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 Complement of max product of intuitionistic fuzzy graphs

2021 ◽  
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Hamming Distance ◽  
Fuzzy Graph ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Graphs ◽  
Degree Of A Vertex ◽  
Intuitionistic Fuzzy Graphs

AbstractIn this paper, the complement of max product of two intuitionistic fuzzy graphs is defined. The degree of a vertex in the complement of max product of intuitionistic fuzzy graph is studied. Some results on complement of max product of two regular intuitionistic fuzzy graphs are stated and proved. Finally, we provide an application of intuitionistic fuzzy graphs in school determination using normalized Hamming distance.

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 Decision-Making Analysis Based on Fuzzy Graph Structures

2020 ◽  
Vol 2020 ◽  
pp. 1-30 ◽  
Author(s):  
Ali N. A. Koam ◽  
Muhammad Akram ◽  
Peide Liu
Keyword(s):  
Decision Making ◽  
General Procedure ◽  
Combinatorial Problems ◽  
Graph Structure ◽  
Fuzzy Graph ◽  
The Road ◽  
Research Article ◽  
Graph Structures ◽  
Fuzzy Graphs ◽  
Degree Of A Vertex

A graph structure is a useful framework to solve the combinatorial problems in various fields of computational intelligence systems and computer science. In this research article, the concept of fuzzy sets is applied to the graph structure to define certain notions of fuzzy graph structures. Fuzzy graph structures can be very useful in the study of various structures, including fuzzy graphs, signed graphs, and the graphs having labeled or colored edges. The notions of the fuzzy graph structure, lexicographic-max product, and degree and total degree of a vertex in the lexicographic-max product are introduced. Further, the proposed concepts are explained through several numerical examples. In particular, applications of the fuzzy graph structures in decision-making process, regarding detection of marine crimes and detection of the road crimes, are presented. Finally, the general procedure of these applications is described by an algorithm.

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 Clustering algorithm with strength of connectedness for $ m $-polar fuzzy network models

2021 ◽  
Vol 19 (1) ◽  
pp. 420-455
Author(s):  
Muhammad Akram ◽  
◽  
Saba Siddique ◽  
Majed G. Alharbi ◽  
Keyword(s):  
Clustering Algorithm ◽  
Research Study ◽  
Research Work ◽  
Network Models ◽  
Uncertain Information ◽  
Fuzzy Graph ◽  
Clustering Method ◽  
Fuzzy Graphs ◽  
Degree Of A Vertex ◽  
Fuzzy Network

<abstract><p>In this research study, we first define the strong degree of a vertex in an $ m $-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete $ m $-polar fuzzy graphs. Further, we introduce the concept of $ m $-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete $ m $-polar fuzzy graphs. We discuss connectivity parameters in $ m $-polar fuzzy graphs with precise examples, and we investigate the $ m $-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an $ m $-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as $ \epsilon_A $-reachable vertices in $ m $-polar fuzzy graphs, $ \epsilon_A $-connected $ m $-polar fuzzy graphs, or $ \epsilon_A $-connected $ m $-polar fuzzy subgraphs (in case the $ m $-polar fuzzy graph itself is not $ \epsilon_A $-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.</p></abstract>

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.

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.

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.

Some topological indices in fuzzy graphs

2020 ◽  
Vol 39 (5) ◽  
pp. 6033-6046
Author(s):  
Shriram Kalathian ◽  
Sujatha Ramalingam ◽  
Sundareswaran Raman ◽  
Narasimman Srinivasan
Keyword(s):  
Graph Theory ◽  
Classical Model ◽  
Distance Matrix ◽  
Wiener Index ◽  
Topological Indices ◽  
Fuzzy Graph ◽  
Structural Graph ◽  
Chemical Graph Theory ◽  
Fuzzy Graphs ◽  
Schultz Index

A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

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