A Survey on Soft Directed Graphs: Definitions, Structural Properties, and Applications in Uncertain Systems

This article presents a review of the main concepts and results in the theory of soft directed graphs. We first recall the definition of a soft directed graph and describe its basic structure. We then review soft indegree, soft outdegree, degree sums, order, size, frequency, and related measures. Adjacency and incidence matrices used to represent soft directed graphs are discussed. We also present soft walks, trails, paths, cycles, distance, and different types of connectedness. The product operations studied for soft directed graphs are reviewed, including Cartesian, lexicographic, categorical, modular, disjunctive, homomorphic, corona, rooted, and normal products, together with their restricted forms wherever defined. The main results concerning the order, size, and degrees of these product structures are also presented. The article brings together the definitions, structural properties, matrix representations, connectedness results, and product operations of soft directed graphs in a single review.