The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. ...The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.展开更多
This paper focuses graph theory method for the problem of decomposition w.r.t. outputs for Boolean control networks(BCNs). First, by resorting to the semi-tensor product of matrices and the matrix expression of BCNs, ...This paper focuses graph theory method for the problem of decomposition w.r.t. outputs for Boolean control networks(BCNs). First, by resorting to the semi-tensor product of matrices and the matrix expression of BCNs, the definition of decomposition w.r.t. outputs is introduced. Second, by referring to the graphical structure of BCNs, a necessary and sufficient condition for the decomposition w.r.t. outputs is obtained based on graph theory method. Third, an effective algorithm to realize the maximum decomposition w.r.t. outputs is proposed. Finally, some examples are addressed to validate the theoretical results.展开更多
文摘The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.
基金supported in part by the National Natural Science Foundation of China under Grant Nos.61673012,11271194a Project on the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD)
文摘This paper focuses graph theory method for the problem of decomposition w.r.t. outputs for Boolean control networks(BCNs). First, by resorting to the semi-tensor product of matrices and the matrix expression of BCNs, the definition of decomposition w.r.t. outputs is introduced. Second, by referring to the graphical structure of BCNs, a necessary and sufficient condition for the decomposition w.r.t. outputs is obtained based on graph theory method. Third, an effective algorithm to realize the maximum decomposition w.r.t. outputs is proposed. Finally, some examples are addressed to validate the theoretical results.
