First Zagreb index on a fuzzy graph and its application

2021 ◽  
pp. 1-13
Author(s):  
Sk Rabiul Islam ◽  
Madhumangal Pal
Keyword(s):  
Network Theory ◽  
Cartesian Product ◽  
Spectral Graph Theory ◽  
Fuzzy Graph ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Molecular Chemistry ◽  
Spectral Graph ◽  
Fuzzy Graphs ◽  
A Company

The Zagreb index (ZI) is a very important graph parameter and it is extensively used in molecular chemistry, spectral graph theory, network theory and several fields of mathematics and chemistry. In this article, the first ZI is studied for several fuzzy graphs like path, cycle, star, fuzzy subgraph, etc. and presented an ample number of results. Also, it is established that the complete fuzzy graph has maximal first ZI among n-vertex fuzzy graphs. Some bounds of first ZI are discussed for Cartesian product, composition, union and join of two fuzzy graphs. An algorithm has been designed to calculate the first ZI of a fuzzy graph. At the end of the article, a multi-criteria decision making (MCDM) method is provided using the first ZI of a fuzzy graph to find the best employee in a company. Also a comparison is provided among related indices on the result of application and shown that our method gives better results.

Hyper-Wiener index for fuzzy graph and its application in share market

2021 ◽  
pp. 1-11
Author(s):  
Sk Rabiul Islam ◽  
Madhumangal Pal
Keyword(s):  
Network Theory ◽  
Spanning Tree ◽  
Wiener Index ◽  
Topological Indices ◽  
Spectral Graph Theory ◽  
Fuzzy Graph ◽  
Molecular Chemistry ◽  
Spectral Graph ◽  
Fuzzy Graphs ◽  
Maximum Spanning Tree

Topological indices have an important role in molecular chemistry, network theory, spectral graph theory and several physical worlds. Most of the topological indices are defined in a crisp graph. As fuzzy graphs are more generalization of crisp graphs, those indices have more application in fuzzy graphs also. In this article, we introduced the fuzzy hyper-Wiener index (FHWI) and studied this index for various fuzzy graphs like path, cycle, star, etc and provided some interesting bounds of FHWI for that fuzzy graph. A lower bound of FHWI is established for n-vertex connected fuzzy graph depending on strength of a strong edges. A relation between FHWI of a tree and its maximum spanning tree is established and this index is calculated for the saturated cycle. Also, at the end of the article, an application in the share market of this index is presented.

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 Dominating Energy of Operations on Intuitionistic Fuzzy Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 328 ◽  
Author(s):  
A. Kalimulla ◽  
R. Vijayaragavan ◽  
S. Sharief Basha
Keyword(s):  
Cartesian Product ◽  
Fuzzy Graph ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Graphs ◽  
Intuitionistic Fuzzy Graphs

The concept of energy of an Intuitionistic Fuzzy Graph is extended to dominating Energy in operations on Intuitionistic Fuzzy Graph. In this paper, We have obtained the value of dominating Energy in different operations such as complement, Union, Join, Cartesian product and composition between two intuitionistic Fuzzy graphs. Also we study the relation between the dominating Energy in the operations on two Intuitionistic Fuzzy Graphs.  

scholarly journals New Concepts of Picture Fuzzy Graphs with Application

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 470 ◽  
Author(s):  
Cen Zuo ◽  
Anita Pal ◽  
Arindam Dey
Keyword(s):  
Fuzzy Set ◽  
Cartesian Product ◽  
Intuitionistic Fuzzy Set ◽  
Real Life ◽  
Fuzzy Graph ◽  
Intuitionistic Fuzzy ◽  
Strong Product ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Picture Fuzzy Set

The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set. It can work very efficiently in uncertain scenarios which involve more answers to these type: yes, no, abstain and refusal. In this paper, we introduce the idea of the picture fuzzy graph based on the picture fuzzy relation. Some types of picture fuzzy graph such as a regular picture fuzzy graph, strong picture fuzzy graph, complete picture fuzzy graph, and complement picture fuzzy graph are introduced and some properties are also described. The idea of an isomorphic picture fuzzy graph is also introduced in this paper. We also define six operations such as Cartesian product, composition, join, direct product, lexicographic and strong product on picture fuzzy graph. Finally, we describe the utility of the picture fuzzy graph and its application in a social network.

scholarly journals Decision Making Approach under Pythagorean Dombi Fuzzy Graphs for Selection of Leading Textile Industry

2019 ◽  
Vol 24 (4) ◽  
pp. 102
Author(s):  
Muhammad Akram ◽  
Jawaria Mohsan Dar ◽  
Sundas Shahzadi
Keyword(s):  
Decision Making ◽  
Textile Industry ◽  
Research Study ◽  
Cartesian Product ◽  
Multi Criteria Decision Making ◽  
Fuzzy Graph ◽  
Operational Parameter ◽  
Strong Product ◽  
Fuzzy Graphs ◽  
Selection Of

Graphs play a pivotal role in structuring real-world scenarios such as network security and expert systems. Numerous extensions of graph theoretical conceptions have been established for modeling uncertainty in graphical network situations. The Pythagorean Dombi fuzzy graph (PDFG), a generalization of the fuzzy Dombi graph (FDG), is very useful in representing vague relations between several objects, whereas the operational parameter has a flexible nature in decision-making problems. The main objective of this research study is to expand the area of discussion on PDFGs by establishing fruitful results and notions related to operations such as the direct product, Cartesian product, semi-strong product, strong product, and composition on PDFGs. Certain concepts, including the degree of vertices and total degree, are discussed as its modifications. Meanwhile, these outcomes are considered on PDFGs maintaining the strongness property. At the end, an algorithm for Pythagorean Dombi fuzzy multi-criteria decision-making is given, and a numerical example based on the selection of a leading textile industry is put forward to clarify the suitability of the proposed approach.

