Soft Graph Products: A Soft Subgraph Perspective

Soft set theory provides a flexible framework for modeling uncertainty, and its application to graph theory has led to the development of soft graphs and their various operations. In this paper, we explore the relationships among different soft graph products from a soft subgraph perspective. Specifically, we establish and prove soft subgraph relations among major soft graph products, including Cartesian, lexicographic, co-normal, strong, tensor, modular, and rooted products. The study shows that the Cartesian product of two soft graphs is a soft subgraph of the lexicographic, co-normal, and strong products; the lexicographic product is a soft subgraph of the co-normal product; the tensor product is a soft subgraph of the lexicographic, modular, normal, and co-normal products; the strong product is a soft subgraph of the lexicographic and co-normal products; and the rooted product is a soft subgraph of the Cartesian, lexicographic, co-normal, and strong products. Illustrative examples are provided to validate these theoretical results. The findings contribute to a clearer structural understanding of interrelations among soft graph products.