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