On the matrix equation Al + Al+k = Jn Original Research Article

This paper studies the matrix equation Al + Al+k = Jn, where l,k are nonnegative integers, Jn is the n × n matrix of all lʹs, and A is an unknown (0,1)-matrix. We shall provide a solution for every odd k and every n which is feasible, i.e. n = dl + dl+k for some nonnegative integer d, and show that the equation has no solution in other cases with some trivial cases excluded. We also show that for any solution A to this equation there must be a (0,1)-matrix C satisfying I + Ck = Jdk+1, and Γ(A), the associated digraph of A, is the lth iterated line digraph of Γ(C). In particular, the well-known Kautz digraph K(d, l + 1) can be characterized as Γ(A), where A satisfies Al + Al+1 = Jn for n = dl + dl+1.