On the Digraph of a Unitary Matrix

Given a matrix M of size n, the digraph D on n vertices is said to be the digraph of M , when Mij 6= 0 if and only if (vi, vj) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, − → L D, is the digraph of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and − → LD is the digraph of a unitary matrix then − → L D is Hamiltonian. We conclude with some elementary observations. Among the motivations of this paper are coined quantum random walks, and, more generally, discrete quantum evolution on digraphs.