  A new class of simple symmetric digraphs called $\mathbb{D}$ is defined and
studied here. Any digraph in $\mathbb{D}$ 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 $\mathbb{D}$, that is the digraph related to $A$ is either a corona or a
star digraph.
