Group inverses of matrices of simple symmetric directed graphs

A new class of simple symmetric digraphs called D is defined and studied here. Any digraph in D has the property that each non-pendant is adjacent to at least one pendant vertex. A graph theoretical description for the entries of the group inverse of a real square matrix A with any digraph belonging to this class is given. We classify all the real square matrix A such that the digraphs associated with A and A# both are in D, that is the digraph related to A is either a corona or a star digraph.