A Note on “Wiener Index of a Fuzzy Graph and Application to Illegal Immigration Networks”

Connectivity parameters have an important role in the study of communication networks. Wiener index is such a parameter with several applications in networking, facility location, cryptology, chemistry, and molecular biology, etc. In this paper, we show two notes related to the Wiener index of a fuzzy graph. First, we argue that Theorem 3.10 in the paper “Wiener index of a fuzzy graph and application to illegal immigration networks, Fuzzy Sets and Syst. 384 (2020) 132–147” is not correct. We give a correct statement of Theorem 3.10. Second, by using a new operator on matrix, we propose a simple and polynomial-time algorithm to compute the Wiener index of a fuzzy graph.

q -Rung Orthopair Fuzzy Matroids with Application to Human Trafficking

The theory of q -rung orthopair fuzzy sets ( q -ROFSs) is emerging for the provision of more comprehensive and useful information in comparison to their counterparts like intuitionistic and Pythagorean fuzzy sets, especially when responding to the models of vague data with membership and non-membership grades of elements. In this study, a significant generalized model q -ROFS is used to introduce the concept of q -rung orthopair fuzzy vector spaces ( q -ROFVSs) and illustrated by an example. We further elaborate the q -rung orthopair fuzzy linearly independent vectors. The study also involves the results regarding q -rung orthopair fuzzy basis and dimensions of q -ROFVSs. The main focus of this study is to define the concepts of q -rung orthopair fuzzy matroids ( q -ROFMs) and apply them to explore the characteristics of their basis, dimensions, and rank function. Ultimately, to show the significance of our proposed work, we combine these ideas and offer an application. We provide an algorithm to solve the numerical problems related to human flow between particular regions to ensure the increased government response action against frequently used path (heavy path) for the countries involved via directed q -rung orthopair fuzzy graph ( q -ROFG). At last, a comparative study of the proposed work with the existing theory of Pythagorean fuzzy matroids is also presented.

Certain Operations on Picture Fuzzy Graph with Application

Fuzzy graphs (FGs) can play a useful role in natural and human-made structures, including process dynamics in physical, biological, and social systems. Since issues in everyday life are often uncertain due to inconsistent and ambiguous information, it is extremely difficult for an expert to model those difficulties using an FG. Indeterminate and inconsistent information related to real-valued problems can be studied through a picture of the fuzzy graph (PFG), while the FG does not provide mathematically acceptable information. In this regard, we are interested in reducing the limitations of FGs by introducing some new definitions and results for the PFG. This paper aims to describe and explore a few properties of PFGs, including the maximal product (MP), symmetric difference (SD), rejection (RJ), and residue product (RP). Furthermore, we also discuss the degree and total degree of nodes in a PFG. This study also demonstrates the application of a PFG in digital marketing and social networking.

A Novel Description on Vague Graph with Application in Transportation Systems

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.

Application of Pythagorean Fuzzy Digraphs in Package Delivery Robots

The fuzzy set concept was developed to cope with uncertainty, whereas traditional sets are intended to deal with certainty. To address flaws in fuzzy set theory, extensions such as Intuitionistic Fuzzy Set (IFS), neutrosophic fuzzy sets, image fuzzy sets, and Pythagorean fuzzy set (PyFS) were developed. Pythagorean fuzzy set is useful tool for more clearly defining hazy concepts. In comparison to other fuzzy models, Pythagorean fuzzy set-based models allow more flexibility in handling human judgement information. The fuzzy graph structure is used to deal with the uncertainty in a network and to characterize its relationship with the non-empty vertex set. Pythagorean fuzzy graph (PyFG) was one of the Intuitionistic Fuzzy Graph (IFG) extensions. PyFG was created to cope with the uncertainty of an object and its relationship with other objects. PyFS and PyFG are the driving forces behind this innovative concept. This work defines Pythagorean Fuzzy Digraph (PyFDG), and PyFDG's score function. An algorithm is proposed for an issue to find the Pythagorean shortest path in package delivery robots.

Shortest path algorithm of a network via spherical fuzzy digraphs

Many extension and generalization of fuzzy sets have been introduced and studied in the literature. Picture fuzzy set acquired more concentration in the domain of decision making, as many real time circumstances might have choices of more than one and researchers are looking for optimum choice/decision. Spherical fuzzy digraph is a generalization of intuitionistic fuzzy set and fuzzy graph. In this paper, we redefine some preliminary operations of Spherical fuzzy graph and it is referred as spherical fuzzy digraph. We discuss some arithmetic operations and relations among spherical fuzzy digraph. We further proposed a method to solve a shortest path problem using score function.

New Kinds of Global Domination In Fuzzy Graph Using Strong Arc – New Approach

In this paper we establish the relation between strong arc domination number and global domination number ( new approach ) of some standard graphs using strong arcs. Also various new kinds of global domination number of using strong arc is discussed.

Split Domination In Interval-valued Fuzzy Graphs

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.