On the Relationship between Petri Net Structure and Directed Graph

Petri net is a well-known bipartite graph theory to model and analyze discrete event systems. The properties of Petri net can be classified into two types, i.e., behavioral property and structural property. Many behavioral properties are investigated in association with the markings of Petri nets. On the other hand, the structural properties are just considered based on the Petri net structure without markings. In this meaning, Petri net has been classified to normal, cycle and parallel structures according to its homogenous state matrix equation. As Petri net is a bipartite graph, its structure can be transformed into a directed graph and the Mason's theorem can be applied to know the properties of the original net. In this paper, we discuss the relationship between Petri net structure and directed graph, and describe some results for the normal and cycle structures of Petri nets. Furthermore, several useful concepts, for example, transitive graph and transitive matrix are also defined here, and its characteristic polynomial and characteristic equation are used to find out the cycle structures.