Moore-penrose inverse of the incidence matrix of a tree

Abstract

Let T be a tree with n vertices, where each edge is given an orientation, and let Q be its vertex-edge incidence matrix. It is shown that the Moore-Penrose inverse of Q is the (n-1)×n matrix M obtained as follows. The rows and the columns of M are indexed by the edges and the vertices of T respectively. If e,v are an edge and a vertex of T respectively, then the (e,v)-entry of M is, upto a sign, the number of vertices in the connected component of T\e which does not contain v. Furthermore, the sign of the entry is positive or negative, depending on whether e is oriented away from or towards v. This result is then used to obtain an expression for the Moore-Penrose inverse of the incidence matrix of an arbitrary directed graph. A recent result due to Moon is also derived as a consequence.