Soft adjacency matrix and soft planer graphs and its applications

Using soft graphs along with fuzzy graphs, interval valued fuzzy graphs, bipolar fuzzy graphs and vague graphs is another way to solve problems that are faced with uncertainties. Since a soft graph is a set of subsets of a simple graph, it is necessary to generalize some concepts of simple graphs to soft graphs. Therefore, many researchers have studied on soft graphs and defined some concepts and operators such as community, sharing and complement for it. In the continuation of this research, in this article, we first define the definition of soft adjacency matrix and we define union, intersection, addition and difference for soft adjacency matrices and obtain the relationship between these sets. Then we will define the order, size and degree for soft graphs and express the concept of planar soft graph and dual soft graph and then we will examine the relationship between order and size in planar soft graph. At the end of the article, an example of the application of soft planar graphs in the control of urban traffic flows is stated.