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 ofM , when $M_{ij}\neq 0$ if and only if (v,sub>i,v,sub>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 $\overrightarrow{L}D$ is the pattern of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and $\overrightarrow{L}D$ is the digraph of a unitary matrix, then $\overrightarrow{L}D$ is Hamiltonian. We observe that strong quadrangularity is sufficient to show that disconnected strongly regular graphs are the digraphs of unitary matrices and that n-paths, n-paths with loops at each vertex, n-cycles, directed trees, and trees are not.