Null decomposition and a combinatorial-block generalized inversion of trees

Abstract

We study the Drazin inverse of the adjacency matrix of a tree \( T \) through its null decomposition. This decomposition, which partitions the vertex set of \( T \), reveals a close connection between the Drazin inverse and the matching and independence structures of the tree. In addition, we use a Bjerhammar-type condition for the Drazin inverse of a matrix.