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Soft set theory, proposed by D. Molodtsov, is a mathematical framework for dealing with uncertain data. Soft set theory is now widely used to solve decision-making problems and has a wide range of applications in economics, engineering, medicine, and other fields, as demonstrated by Molodtsov’s pioneering work. A directed graph is a graph with directed edges. Directed graphs can be used to study and solve problems involving social networks, shortest paths, electrical circuits, and so on. By extending the notion of the soft set to directed graphs, we presented soft directed graphs. Soft directed graphs offer a parameterized perspective on directed graphs. In this paper, we introduce and investigate the corona product and the restricted corona product of soft directed graphs.

A soft set is a mathematical tool designed for managing uncertainty. Semigraph is a generalization of a graph which is different from a hypergraph. Soft semigraph was introduced by applying the concept of soft set in semigraph. Through parameterization, soft semigraph generates a series of representations of a relationship given by a semigraph. Graph isomorphism serves as a crucial method for matching patterns in a range of areas including image processing, computer and information systems, social network analysis, exploration of chemical bonds, and analysis of protein structures. In this paper, we present various kinds of isomorphisms existing among soft semigraphs, namely, $s$-isomorphism, sev-isomorphism, se-isomorphism and sa-isomorphism. We prove that $s$-isomorphism between soft semigraphs is also sev-isomorphism. Moreover, we establish that sev and $s$-isomorphisms are also se-isomorphism and $s$, se and sev-isomorphisms are also sa-isomorphism. Through specific examples, we illustrate instances where the converse aspect of some of these results may not hold.

The real world is full of uncertainties, which we require appropriate mathematical structures to model and handle them. One such structure is the soft set, which can be combined with various algebraic and topological structures. The paper discusses how soft sets and hyperstructures can be combined to create new soft structures that can handle uncertainty and imprecision in algebraic structures. Soft sets are mappings from parameters to subsets of a universal set, and they can model fuzzy concepts. Hyperstructures are generalizations of classical algebraic structures, where the composition of two elements is a set instead of a single element. They can capture complex and diverse phenomena that cannot be modeled by single-valued operations. The paper gives examples of soft structures, such as soft polygroups, soft topological polygroups, soft topological soft polygroups and their applications in mathematics and science. The paper aims to show the usefulness and versatility of soft structures in dealing with uncertainty and imprecision in algebraic structures. We will define the concept of soft topological soft polygroups and examine its characteristics.

The emergence of soft set theory represented a significant advancement in establishing a holistic mathematical framework for handling uncertain data. Currently, numerous scholars are integrating this theory to address decision-making problems. In graph theory, a directed graph consists of vertices connected by directed edges, commonly referred to as arcs. Soft directed graphs are introduced by integrating the principles of soft sets into directed graphs, offering a parameterized perspective on directed graphs. In this paper, we introduce the fundamental notions of order, size, and frequency within the context of soft directed graphs. Furthermore, we establish the concepts of vertex and arc degree sums and explore their properties. Additionally, we delve into the concepts of strong vertex and strong arc, elucidating their significance. Finally, we define the notion of complement within soft directed graphs.

The introduction of soft set theory represented a crucial advancement in establishing a thorough mathematical structure for handling uncertain data. Currently, numerous researchers are integrating this theory to address decision-making challenges. Within graph theory, a directed graph consists of vertices linked by directed edges, or arcs. The incorporation of soft set concepts into directed graphs leads to the formation of soft directed graphs. These graphs offer a parameterized perspective on directed graphs. This study delves into the examination of Eulerian properties within soft directed graphs.

Molodtsov’s introduction of soft set theory marked a significant advancement in managing uncertainty, addressing the limitations of classical set theory in handling imprecise and vague information. This versatile framework has since been extensively explored, leading to sophisticated models for decision-making and medical diagnosis. Building on the principles of soft set theory, the concept of soft graphs was introduced to enhance the representation and analysis of uncertain information. Soft graphs incorporate parametrization, enabling dynamic modeling of graph-based relationships across various contexts and applications. This innovative approach creates multiple, adaptable representations that better reflect real-world complexities. This survey paper provides a comprehensive guide to the evolution of soft graphs, starting with an overview of the origins and basic concepts of soft set theory, highlighting its strengths and applications. It then delves into the development of soft graphs, explaining their theoretical foundations and methodologies. The paper explores how soft graphs have been used to model and solve problems in different domains, emphasizing their flexibility and effectiveness. By examining both foundational concepts and contemporary research advancements, this paper serves as a valuable resource for researchers interested in the applications of soft set theory and soft graphs in managing uncertainty and imprecision.

