SOFT DISEMIGRAPHS: EXPLORATION OF CERTAIN EXTENDED AND RESTRICTED OPERATIONS

Fuzzy soft graphs are efficient numerical tools for simulating the uncertainty of the real world. A fuzzy soft graph is a perfect fusion of the fuzzy soft set and the graph model that is widely used in a variety of fields. This paper discusses a few unique notions of perfect fuzzy soft tripartite graphs (PFSTG), as well as the concepts of complement of perfect fuzzy soft tripartite graphs (CPFSTGs). Because soft sets are most useful in real‐world applications, the newly developed concepts of perfect soft tripartite fuzzy graphs will lead to many theoretical applications by adding extra fuzziness in analysing. We look at some of their properties and come up with a few results that are related to these concepts. Furthermore, we investigated some fundamental theorems and illustrated an application of size of perfect fuzzy soft tripartite graphs in employee selection for an institution using the perfect fuzzy soft tripartite graph.

A Pythagorean fuzzy set is very effective mathematical framework to represent parameter-wise imprecision which is the property of linguistic communication. A Pythagorean fuzzy soft graph is more potent than the intuitionistic fuzzy soft as well as the fuzzy soft graph as it depicts the interactions among the objects of a system using Pythagorean membership grades with respect to different parameters. This article addresses the content of competition graphs as well as economic competition graphs like k-competition graphs, m-step competition graphs and p-competition graphs in Pythagorean fuzzy soft environment. All these concepts are illustrated with examples and fascinating results. Furthermore, an application which describes the competition among distinct forest trees, that grow together in the mixed conifer forests of California, for plant resources is elaborated graphically. An algorithm is also designed for the construction of Pythagorean fuzzy soft competition graphs. It is worthwhile to express the competing and non-competing interactions in various networks with the help of Pythagorean fuzzy soft competition graphs wherein a variation in competition relative to different attributes is visible.

Topological numbers of fuzzy soft graphs and their applications in globalizing the world by mutual trade

On some new operations in soft set theory

Fuzzy sets and soft sets are two different soft computing models for representing vagueness and uncertainty. In this chapter, we present these soft computing models in a combination applied to graphs. We discuss certain notions, including fuzzy soft graphs, strong fuzzy soft graphs, complete fuzzy soft graphs, regular fuzzy soft graphs and irregular fuzzy soft graphs. We describe the notions of fuzzy soft trees, fuzzy soft cycles, fuzzy soft bridges, fuzzy soft cutnodes and investigate some of their fundamental properties. We also discuss some types of arcs in fuzzy soft graphs. We describe applications of fuzzy soft graphs in social network and road network. This chapter is due to [28, 30, 48].

Fuzzy soft graphs are effective mathematical tools that are used to model the vagueness of the real world. A fuzzy soft graph is a fusion of the fuzzy soft set and the graph model and is widely used across different fields. In this current research, the concept of picture fuzzy soft graphs is presented by combining the theory of picture soft sets with graphs. The introduction of this new picture fuzzy soft graphs is an emerging concept that can be rather developed into various graph theoretical concepts. Since soft sets are most usable in real-life applications, the newly combined concepts of the picture and fuzzy soft sets will lead to many possible applications in the fuzzy set theoretical area by adding extra fuzziness in analyzing. The notions of picture soft graphs, strong and complete picture soft fuzzy graphs, a few types of product picture fuzzy soft graphs, and regular, totally regular picture fuzzy soft graphs are discussed and validated using real-world scenarios. In addition, an application of decision-making for medical diagnosis in the current COVID scenario using the picture fuzzy soft graph has been illustrated.

The soft set theory proposed by D. Molodtsov in 1999 is a general mathematical method for dealing with uncertain data. Now many researchers are applying soft set theory in decision making problems. Graph theory is the mathematical study of objects and their pairwise relationships, known as vertices and edges, respectively. The concept of soft graphs is used to provide a parameterized point of view for graphs. Directed graphs can be used to analyze and resolve problems with electrical circuits, project timelines, shortest routes, social links and many other issues. We introduced the notion of the soft directed graph by applying the concepts of soft set in a directed graph. In this paper, we introduce the concept of soft subdigraph and some soft directed graph operations like AND operation, OR operation, soft union, extended union, extended intersection, restricted union and restricted intersection and investigate some of their properties.

Molodtsov’s theory of soft sets is free from the parameterizations insufficiency of fuzzy set theory. Type-2 soft set as an extension of a soft set has an essential mathematical structure to deal with parametrizations and their primary relationship. Fuzzy type-2 soft models play a key role to study the partial membership and uncertainty of objects along with underlying and primary set of parameters. In this research article, we introduce the concept of fuzzy type-2 soft set by integrating fuzzy set theory and type-2 soft set theory. We also introduce the notions of fuzzy type-2 soft graphs, regular fuzzy type-2 soft graphs, irregular fuzzy type-2 soft graphs, fuzzy type-2 soft trees, and fuzzy type-2 soft cycles. We construct some operations such as union, intersection, AND, and OR on fuzzy type-2 soft graphs and discuss these concepts with numerical examples. The fuzzy type-2 soft graph is an efficient model for dealing with uncertainty occurring in vertex-neighbors structure and is applicable in computational analysis, applied intelligence, and decision-making problems. We study the importance of fuzzy type-2 soft graphs in chemical digestion and national engineering services.

