On the digraph of a unitary matrix

Abstract

Given a matrix M of size n, a digraph D on n vertices is said to be the digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) 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, LD, 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 LD 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.