A soft set functions as a mathematical instrument crafted to handle uncertainties. Semigraph, a broader form of graph distinct from hypergraph, saw the integration of soft set principles to give rise to soft semigraph. Utilizing parameterization, soft semigraph delineates various representations of a semigraph’s relationships. The pivotal role of graph isomorphism lies in its ability to discern patterns across diverse domains such as image processing, computer systems, social network analysis, chemical bonding exploration, and protein structure analysis. Different types of isomorphisms are present within soft semigraphs, including $s$-isomorphism, $sev$-isomorphism, $se$-isomorphism, and $sa$-isomorphism. This paper focuses on elucidating certain characteristics associated with these isomorphisms. Additionally, we investigate the complement of a soft semigraph.

Soft set theory provides a systematic approach for handling imprecision and uncertainty by categorizing elements of a set based on specific parameters. In semigraph theory, soft semigraphs utilize this approach, offering a parameterized perspective that has significantly advanced the field through effective parameter management. Building on this foundation, disemigraphs extend semigraphs by incorporating directional relationships among vertices, making them ideal for modeling scenarios where the sequence and direction of connections are crucial. In this paper, we introduce and explore the lexicographic product, and restricted lexicographic product of soft disemigraphs. We provide formal definitions, illustrative examples, and detailed proofs of several theorems to investigate the properties of these product operations.

Soft set theory is a broad mathematical method for handling uncertain data. Numerous scholars are now using soft set theory to address problems related to decision-making. The concept of soft graphs provides a parameterized point of view for graphs. In graph theory, graph products are binary operations on graphs with various combinatorial purposes. We can define product operations of soft graphs in a manner similar to how graph products are defined. In this paper, we introduce the corona product, the restricted corona product, the rooted product, and the restricted rooted product of soft graphs. We prove that these products of soft graphs are again soft graphs. We also derive methods for computing their vertex count, edge count, and the sum of part degrees.

Soft set theory is a mathematical approach to address the challenges of handling vague or uncertain information. It is a more advanced version of classical set theory that deals with imprecise elements and enables the flexible representation of uncertain data. It involves categorizing the elements of the universe based on specific parameters. Semigraph is a generalization of a graph which is different from a hypergraph. A hypergraph extends the concept of a graph by allowing any subset of vertices to form an edge. Semigraphs, on the other hand, distinguish themselves from hypergraphs by imposing a specific order on the vertices within each edge. Soft semigraphs were developed using the principles of soft set theory applied to semigraphs. This study introduces Eulerian and Hamiltonian soft semigraphs. We establish a necessary and sufficient condition for a soft semigraph to be Eulerian, relying on parameters such as $p$-part consecutive adjacent degree, $p$-part end degree, and the $p$-part consecutive adjacency graph. Additionally, we provide the conditions for a soft semigraph to be Hamiltonian. We introduce the concept of maximal non-Hamiltonian $p$-part. Finally, we define the closure of a soft semigraph and demonstrate the relationship between a Hamiltonian soft semigraph and its closure.

This paper is devoted to the explication of the introduction of rational-type soft fuzzy contraction in soft fuzzy metric space. New fixed point results for these new contractions along with relevant and valid examples have been presented and generalized. We have also presented an application to our new formed results in finding the solution of a nonlinear integral equation.

In this paper, we propose generalized hesitant fuzzy soft sets by integrating generalized fuzzy soft sets with hesitant fuzzy sets. Then we investigate several fundamental properties of generalized hesitant fuzzy soft sets. In particular, a system of all the generalized hesitant fuzzy soft sets with narrow strict union and generalized strict intersection operators turns out to be a lattice. Meanwhile, a system of all the generalized hesitant fuzzy soft sets with generalized strict union and narrow strict intersection operators is a lattice. Based on generalized hesitant fuzzy soft sets, we provide an effective approach to decision making. Finally, three examples are presented to illuminate the feasibility of the method.

