Search

Numerous domains, including all branches of science and technology, database theory, data mining, neural networks, expert systems, cluster analysis, control theory, and image capture, have employed graph theory, which can be used to represent interactions among multiple individuals. An extension of fuzzy sets, specifically intuitionistic fuzzy sets, is the concept of single-valued neutrosophic sets. We extend to graphs the notion of single-valued neutrosophic sets, which is an instance of neutrosophic sets. Neutrosophic fuzzy graphs are generalizations of fuzzy graphs, intuitionistic fuzzy graphs, and interval-valued fuzzy graphs. When in the case of fuzzy or intuitionistic fuzzy graphs relation between atoms in problems is indeterminate, then the neutrosophic fuzzy graph serves as a potent tool for dealing with partial, ambiguous, and inconsistent information in the actual world. In single-valued neutrosophic fuzzy graphs, each vertex is allocated a triplet in which the first element represents membership value, second and third elements as indeterminacy and non-membership value, respectively. In this work, we generalize the conclusions relating crisp graphs and fuzzy graphs by applying graph theory to single-valued neutrosophic sets and exploring a novel type of graph structure known as single-valued neutrosophic graphs. Sombor numbers are an effective and efficient tool in describing the topology of the graph in a numerical value. The main aim of this article is to define the first six Sombor numbers for single-valued neutrosophic fuzzy graphs. Then, degree of vertices of different graph families is determined in single-valued neutrosophic fuzzy framework and calculate the six Sombor numbers for these fundamental graph families for single-valued neutrosophic fuzzy graphs. In the end, an application to show the trade flows between different countries via sea route is presented to prove that single-valued neutrosophic fuzzy graphs are more generalized and efficient. First, we calculated the six Sombor numbers for crisp graph and then in SVNS fuzzy framework. Illustrated cases are then provided to show the applicability, viability, efficacy, and benefits of the suggested ways. [ABSTRACT FROM AUTHOR]

The property of fuzziness is advantageous for any setting where a problem prediction and solution are intended. In comparison to the classical approach (crisp set), fuzzy graphs are beneficial as they provide greater precision and flexibility. Intuitionistic fuzzy graphs (IFGs) are generalizations of fuzzy graphs, in which each vertex is allocated an ordered pair whose first coordinate represents the membership value and whose second coordinate gives the non-membership value. Sombor indices are recently invented indices and are prime indices in describing the topology of a graph. They have demonstrated strong prediction ability in mathematical chemistry. The main aim of this research draft is to define two milestones, the fifth and sixth versions of Sombor indices in an intuitionistic fuzzy environment, IFGs are much more generalized and universal as compared to crisp graph theory, so we converted these two topological indices in this environment and then formulae for some fundamental families of graphs such as path graph, cycle graph, complete graph etc. are being derived. Our manuscript presents a novel application of opening a chain of schools of a popular brand in a certain vicinity, including the cluster center of the chain and applying intuitionistic fuzzy-based topological indices to assess their performance. With the help of this application, we highlight the practical implications and benefits of employing intuitionistic fuzzy-based techniques on the performance of schools through a comparative analysis. We checked the importance of intuitionistic fuzzy Sombor indices by presenting the application and concluded these indices can be applied to decision-making problems. With the help of the application, the efficiency of both of the intuitionistic fuzzy Sombor indices is also characterized, which must be a future efficacious tool in decision-making issues. [ABSTRACT FROM AUTHOR]

A T-spherical fuzzy set (TSFS) is an extended and logical algebraic representation to handle uncertainty, with the help of four functions describing four possible aspects of uncertain information. Aczel-Alsina triangular norm (TN) and conorm (TCN) are novel and proved to be more efficient than other existing TNs and TCNs. In our article, we establish the concept of a T-spherical fuzzy Aczel-Alsina graph (TSFAAG). We described the energy of TSFAAG along with the splitting and shadow energy of TSFAAG. Furthermore, we figured out the Randić energy of TSFAAG and obtained some useful results. Moreover, we give the notion of the Aczel-Alsina digraph (TSFAADG). To see the significance of the proposed TSFAADGs, we employed the energy and Randić energy of TSFAADGs for solving the problem of selecting the best investing company by using a decision-making algorithm. The sensitivity analysis of the variable parameters is also discussed and where the effect on ranking results is studied. To see the effectiveness of the proposed work, we did a comparative study and established some remarks. [ABSTRACT FROM AUTHOR]

