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On the digraph of a unitary matrix

Severini, S; (2003) On the digraph of a unitary matrix. SIAM J MATRIX ANAL A , 25 (1) 295 - 300. 10.1137/S0895479802410293.

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Given a matrix M of size n, the digraph D on n vertices is said to be the digraph of M, when M-ij not equal 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 (L) over right arrowD is the pattern of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and (L) over right arrowD is the digraph of a unitary matrix, then (L) over right arrowD 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.

Type: Article
Title: On the digraph of a unitary matrix
DOI: 10.1137/S0895479802410293
Keywords: digraphs, unitary matrices, quantum random walks, GRAPHS
UCL classification: UCL > Office of the President and Provost
UCL > School of BEAMS
UCL > School of BEAMS > Faculty of Engineering Science
URI: http://discovery.ucl.ac.uk/id/eprint/404727
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