This paper introduces a method for detection and identification of IGBT-based drive open-circuit fault of DTC induction motor drives. The detection mechanism is based on soft set theory and wavelet decomposition, if it is detailed, ⊼-product decision making method and sym2 wavelet decomposition have been used in the detection mechanism. In this method, the stator currents have been used as an input to the system. The stator current has been used for the detection of the fault. The signal analysis has been performed up to the six level details wavelets decomposition. Faulty switch is detected by applying soft set theory to sixth level wavelets transformation. This is the first time applied to inverter in induction motor drives fault detection. The results demonstrate that the proposed fault detection and diagnosis system has very good capabilities.

In this paper, we introduce the concept of soft intervals, soft ordering and sequences of soft real numbers, and some of their structural properties are studied. The notion of soft Lebesgue measure on the soft real numbers has been introduced. Also, a correspondence relationship has been established between the soft Lebesgue measure and the classical Lebesgue measure. Furthermore, we have studied some exciting results and relations between the soft Lebesgue measure and the Lebesgue measure of soft real sets.

In this paper, we define the notion of a mapping on soft classes and study several properties of images and inverse images of soft sets supported by examples and counterexamples. Finally, these notions have been applied to the problem of medical diagnosis in medical expert systems.

It is well known that statistics deals with quantitative analysis. Thus, there is a lack of approach to do quantitative analysis in the presence of qualitative attributes. Soft set theory has the freedom to deal with attributes, [0, 1], etc. along with quantity. Thus, we introduce some fundamental ideas of soft statistics.

Here, soft mean, soft standard deviation, soft coefficient of variation, soft correlation coefficient are introduced and some theorems are proved with respect to utility. Utility theory provides an analysis of choice behavior. As an application of our notions and results, we find soft correlation coefficients between vulnerability and government responses of various regions across the world. The data from “The Global Slavery Index 2016” are considered for application purposes.

In this paper, we prove that Lemma 5.7, Proposition 6.8 and Proposition 6.9 of Kharal [A. Kharal, Distance and similarity measures for soft sets, New Mathematics and Natural Computation6(3) (2010) 321–334] are not correct w.r.t his Definition 5.5. We give suitable examples to show that Lemma 5.7 is not correct. Moreover, we rectify Lemma 5.7, Proposition 6.8 and Proposition 6.9 by providing suitable proofs.

This paper focuses on two very important questions: “what is the future of a hybrid mathematical structure of soft set in science and social science?” and “why should we take care to use hybrid structures of soft set?”. At present, these are the most fundamental questions; which encircle a few prominent areas of mathematics of uncertainties viz. fuzzy set theory, rough set theory, vague set theory, hesitant fuzzy set theory, IVFS theory, IT2FS theory, etc. In this paper, we review connections of soft set theory and hybrid structures in a non-technical manner; so that it may be helpful for a non-mathematician to think carefully to apply hybrid structures in his research areas. Moreover, we must express that we do not have any intention to nullify contributions of fuzzy set theory or rough set theory, etc. to mankind; but our main intention is to show that we must be careful to develop any new hybrid structure with soft set. Here, we have a short discussion on needs of artificial psychology and artificial philosophy to enrich artificial intelligence.

Soft set theory introduced by D. Molodstov is a new theory which deals with uncertainty. Connected graphs can be represented by using soft sets called soft graphs. In the present paper, we introduce the tabular representation of soft graph and define radius, diameter, center and degree of soft graph. We also define union, product of soft graphs and soft trees. We then derive some properties of radius, degree of vertex in soft graph and soft trees.

Molodtsov initiated the soft set theory, providing a general mathematical framework for handling uncertainties that we encounter in various real-life problems. The main object of this paper is to lay a foundation for providing a new soft algebraic tool for considering many problems that contain uncertainties. In this paper, we introduce a new kind of soft ring structure called $(\alpha ,\beta )$-soft-intersectional ring based on some results of soft sets and intersection operations on sets. We also define $(\alpha ,\beta )$-soft-intersectional ideal and $(\alpha ,\beta )$-soft-intersectional subring, and investigate some of their properties using these new concepts. We obtain some results in ring theory based on $(\alpha ,\beta )$-soft intersection sense and its application in ring structures. Furthermore, we provide relationships between soft-intersectional ring and $(\alpha ,\beta )$-soft-intersectional ring, soft-intersectional ideal and $(\alpha ,\beta )$-soft-intersectional ideal.