The M-matrix group inverse problem for star networks

We study the inverse problem for singular, irreducible, symmetric M -matrices that consists in characterizing those for which the group inverse is again an M -matrix. We solve the problem for a structured class of such matrices arising from star graphs by employing both matrix-theoretic tools and potential theory on networks. Our approach yields explicit criteria for the M -property of the group inverse in terms of conductances and Doob potentials, and provides a constructive procedure to obtain such families of matrices with the desired property.