scholarly journals Analysis of the Shortest Path in Spherical Fuzzy Networks Using the Novel Dijkstra Algorithm

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Zafar Ullah ◽  
Huma Bashir ◽  
Rukhshanda Anjum ◽  
Salman A. AlQahtani ◽  
Suheer Al-Hadhrami ◽  
...  
Keyword(s):  
Shortest Path ◽  
Real Life ◽  
Graphical Analysis ◽  
The Novel ◽  
Fuzzy Graph ◽  
Dijkstra Algorithm ◽  
Shortest Path Problems ◽  
Graph Theoretic ◽  
Life Problems ◽  
Novel Concept

The concept of fuzzy graph (FG) and its generalized forms has been developed to cope with several real-life problems having some sort of imprecision like networking problems, decision making, shortest path problems, and so on. This paper is based on some developments in generalization of FG theory to deal with situation where imprecision is characterized by four types of membership grades. A novel concept of T-spherical fuzzy graph (TSFG) is proposed as a common generalization of FG, intuitionistic fuzzy graph (IFG), and picture fuzzy graph (PFG) based on the recently introduced concept of T-spherical fuzzy set (TSFS). The significance and novelty of proposed concept is elaborated with the help of some examples, graphical analysis, and results. Some graph theoretic terms are defined and their properties are studied. Specially, the famous Dijkstra algorithm is proposed in the environment of TSFGs and is applied to solve a shortest path problem. The comparative analysis of the proposed concept and existing theory is made. In addition, the advantages of the proposed work are discussed over the existing tools.

Introduction

2021 ◽  
pp. 1-36
Author(s):  
Vadim Zverovich
Keyword(s):  
Graph Theory ◽  
Biological Networks ◽  
Real Life ◽  
Scale Free ◽  
Graph Theoretic ◽  
Life Problems ◽  
Scale Free Networks ◽  
Web Graph ◽  
The Web

This chapter gives a brief overview of selected applications of graph theory, many of which gave rise to the development of graph theory itself. A range of such applications extends from puzzles and games to serious scientific and real-life problems, thus illustrating the diversity of applications. The first section is devoted to the six earliest applications of graph theory. The next section introduces so-called scale-free networks, which include the web graph, social and biological networks. The last section describes a number of graph-theoretic algorithms, which can be used to tackle a number of interesting applications and problems of graph theory.

scholarly journals Graphical Analysis of Covering and Paired Domination in the Environment of Neutrosophic Information

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Sami Ullah Khan ◽  
Abdul Nasir ◽  
Naeem Jan ◽  
Zhen-Hua Ma
Keyword(s):  
Graph Theory ◽  
Optimization Problems ◽  
Domination Number ◽  
Real Life ◽  
Graphical Analysis ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Neutrosophic Graph ◽  
Paired Domination ◽  
Intuitionistic Fuzzy Graphs

Neutrosophic graph (NG) is a powerful tool in graph theory, which is capable of modeling many real-life problems with uncertainty due to unclear, varying, and indeterminate information. Meanwhile, the fuzzy graphs (FGs) and intuitionistic fuzzy graphs (IFGs) may not handle these problems as efficiently as NGs. It is difficult to model uncertainty due to imprecise information and vagueness in real-world scenarios. Many real-life optimization problems are modeled and solved using the well-known fuzzy graph theory. The concepts of covering, matching, and paired domination play a major role in theoretical and applied neutrosophic environments of graph theory. Henceforth, the current study covers this void by introducing the notions of covering, matching, and paired domination in single-valued neutrosophic graph (SVNG) using the strong edges. Also, many attention-grabbing properties of these concepts are studied. Moreover, the strong covering number, strong matching number, and the strong paired domination number of complete SVNG, complete single-valued neutrosophic cycle (SVNC), and complete bipartite SVNG are worked out along with their fascinating properties.

scholarly journals Jaccard and Dice Similarity Measures Based on Novel Complex Dual Hesitant Fuzzy Sets and Their Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-25
Author(s):  
Tahir Mahmood ◽  
Ubaid Ur Rehman ◽  
Zeeshan Ali ◽  
Ronnason Chinram
Keyword(s):  
Similarity Measure ◽  
Fuzzy Set ◽  
Real Life ◽  
Similarity Measures ◽  
Hesitant Fuzzy Set ◽  
The Novel ◽  
Dual Hesitant Fuzzy Set ◽  
Life Problems ◽  
Novel Approach ◽  
Operational Laws

