 
Summary: Linear Algebra and its Applications 428 (2008) 16851695
Available online at www.sciencedirect.com
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Unitary matrix digraphs and minimum semidefinite rank
Yunjiang Jiang 1, Lon H. Mitchell, Sivaram K. Narayan ,1
Department of Mathematics, University of Georgia, Athens, GA 30602, United States
Department of Mathematics, Virginia Commonwealth University, Richmond, VA 23284, United States
Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, United States
Received 31 March 2006; accepted 13 October 2007
Submitted by S. Fallat
Abstract
For an undirected simple graph G, the minimum rank among all positive semidefinite matrices with graph
G is called the minimum semidefinite rank (msr) of G. In this paper, we show that the msr of a given graph
may be determined from the msr of a related bipartite graph. Finding the msr of a given bipartite graph is
then shown to be equivalent to determining which digraphs encode the zero/nonzero pattern of a unitary
matrix. We provide an algorithm to construct unitary matrices with a certain pattern, and use previous results
to give a lower bound for the msr of certain bipartite graphs.
© 2007 Elsevier Inc. All rights reserved.
AMS classification: 15A18; 15A57; 05C50
Keywords: Rank; Positive semidefinite; Digraph; Unitary; Graph; Quadrangular
