
Hao Li ◽  
Kun Zhang
Keyword(s):  
Graph Theory ◽  
Probability Measure ◽  
Complex Networks ◽  
Shortest Path ◽  
Shortest Path Problem ◽  
Uncertain Measure ◽  
Chance Theory ◽  
Numerical Examples ◽  
Random Digraphs

Abstract In the field of graph theory, the shortest path problem is one of the most significant problems. However, since varieties of indeterminated factors appear in complex networks, determining of the shortest path from one vertex to another in complex networks may be a lot more complicated than the cases in deterministic networks. To illustrate this problem, the model of uncertain random digraph will be proposed via chance theory, in which some arcs exist with degrees in probability measure and others exist with degrees in uncertain measure. The main focus of this paper is to investigate the main properties of the shortest path in uncetain random digraph. Methods and algorithms are designed to calculate the distribution of shortest path more efficiently. Besides, some numerical examples are presented to show the efficiency of these methods and algorithms.

scholarly journals Applications of Fuzzy Graph Theory Portrayed in Various Fields

2021 ◽  
Author(s):  
Abdul Muneera ◽  
T. Nageswara Rao ◽  
R. V. N. Srinivasa ◽  
J. Venkateswara Rao
Keyword(s):  
Graph Theory ◽  
Wireless Sensor Network ◽  
Sensor Network ◽  
Material Science ◽  
Natural Sciences ◽  
Wireless Sensor ◽  
Fuzzy Graph ◽  
Ecological Conditions ◽  
Fuzzy Graphs ◽  
Remote System

Abstract The intend of the paper is to grant the centrality of fuzzy graph (f-graph) hypothetical ideas and the uses of dominations in fuzzy graphs to different genuine circumstances in the territories of science and designing. It is seen an eminent development because of various applications in PC and correspondence, biomedical, atomic material science and science, interpersonal organizations, natural sciences and in different various regions. Interpersonal organizations are the zones where countless individuals are associated. A wireless sensor Network (WSN) remote system which comprises of spatially circulated independent sensors to screen the physical or ecological conditions, for example, pressure, temperature, sound and so forth and to communicate their data through the remote system to a fundamental area. This paper gives an audit of the employments of Fuzzy Graph theory in different sorts of fields.

scholarly journals I-v Fuzzy Shortest Path in a Multigraph

2017 ◽  
Vol 10 (2) ◽  
pp. 364-370
Author(s):  
Siddhartha Biswas
Keyword(s):  
Graph Theory ◽  
Shortest Path ◽  
Research Paper ◽  
Fuzzy Graph ◽  
Algorithmic Rule

In this research paper the author introduces the notion of i-v fuzzy multigraph. The classical Dijkstra’s algorithmic rule to search out the shortest path in graphs isn't applicable to fuzzy graph theory. Consequently the author proposes a brand new algorithmic rule referred to as by IVF-Dijkstra's algorithmic rule with the philosophy of the classical Dijkstra's formula to unravel the SPP in an i-v fuzzy multigraph

Recent Developments on the Basics of Fuzzy Graph Theory

2020 ◽  
pp. 419-436 ◽  
Author(s):  
Ganesh Ghorai ◽  
Kavikumar Jacob
Keyword(s):  
Graph Theory ◽  
Planar Graphs ◽  
Fuzzy Graph ◽  
Threshold Graphs ◽  
Fuzzy Graphs ◽  
Degree Of A Vertex ◽  
Recent Developments

In this chapter, the authors introduce some basic definitions related to fuzzy graphs like directed and undirected fuzzy graph, walk, path and circuit of a fuzzy graph, complete and strong fuzzy graph, bipartite fuzzy graph, degree of a vertex in fuzzy graphs, fuzzy subgraph, etc. These concepts are illustrated with some examples. The recently developed concepts like fuzzy planar graphs are discussed where the crossing of two edges are considered. Finally, the concepts of fuzzy threshold graphs and fuzzy competitions graphs are also given as a generalization of threshold and competition graphs.