Complex dual hesitant fuzzy set (CDHFS) is a combination of two modifications, called complex fuzzy set (CFS) and dual hesitant fuzzy set (DHFS). CDHFS makes two degrees, called membership valued and nonmembership valued in the form of a finite subset of a unit disc in the complex plane, and is a capable method to solve uncertain and unpredictable information in real-life problems. The goal of this study is to describe the notion of CDHFS and its operational laws. The novel approach of the complex interval-valued dual hesitant fuzzy set (CIvDHFS) and its fundamental laws are also described and defended with the help of an example. Further, the vector similarity measures (VSMs), weighted vector similarity measures (WVSMs), hybrid vector similarity measure, and weighted hybrid vector similarity measure are additionally explored. These similarity measures (SM) are applied to the environment of pattern recognition and medical diagnosis to assess the capability and feasibility of the interpreted measures. We additionally solved some numerical examples utilizing the established measures. We examine the dependability and validity of the proposed measures by comparing them with some existing measures. The advantages, comparative analysis, and graphical portrayal of the investigated interpreted measures and existing measures are additionally described in detail.

scholarly journals Certain Properties of Vague Graphs with a Novel Application

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1647
Author(s):  
Yongsheng Rao ◽  
Saeed Kosari ◽  
Zehui Shao
Keyword(s):  
Social Systems ◽  
Sufficient Conditions ◽  
Real Life ◽  
Laplacian Matrix ◽  
Fuzzy Graph ◽  
Fuzzy Graphs ◽  
Life Problems ◽  
Real World Problem ◽  
Laplacian Energy ◽  
Definition Of

Fuzzy graph models enjoy the ubiquity of being present in nature and man-made structures, such as the dynamic processes in physical, biological, and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems that are often uncertain, for an expert, it is highly difficult to demonstrate those problems through a fuzzy graph. Resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem can be done using a vague graph (VG), with which the fuzzy graphs may not generate satisfactory results. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGs. The objective of this paper is to present certain types of vague graphs (VGs), including strongly irregular (SI), strongly totally irregular (STI), neighborly edge irregular (NEI), and neighborly edge totally irregular vague graphs (NETIVGs), which are introduced for the first time here. Some remarkable properties associated with these new VGs were investigated, and necessary and sufficient conditions under which strongly irregular vague graphs (SIVGs) and highly irregular vague graphs (HIVGs) are equivalent were obtained. The relation among strongly, highly, and neighborly irregular vague graphs was established. A comparative study between NEI and NETIVGs was performed. Different examples are provided to evaluate the validity of the new definitions. A new definition of energy called the Laplacian energy (LE) is presented, and its calculation is shown with some examples. Likewise, we introduce the notions of the adjacency matrix (AM), degree matrix (DM), and Laplacian matrix (LM) of VGs. The lower and upper bounds for the Laplacian energy of a VG are derived. Furthermore, this study discusses the VG energy concept by providing a real-time example. Finally, an application of the proposed concepts is presented to find the most effective person in a hospital.

scholarly journals A Novel Description on Vague Graph with Application in Transportation Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Zheng Kou ◽  
Saeed Kosari ◽  
Maryam Akhoundi
Keyword(s):  
Social Systems ◽  
Real Life ◽  
Transportation Systems ◽  
Fuzzy Graph ◽  
Network Heterogeneity ◽  
Life Problems ◽  
Real World Problem ◽  
Vague Graph ◽  
Ecology And Economy

Fuzzy graph (FG) models embrace the ubiquity of existing in natural and man-made structures, specifically dynamic processes in physical, biological, and social systems. It is exceedingly difficult for an expert to model those problems based on a FG because of the inconsistent and indeterminate information inherent in real-life problems being often uncertain. Vague graph (VG) can deal with the uncertainty associated with the inconsistent and determinate information of any real-world problem, where FGs many fail to reveal satisfactory results. Regularity definitions have been of high significance in the network heterogeneity study, which have implications in networks found across biology, ecology, and economy; so, adjacency sequence (AS) and fundamental sequences (FS) of regular vague graphs (RVGs) are defined with examples. One essential and adequate prerequisite has been ascribed to a VG with maximum four vertices is that it should be regular based on the adjacency sequences concept. Likewise, it is described that if ζ and its principal crisp graph (CG) are regular, then all the nodes do not have to have the similar AS. In the following, we obtain a characterization of vague detour (VD) g-eccentric node, and the concepts of vague detour g-boundary nodes and vague detour g-interior nodes in a VG are examined. Finally, an application of vague detour g-distance in transportation systems is given.

