Unitary matrix digraphs and minimum semidefinite rank |
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Authors: | Yunjiang Jiang Lon H Mitchell |
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Institution: | a Department of Mathematics, University of Georgia, Athens, GA 30602, United States b Department of Mathematics, Virginia Commonwealth University, Richmond, VA 23284, United States c Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, United States |
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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. |
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Keywords: | 15A18 15A57 05C50 |
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