Directed hypergraphs represent a natural extension of directed graphs, while soft set theory provides a method for addressing vagueness and uncertainty. This paper introduces the notion of soft directed hypergraphs by integrating soft set principles into directed hypergraphs. Through parameterization, soft directed hypergraphs yield a sequence of relation descriptions derived from a directed hypergraph. Additionally, we present several operations for soft directed hypergraphs, including extended union, restricted union, extended intersection, and restricted intersection, and explore their characteristics.

Soft set theory has a rich potential for application in many scientific areas such as medical science, engineering and computer science. This theory can deal uncertainties in nature by parametrization process. In this article, the authors explore the concepts of soft relation on a soft set, soft equivalence relation on a soft set, soft graphs using soft relation, vertex chained soft graphs and edge chained soft graphs and investigate various types of operations on soft graphs such as union, join and complement. Also, it is established that every fuzzy graph is an edge chained soft graph.

The soft sets and fuzzy soft sets have been used recently and since 2015 to define the fuzzy soft graphs. In this paper, we are going to extend the properties of fuzzy soft graphs by using its known parameters and properties as degree, totall degree, regularity and total regularity. Indeed, the direct product, combination and join of fuzzy soft graphs will be introduced and some main properties will be studied.

On operations of soft sets

In this paper, we define the notion of fuzzy soft graph and then introduce the concepts of homomorphism, isomorphism, weak isomorphism, co-weak isomorphism of fuzzy soft graphs. Also, we study some properties of isomorphism on fuzzy soft graphs.

Molodtsov developed soft set theory in 1999 as an approach for modeling vagueness and uncertainty. Many academics currently employ soft set theory to solve decision-making problems. A parameterized point of view for graphs is provided using the idea of soft graphs. Soft graph theory is a rapidly growing field in graph theory because of its adaptability to parameterization techniques. In graph theory, the topic of graph products has attracted much attention. It is a binary operation on graphs that has many combinatorial applications. Analogous to the definitions of graph products, we can define product operations on soft graphs. In this paper, we introduce the tensor product, the restricted tensor product, the strong product, and the restricted strong product of soft graphs. We prove that these products of soft graphs are also soft graphs, and we derive formulas for finding vertex count, edge count, and the sum of part degrees in them.

Soft set theory has gained prominence as a revolutionary approach for handling uncertainty-related problems and modeling uncertainty since it was proposed by Molodtsov. The concept of soft set operations, which is the major notion for the theory, has served as the foundation for theoretical and practical advances in the theory, therefore deriving the algebraic properties of the soft set operations and studying the algebraic structure of soft sets associated with soft set operations have attracted the researchers’ interest continuously. In the theory of soft set, many soft intersection operations have been defined up to now among which there are some differences, and some of which are no longer preferred for use as they are essentially not useful and functional. Although the definition of restricted intersection is widely accepted in the literature and used in the studies, it is still incomplete with its current form suffering from certain cases where the parameter sets of the soft sets may be disjoint is ignored, thus all the circumstances in the theorems are not considered in the related proofs causing to the incorrectness or deficiency in the studies where this operation is used or its properties are investigated. In this regard, in the existing literature, there is a critical lack of comprehensive study on the correct defined restricted intersection operation together with extended intersection including their correct properties and distributions and the correct algebraic structures assoiciated with these soft set operations. In this study, we primarly intend to fill this crucial gap by first correcting the deficiencies in the presentation of the definition of restiricted intersection and revising it. Moreover, in many papers related to these operation, several theorems were presented without their proofs, or there were some incorrect parts in the proofs. In this study, all the proofs based on the function-equality are regularly provided and besides, the relationships between the concept of soft subset and restricted and extended intersection operations are presented for the first time with their detailed proofs. Furhermore, we obtain many new properties of these operations as analogy and counterpart of intersection operation in classical set theory. Moreover, the operations’ full properties and distributions over other soft set operations are throughly investigated to determine the correct algebraic structures the operations form individually and in combination with other soft set operations both in the set of soft sets over the universe and with a fixed parameter set. We demonstrate that the restricted/extended intersection operations, when combined with other kinds of soft set operations, form several significant algebraic structures, such as monoid, bounded semi-lattice, semiring, hemiring, bounded distributive lattice, Bool algebra, De Morgan Algebra, Kleene Algebra, Stone algebra and MV-algebra but with deteailed explanations. In this regard, this overall study represents the most comprehensive analysis of restricted intersection and extended intersection in the literature to date as it covers all of the earlier important research on this topic with the corrected theorems and their proofs, thus advancing the theory by filling the significant gap in the literature, acting as a guide for the beginners of this popular theory, and besides shedding light on the future studies on soft sets.