2011
DOI: 10.1080/03081080903542791
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On the minimum semidefinite rank of a simple graph

Abstract: The minimum semidefinite rank (msr) of a graph is defined to be the minimum rank among all positive semidefinite matrices whose zero/ nonzero pattern corresponds to that graph. We recall some known facts and present new results, including results concerning the effects of vertex or edge removal from a graph on msr.

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Cited by 25 publications
(25 citation statements)
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“…Booth et al [3] and van der Holst [18] have independently shown that the positive semidefinite minimum rank is completely determined by the blocks of G independently of how they are joined. More specifically: if G 1 , ..., G k are the blocks of G then mr K …”
Section: + (G) D |G| − 1 Then There Exists a Sign Pattern F Such That Mrmentioning
confidence: 98%
“…Booth et al [3] and van der Holst [18] have independently shown that the positive semidefinite minimum rank is completely determined by the blocks of G independently of how they are joined. More specifically: if G 1 , ..., G k are the blocks of G then mr K …”
Section: + (G) D |G| − 1 Then There Exists a Sign Pattern F Such That Mrmentioning
confidence: 98%
“…An orthogonal representation in C d naturally generates a (d; 1) orthogonal subspace representation, and vice versa, so ξ(G) = ξ [1] (G).…”
Section: Orthogonal Subspace Representations and R-fold Orthogonal Rankmentioning
confidence: 99%
“…Remark 2.11. Since ξ [1] (G) = ξ(G) for every graph G, the previous properties of r-fold orthogonal rank also apply to orthogonal rank, where appropriate.…”
Section: Proof Without Loss Of Generality Let Dmentioning
confidence: 99%
“…The positive semidefinite case. A variant of the minimum rank problem which has been well-studied (see, e.g., [7], [8], [13,Section 46.3] and [19]) is that in which only positive semidefinite matrices are considered. For convenience in this context, we set up the following notation, analogous to Notation 1.2.…”
Section: Circulant Matricesmentioning
confidence: 99%