scholarly journals Product Operations on q-Rung Orthopair Fuzzy Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 588 ◽  
Author(s):  
Songyi Yin ◽  
Hongxu Li ◽  
Yang Yang
Keyword(s):  
Direct Product ◽  
Cartesian Product ◽  
Total Degree ◽  
Fuzzy Graph ◽  
Lexicographic Product ◽  
Intuitionistic Fuzzy ◽  
Strong Product ◽  
Fuzzy Graphs ◽  
Degree Of A Vertex

The q-rung orthopair fuzzy graph is an extension of intuitionistic fuzzy graph and Pythagorean fuzzy graph. In this paper, the degree and total degree of a vertex in q-rung orthopair fuzzy graphs are firstly defined. Then, some product operations on q-rung orthopair fuzzy graphs, including direct product, Cartesian product, semi-strong product, strong product, and lexicographic product, are defined. Furthermore, some theorems about the degree and total degree under these product operations are put forward and elaborated with several examples. In particular, these theorems improve the similar results in single-valued neutrosophic graphs and Pythagorean fuzzy graphs.

scholarly journals Revision of Pseudo-Ultrametric Spaces Based on m-Polar T-Equivalences and Its Application in Decision Making

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1232
Author(s):  
Azadeh Zahedi Khameneh ◽  
Adem Kilicman ◽  
Fadzilah Md Ali
Keyword(s):  
Metric Spaces ◽  
Cartesian Product ◽  
Equivalence Relations ◽  
Fuzzy Graph ◽  
Triangular Norm ◽  
Ultrametric Spaces ◽  
Fuzzy Environment ◽  
Fuzzy Similarity ◽  
Fuzzy Graphs ◽  
New Type

In mathematics, distance and similarity are known as dual concepts. However, the concept of similarity is interpreted as fuzzy similarity or T-equivalence relation, where T is a triangular norm (t-norm in brief), when we discuss a fuzzy environment. Dealing with multi-polarity in practical examples with fuzzy data leadsus to introduce a new concept called m-polar T-equivalence relations based on a finitely multivalued t-norm T, and to study the metric behavior of such relations. First, we study the new operators including the m-polar triangular norm T and conorm S as well as m-polar implication I and m-polar negation N, acting on the Cartesian product of [0,1]m-times.Then, using the m-polar negations N, we provide a method to construct a new type of metric spaces, called m-polar S-pseudo-ultrametric, from the m-polar T-equivalences, and reciprocally for constructing m-polar T-equivalences based on the m-polar S-pseudo-ultrametrics. Finally, the link between fuzzy graphs and m-polar S-pseudo-ultrametrics is considered. An algorithm is designed to plot a fuzzy graph based on the m-polar SL-pseudo-ultrametric, where SL is the m-polar Lukasiewicz t-conorm, and is illustrated by a numerical example which verifies our method.

scholarly journals The Kirchhoff Index of Toroidal Meshes and Variant Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jia-Bao Liu ◽  
Xiang-Feng Pan ◽  
Jinde Cao ◽  
Xia Huang
Keyword(s):  
Asymptotic Behavior ◽  
Graph Theory ◽  
Distance Function ◽  
Network Theory ◽  
Spectral Graph Theory ◽  
Electrical Network ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Spectral Graph ◽  
Explicit Formulae

The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices inG. In this paper, we established the relationships between the toroidal meshes networkTm×nand its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes ofL(Tm×n),S(Tm×n),T(Tm×n), andC(Tm×n)were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach.

scholarly journals Signless Laplacian Energy in Products of Intuitionistic Fuzzy Graphs

2019 ◽  
Vol 8 (3) ◽  
pp. 8536-8545
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Fuzzy Graph ◽  
Lexicographic Product ◽  
Intuitionistic Fuzzy ◽  
Strong Product ◽  
Fuzzy Graphs ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Intuitionistic Fuzzy Graphs

The observation of an Intuitionistic Fuzzy Graph’s signless laplacian energy is expanded innumerous products in Intuitionistic Fuzzy Graph. During this paper, we have got the value of signless laplacian Energy in unrelated products such as Cartesian product, Lexicographic Product, Tensor product and Strong Product, product, product and product amongst 2 intuitionistic Fuzzy graphs. Additionally we tend to study the relation between the Signless laplacian Energy within the varied products in 2 Intuitionistic Fuzzy Graphs

scholarly journals Wiener Index of Some Operations on Fuzzy Graphs

2021 ◽  
pp. 1-11
Author(s):  
Saqr H. Al-Emrany ◽  
Mahiuob M. Q. Shubatah ◽  
Abdullah Q. Al-Mekhlafi
Keyword(s):  
Cartesian Product ◽  
Wiener Index ◽  
Fuzzy Graph ◽  
Product Composition ◽  
Fuzzy Graphs

In this paper, the concept of the Wiener index in some operations on fuzzy graphs was introduced and investigated. The bound of W(G) of some operations on fuzzy graphs are obtained like union, join, Cartesian product, composition, complete fuzzy graph, and bipartite complete fuzzy graph.

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