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.

Molodtsov pioneered the notion of soft set theory, presenting it as a mathematical tool for dealing with uncertainty. Numerous researchers have subsequently developed models leveraging this theory to tackle challenges in decision-making and medical diagnosis. Soft set theory emerges as a flexible framework adept at handling uncertain and imprecise information, a domain where classical set theory often struggles. Expanding on the soft set concept, researchers have introduced the idea of a soft graph. This innovative concept allows for the creation of diverse representations of graph-based relations by incorporating parameterisation. In this work, we present and investigate some of the features of the homomorphic and restricted homomorphic products of soft graphs. This paper establishes the structural properties of these products, ensuring that they are well-defined and maintain the essential characteristics of soft graphs. Additionally, we derive combinatorial identities related to the counts of vertices and edges, as well as the degree sums, offering deeper insights into the composition and behaviour of these graph products.

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.

This paper extends the framework of soft set theory to directed graphs by introducing and analyzing the concepts of the normal product and restricted normal product of soft directed graphs. Building on Molodtsov’s foundational work in developing soft set theory to address uncertainty in data, this study presents new methods for modeling and understanding complex systems where uncertainty plays a significant role. Soft directed graphs, which enhance traditional graph models by incorporating parameters and uncertain relationships, serve as the foundation for this investigation. The normal product, defined as a combination of two soft directed graphs based on their respective parameter sets, and the restricted normal product, which combines soft directed graphs only where their parameter sets intersect, provide a comprehensive framework for these new operations. This paper also establishes the structural properties of these products, ensuring they are well-defined and retain the key features of soft directed graphs. Furthermore, we derive combinatorial identities related to vertex and arc counts, as well as degree sums, offering deeper insights into the composition and behavior of these graph products.

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 [Formula: see text]-part consecutive adjacent degree, [Formula: see text]-part end degree, and the [Formula: see text]-part consecutive adjacency graph. Additionally, we provide the conditions for a soft semigraph to be Hamiltonian. We introduce the concept of maximal non-Hamiltonian [Formula: see text]-part. Finally, we define the closure of a soft semigraph and demonstrate the relationship between a Hamiltonian soft semigraph and its closure.

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 general mathematical method for dealing with uncertain data. It is a classification of elements of the universe with respect to some given set of parameters. Semigraph is a generalization of graph which is different from hypergraph. Soft semigraph was introduced by applying the concept of soft set in semigraph. Connectedness is one of the basic concepts of graph theory. In this paper, we introduce the concepts of [Formula: see text]-walk, [Formula: see text]-trail, [Formula: see text]-path and [Formula: see text]-cycle in soft semigraphs and we define an [Formula: see text]-connected soft semigraph. We explain the procedures of vertex and [Formula: see text]-edge deletions from a soft semigraph. Also, we introduce [Formula: see text]-part cut vertex, [Formula: see text]-cut vertex, [Formula: see text]-part bridge and [Formula: see text]-bridge of a soft semigraph and investigate some of their properties.

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, [Formula: see text]-isomorphism, sev-isomorphism, se-isomorphism and sa-isomorphism. We prove that [Formula: see text]-isomorphism between soft semigraphs is also sev-isomorphism. Moreover, we establish that sev and [Formula: see text]-isomorphisms are also se-isomorphism and [Formula: see text], 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.

Soft set theory, introduced by D. Molodtsov, is a mathematical approach that tackles uncertain data. Semigraph is a generalization of the concept of a graph. By incorporating soft sets into semigraphs, a new concept called soft semigraphs has emerged. The study of soft semigraphs has gained significance in graph theory due to their valuable applications in parametrization. This paper explores the properties of various vertex degrees in soft semigraphs, including degrees, end degrees, edge degrees, adjacent degrees, and consecutive adjacent degrees.

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.

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.

Most of our conventional tools for formal reasoning, computing, and modeling are precise, deterministic, and crisp. However, many complicated problems in the domains of economics, medicine, engineering, the environment, social science, and other disciplines demand data that is not always precise. We cannot always use traditional approaches because there are so many different types of uncertainty present in these problems. This difficulty could be caused by the parameterization tool's insufficiency. Soft set theory, which Molodtsov presented in 1999, is a generic mathematical approach for handling uncertain data. Many researchers are currently using soft set theory to solve problems involving decision-making. The concept of soft graphs is used to provide a parameterized point of view for graphs. The topic of graph products has received a lot of interest in graph theory. It is a binary operation on graphs with numerous combinatorial uses. On soft graphs, we can define product operations in a manner similar to how graph products are defined. The co-normal product, the restricted co-normal product, the modular product, and the restricted modular product of soft graphs are all introduced in this study. We prove that these products of soft graphs are again soft graphs and derive methods for computing their vertex count, edge count, and the sum of part degrees.

This paper presents an introduction to soft hypergraph operations, namely extended union, extended intersection, restricted union, and restricted intersection, along with the concept of soft semisubhypergraph. We explore various properties of these operations and provide illustrative examples.

D. Molodtsov proposed the soft set theory as a mathematical framework to handle uncertain data, which has now become a popular approach for solving decision-making problems. A directed graph is a type of graph where edges have a specific direction. Directed graphs are useful for analyzing social connections, and electrical circuits, and finding the shortest paths. Soft directed graphs were introduced to extend the concept of soft set theory to directed graphs. Soft directed graphs provide a parameterized perspective on directed graphs. This study examines the disjunctive product and restricted disjunctive product of soft directed graphs and their properties.

A soft set is a classification of elements of the universe with respect to some given set of parameters. It is an approach for modeling vagueness and uncertainty. Soft semigraphs are used to provide a parametrized point of view for semigraphs. The theory of soft semigraphs is a fast-developing area in semigraph theory due to its capability to deal with the parametrization tool. In this paper, we introduce soft disemigraph by applying the concept of soft set in disemigraph and the concepts of degrees and digraphs associated with a soft disemigraph. Also we introduce AND and OR operations on them and investigate some of their properties.