To imitate the uncertainty and ambiguity in various decision-making problems, the T-spherical fuzzy set is more pragmatic and influential than the picture fuzzy set and q-rung orthopair fuzzy set. Because of the vagueness and fuzziness found in real-life problems, in which intuitionistic fuzzy sets may not give adequate results, the T-spherical fuzzy set is an efficient mathematical model to deal with them and can be handled effectively by using the notion of T-spherical fuzzy sets. In a different framework which is based on more opinions like yes, no, refusal, and abstaining, the T-spherical fuzzy set has been proven to be the most beneficial. In this article, we proposed the idea of T-spherical fuzzy Hamacher graphs (TSFHGs) based on Hamacher t-norm and t-conorm. We investigate the notion of the energy of TSFHGs, splitting TSFHGs, and shadow TSFHGs. Further, we introduce the Randić energy of TSFHG and studied its fundamental results. Moreover, we introduced the T-spherical fuzzy Hamacher digraphs (TSFHDGs) and discussed various results. We studied the applications of the proposed energies of TSFHGs in a decision-making problem by using an algorithm involving TSFHDGs and Hamacher aggregation operators. We also established a comparative study to see the significance of our proposed results. [ABSTRACT FROM AUTHOR]

The Wiener index WI is one of the connectivity parameters used to know the biochemical and physicochemical properties of compounds depending upon their molecular structures. Intuitionistic fuzzy graphs IFG s are a convenient tool to represent the objects and relations between them with two types of information using truth membership degree and falsity membership degree. This research work presents the concept of WI under the structure IFG s , I F trees, and I F cycles. Some bounds on WI are investigated. The relationship between WI and connectivity index CI is also studied. In the end of this study, an application of the WI in transport network flow is proposed. [ABSTRACT FROM AUTHOR]

In this research draft, by keeping in view, the importance of Zagreb numbers, first we defined some familiar graph families and their degrees in fuzzy soft environment, and then by converting these two novel topological numbers in fuzzy soft framework, their formulae are calculated for these graphical networks. An application of international trade using two kinds of routes is also discussed in the article, and by comparing the results efficiency of both the numbers is compared and analyzed how topological descriptors are useful in examining the progress and profit in the countries, that are a part of the project. Graphs are used in the modeling of links and processes within physical, biological, social, and information systems. A fuzzy graph is the most adaptable tool in mathematics, which describes the fuzzy relationship between objects. Fuzzy graphs are very helpful to give more accuracy to the system as compared to crisp graphs. A classification of the universe with respect to certain specified parameters is known as a soft set. It is a new method for simulating ambiguity and uncertainty. Soft set theory was created by Molodtsov in 1999 as a method for simulating vagueness and uncertainty. Soft set theory is presently used widely in academia to address decision-making issues. The concept of soft graphs is used to provide a parameterized point of view for graphs. In many areas of mathematics and chemistry, as well as in network theory, spectral graph theory, and molecular chemistry, the Zagreb numbers are crucial graph parameters. Due to the importance and significance of Zagreb numbers, in this research article we define these graph parameters in fuzzy soft environment and then derived the fundamental formula for first and second Zagreb numbers of some popular graph families. Also by keeping in view the importance of graph theory in solving decision making problems in business industry, and the most generic behavior of fuzzy soft graph due to its parameterized nature we use our work in application of decision making in trade among different nations. • We introduced two novel topological numbers in the framework of soft fuzzy graph. • Calculated general results of different graphs for these indices. • Then discussed an application related to China–Pakistan Economic Corridor. [ABSTRACT FROM AUTHOR]

Planar graphs play an effective role in many practical applications where the crossing of edges becomes problematic. This paper aims to investigate the complex q-rung orthopair fuzzy (CQROF) planar graphs (CQROFPGs). In a CQROFPG, the nodes and edges are based on complex QROF information that represents the uncertain knowledge in the range of unit circles in terms of complex numbers. The motivation in discussing such a topic is the wide flexibility of QROF information in the expression of uncertain knowledge compared to intuitionistic and Pythagorean fuzzy settings. We discussed the complex QROF graphs (CQROFGs), complex QROF multigraphs (CQROFMGs), and related terms followed by examples. Furthermore, the notion of strength and planarity index (PI) of the CQROFPGs is defined and exemplified followed by a study of strong and weak edges. We further defined the notion of complex QROF face (CQROFF) and complex QROF dual graph (CQROFDG) and exemplified these concepts. A study of isomorphism, coweak and weak isomorphism, is set up, and some results relating to the CQROFPG and isomorphisms are explored using examples. Furthermore, the problem of short circuits that results due to crossing is discussed because of the proposed study where an algorithm based on complex QROF (CQROF) information is presented for reducing the crossing in networks. Some advantages of the projected study over the previous study are observed, and some future study is predicted. [ABSTRACT FROM AUTHOR]