Nonnegative alternating circulants leading to \(M\)-matrix group inverses (Q1906772)

Let \(\mathcal C\) denote the set of all irreducible nonnegative alternating circulant matrices, i.e. all matrices \(A= \sum^{n- 1}_{i= 0} \alpha_i P^i\), when \(P\) is an \(n\)th order cyclic permutation matrix, where \(\alpha_i> 0\) and \(\alpha_i= \alpha_j\) for \(i= j\pmod 2\). Those matrices \(B\) in \(\mathcal C\) are characterized for which the group inverse \((rI- B)^{\neq}\), where \(r= \rho(B)\), is also an \(M\)-matrix. For those matrices \(\rho(B)\) is a concave function of the off-diagonal entries.