The Research On Representations Of Drazin Inverse Of Adjacency Matrix For Some Directed Graphs

The generalized inverses of the matrix have widely applications in many fields, research on the representations of generalized inverses of the matrix is very important. The representations for the Drazin inverse of the block matrices is a very important branch of generalized inverses of the matrix, and have a large number important applications in solving singular differential equation and difference equation, iterative method, cryptography, Markov chain and many other important areas, so the Drazin inverse of matrix is attracted more attention of scholars.In recent decades, graph theory as a branch of mathematics developed very rapidly. The graph in graph theory consists of a set of points together with lines joining certain pairs of these points. In numerous studies graph theory methods, matrix theory has important appli-cations. The matrices of graph play an important role in characterization the structure and properties of graphs. For example: the second small eigenvalues of Laplacian matrix for graph,ie. algebraic connectivity of graph, can be used to determine the connectivity of graph. In recent years,many researchers focus on the study of the generalized inverses of the matrix for graphs, and obtained many research results. Scholars presented the group inverses of laplace matrix for weighted trees?Moore-Penrose inverse of matrices with an acyclic bipartite graph and group inverse of adjacency matrices with bipartite digraph.In this paper, we mainly give the representations of the Drazin inverse for the adjacency matrix of the following digraphs. At the same time, the article also depicted the index of directed graph adjacency matrix, whose vertex degree of 1 .(1) The digraph of a vertex of the cycle connected with several chains;(2) The digraph of a vertex of the cycle connected with the chain which has s subchain;(3) The digraph of every vertex of the cycle connected with same chains;In addition, we give the representations of Group inverse for adjacency matrix of windmill digraph?digraph of cycle and chain by a cartesian product?sunflower digraph?conical digraph and digraph of multiple cycle connected.