Graphical description of group inverses of certain bipartite matrices

Abstract

A necessary and sufficient condition for the existence of the group inverse of a certain bipartite matrix is given, and an explicit formula is obtained for the group inverse in terms of its block submatrices. This form is used to derive a graph–theoretic description of the entries of the group inverse of some examples of such a matrix, including those corresponding to broom graphs. If the group inverse of a nonnegative matrix corresponding to a broom graph exists, then it is shown that this group inverse is signed. An open question about group inverses of more general bipartite matrices is posed and a summary of cases for which its answer is known is given.