scholarly journals A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 551 ◽  
Author(s):  
Liangsong Huang ◽  
Yu Hu ◽  
Yuxia Li ◽  
P. K. Kishore Kumar ◽  
Dipak Koley ◽  
...  
Keyword(s):  
Graph Theory ◽  
Optimization Problems ◽  
Real Life ◽  
Road Transport ◽  
Fuzzy Graph ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Real World Problem ◽  
Neutrosophic Graph ◽  
Definition Of

Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the d m - and t d m -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely d m -regular, t d m -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the μ -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here.

scholarly journals A study on labelling of pythagorean neutrosophic fuzzy graphs

10.26524/cm97 ◽  
2021 ◽  
Vol 5 (1) ◽  
Author(s):  
Ajay D ◽  
Karthiga S ◽  
Chellamani P
Keyword(s):  
Graph Theory ◽  
Fuzzy Set ◽  
Fuzzy Graph ◽  
Fuzzy Graphs

Pythagorean neutrosophic fuzzy set comprises elements with dependent membership (µ), non-membership (σ)  and  independent  indeterminacy  (β)  functions  with  the  flexibility 0 ≤ µ2 + β2 + σ2 ≤ 2. Pythagorean neutrosophic fuzzy graph is a new concept emerged by combining the concept of Pythagorean neutrosophic fuzzy set and fuzzy graph theory. In this paper, the authors present the labelling of Pythagorean neutrosophic fuzzy graphs and investigate their properties.

scholarly journals Split Domination In Interval-valued Fuzzy Graphs

2021 ◽  
pp. 10-20
Author(s):  
Nojood A. AL-Khadari ◽  
Mahiuob M. Q. Shubatah
Keyword(s):  
Fuzzy Graph ◽  
Standard Interval ◽  
Fuzzy Graphs ◽  
Relationship Of ◽  
The Relationship ◽  
Interval Valued

Aims / Objectives: In this paper, we introduced and investigated the concept of split domination in interval-valued fuzzy graph and denoted by γs. We obtained many results related to γs. We investigated and study the relationship of γs with other known parameters in interval-valued fuzzy graph. Finally we calculated γs(G) for some standard interval valued fuzzy graphs.

scholarly journals Nikfar Domination in Fuzzy Graphs

2019 ◽  
Author(s):  
Mohammadesmail Nikfar
Keyword(s):  
Graph Theory ◽  
Real World ◽  
Dynamic State ◽  
Fuzzy Graph ◽  
Transport Planning ◽  
Open Problems ◽  
Static State ◽  
Fuzzy Graphs ◽  
Real World Problem ◽  
Expository Article

The aim of this expository article is to present recent developments in the centuries-old discussion on the interrelations between several types of domination in graphs. However, the novelty even more prominent in the newly discovered simplified presentations of several older results. The main part of this article, concerning a new domination and older one, is presented in a narrative that answers two classical questions: (i) To what extend must closing set be dominating? (ii) How strong is the assumption of domination of a closing set? In a addition, we give an overview of the results concerning domination. The problem asks how small can a subset of vertices be and contain no edges or, more generally how can small a subset of vertices be and contain other ones. Our work was as elegant as it was unexpected being a departure from the tried and true methods of this theory that had dominated the field for one fifth a century. This expository article covers all previous definitions. The inability of previous definitions in solving even one case of real-world problems due to the lack of simultaneous attentions to the worthy both of vertices and edges causing us to make the new one. The concept of domination in a variety of graphs models such as crisp, weighted and fuzzy, has been in a spotlight. We turn our attention to sets of vertices in a fuzzy graph that are so close to all vertices, in a variety of ways, and study minimum such sets and their cardinality. A natural way to introduce and motivate our subject is to view it as a real-world problem. In its most elementary form, we consider the problem of reducing waste of time in transport planning. Our goal here is to first describe the previous definitions and the results, and then to provide an overview of the flows ideas in their articles. The final outcome of this article is twofold: (i) Solving the problem of reducing waste of time in transport planning at static state; (ii) Solving and having a gentle discussions on problem of reducing waste of time in transport planning at dynamic state. Finally, we discuss the results concerning holding domination that are independent of fuzzy graphs. We close with a list of currently open problems related to this subject. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, fuzzy graph theory and graph theory.

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