Research On Weak Connectedness Of Tensor Product Of Some Digraphs

Tensor product of graphs is derived from tensor product of matrices,which has many applications in theory of games and theory of automata.As an operation of graphs,more classes of graphs can be obtained by tensor product.Tensor product of graphs is a tool to study the spectrum of graphs and plays an important role in the spectral graph theory.Kronecker canonical form is the canonical form for a pair of matrices under simultaneous equivalence,which comes from solving linear differential equations and has many applications in generalized eigenvalues problem.The classification of tensor product of graphs is an important problem,some invariants of graphs can be obtained through the incidence matrix of graphs.Therefore,the problems related to the tensor product of graphs and Kronecker canonical form of matrix pencil have attracted extensive attention.The thesis researches weakly connected property of tensor product of digraphs and the relationship between tensor product of graphs and Kronecker canonical form of incidence matrix pencil.It mainly studies how to produce trees from tensor product of two digraphs and how to transform incidence matrix pencil of digraph into its Kronecker canonical form.The thesis first proves that trees can be produced by taking tensor product of directed path and a class of tricyclic graph.And then,it is proved that whether underlying graph of tensor product of directed path and digraph is a tree completely determined by Kronecker canonical form of incidence matrix pencil.Finally,the method of transforming the incidence matrix pencil of a class of tricyclic of T-type digraphs into Kronecker canonical form is presented,and it is proved that incidence matrix pencil of a class of T-type tricyclic digraphs can be transformed into Kronecker canonical form through invertible matrices over the ring of integers.