Characterization For The Entries Of Moore-penrose Inverse Of Some Graph Matrices

The generalized inverse of matrix is an important branch in the field of matrix theory,and it is an important tool to study the Least Square problem,System Identification,Markov Chain and other problems.The generalized inverse of matrix is widely used in many fields,such as Neural Network,Optimization,Cybernetics,Differential equation and Integral equation.Moore-Penrose inverse is a kind of generalized inverse,in recent years,R.B.Bapat has given entries of Moore-Penrose inverse of the incidence matrix such as directed tree and directed complete multipartite graphs and the expression of Moore-Penrose inverse entries of all-paths matrix of paths,on this basis,the main work of this paper is as follows:1.In this paper,we mainly use the distance between vertices and edges,the number of vertices and edges of a graph to give the expression of Moore-Penrose inverse entries of the incidence matrix of a directed acyclic graph.2.In this paper,we give the expression of Moore-Penrose inverse entries of the incidence matrix of some bipartite graphs by using the number of edges of spanning trees,the degree of vertices and where a vertex is associated with an edge.3.Based on the definition of all-paths matrix of undirected tree,this paper gives the expression of Moore-Penrose inverse entries of all-paths matrix of undirected star graph.The definition of the all-paths matrix of the directed tree is given,and the relationship between the Moore-Penrose inverse of the incidence matrix of the directed tree and the Moore-Penrose inverse of the all-paths matrix is given in combining the path passing edge and the direction of the edge in the tree.