On the matrix equation \(A^k=J-I\) (Q1124912)

The paper deals with the (0,1)-matrix equation (1) \(X^k= L_n\), where \(k\) is a positive integer, the main diagonal of the \(n\times n\) matrix \(L_n\) is zero and the off-diagonal elements of \(L_n\) are equal to 1. It is known that equation (1) is solvable if and only if \(k\) is odd and \(n=d^k+1\) for an integer \(d\geq 0\). The authors present the complete solution for the case \(d=2\). Several open problems are also posed.