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 Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 945
Author(s):  
Saba Ayub ◽  
Muhammad Shabir ◽  
Muhammad Riaz ◽  
Muhammad Aslam ◽  
Ronnason Chinram
Keyword(s):  
Decision Making ◽  
Real Life ◽  
Score Function ◽  
Fuzzy Relation ◽  
Optimal Decision ◽  
Fuzzy Relations ◽  
Binary Relations ◽  
Life Problems ◽  
Modeling Uncertainties ◽  
Novel Concept

Binary relations are most important in various fields of pure and applied sciences. The concept of linear Diophantine fuzzy sets (LDFSs) proposed by Riaz and Hashmi is a novel mathematical approach to model vagueness and uncertainty in decision-making problems. In LDFS theory, the use of reference or control parameters corresponding to membership and non-membership grades makes it most accommodating towards modeling uncertainties in real-life problems. The main purpose of this paper is to establish a robust fusion of binary relations and LDFSs, and to introduce the concept of linear Diophantine fuzzy relation (LDF-relation) by making the use of reference parameters corresponding to the membership and non-membership fuzzy relations. The novel concept of LDF-relation is more flexible to discuss the symmetry between two or more objects that is superior to the prevailing notion of intuitionistic fuzzy relation (IF-relation). Certain basic operations are defined to investigate some significant results which are very useful in solving real-life problems. Based on these operations and their related results, it is analyzed that the collection of all LDF-relations gives rise to some algebraic structures such as semi-group, semi-ring and hemi-ring. Furthermore, the notion of score function of LDF-relations is introduced to analyze the symmetry of the optimal decision and ranking of feasible alternatives. Additionally, a new algorithm for modeling uncertainty in decision-making problems is proposed based on LDFSs and LDF-relations. A practical application of proposed decision-making approach is illustrated by a numerical example. Proposed LDF-relations, their operations, and related results may serve as a foundation for computational intelligence and modeling uncertainties in decision-making problems.

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 New Concepts of Intuitionistic Fuzzy Trees with Applications

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Yongsheng Rao ◽  
Saeed Kosari ◽  
Zehui Shao ◽  
A. A. Talebi ◽  
A. Mahdavi ◽  
...  
Keyword(s):  
Neural Networks ◽  
Computer Networks ◽  
Real Life ◽  
The Novel ◽  
Fuzzy Graph ◽  
Fuzzy Models ◽  
Intuitionistic Fuzzy ◽  
New Concepts ◽  
The Relationship

AbstractIt is known that Intuitionistic fuzzy models give more precision, flexibility and compatibility to the system as compared to the classic and fuzzy models. Intuitionistic fuzzy tree has an important role in neural networks, computer networks, and clustering. In the design of a network, it is important to analyze connections between the levels. In addition, the intuitionistic fuzzy tree is becoming increasingly significant as it is applied to different areas in real life. The study proposes the novel concepts of intuitionistic fuzzy graph (IFG) and some basic definitions. We investigate the types of arcs, for example, $$\alpha _{\mu }$$ α μ -strong, $$\beta _{\mu }$$ β μ -strong, and $$\delta _{\mu }$$ δ μ -arc in an intuitionistic fuzzy graph, and introduce some of their properties. In particular, the present work develops the concepts of intuitionistic fuzzy bridge (IFB), intuitionistic fuzzy cut nodes (IFCN) and some important properties of an intuitionistic fuzzy bridge. Next, we define an intuitionistic fuzzy cycle (IFC) and an intuitionistic fuzzy tree (IFT). Likewise, we discuss some properties of the IFT and the relationship between an intuitionistic fuzzy tree and an intuitionistic fuzzy cycle. Finally, an application of intuitionistic fuzzy tree is illustrated in other sciences.

scholarly journals Solving Multiple Traveling Salesman Problem by Meerkat Swarm Optimization Algorithm

2019 ◽  
Vol 54 (3) ◽  
Author(s):  
Belal Al-Khateeb ◽  
Mohammed Yousif
Keyword(s):  
Traveling Salesman Problem ◽  
Shortest Path ◽  
Optimization Algorithm ◽  
Real Life ◽  
Traveling Salesman ◽  
Swarm Optimization ◽  
Total Cost ◽  
Life Problems ◽  
Multiple Traveling Salesman Problem ◽  
The Traveling Salesman Problem

Multiple Traveling Salesman Problem (MTSP) is one of various real-life applications, MTSP is the extension of the Traveling Salesman Problem (TSP). TSP focuses on searching of minimum or shortest path (traveling distance) to visit all cities by salesman, while the primary goal of MTSP is to find shortest path for m paths by n salesmen with minimized total cost. Wherever, total cost means the sum of distances of all salesmen. In this work, we proposed metaheuristic algorithm is called Meerkat Swarm Optimization (MSO) algorithm for solving MTSP and guarantee good quality solution in reasonable time for real-life problems. MSO is a metaheuristic optimization algorithm that is derived from the behavior of Meerkat in finding the shortest path. The implementation is done using many dataset from TSPLIB95. The results demonstrate that MSO in most results is better than another results that compared in average cost that means the MSO superior to other results of MTSP.

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