On the minimum semidefinite rank of a simple graph

Booth, Matthew; Hackney, Philip; Harris, Benjamin; Johnson, Charles R.; Lay, Margaret; Lenker, Terry D.; Mitchell, Lon H.; Narayan, Seema; Pascoe, Amanda; Sutton, Brian D.

doi:10.1080/03081080903542791

Cited by 25 publications

(25 citation statements)

References 17 publications

(18 reference 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%

A new lower bound for the positive semidefinite minimum rank of a graph

Zimmer

2013

Linear Algebra and its Applications

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Section: + (G) D |G| − 1 Then There Exists a Sign Pattern F Such That Mrmentioning

confidence: 98%

A new lower bound for the positive semidefinite minimum rank of a graph

Zimmer

2013

Linear Algebra and its Applications

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“…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%

Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions

Hogben

Palmowski

Roberson

et al. 2017

ELA

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Abstract. Fractional minimum positive semidefinite rank is defined from r-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An r-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. This paper also introduces r-fold orthogonal representations of graphs and formalizes the understanding of projective rank as fractional orthogonal rank. Connections of these concepts to quantum theory, including Tsirelson's problem, are discussed.

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“…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%

The minimum rank problem for circulants

Deaett

Meyer

2016

Linear Algebra and its Applications

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The minimum rank problem is to determine for a graph $G$ the smallest rank of a Hermitian (or real symmetric) matrix whose off-diagonal zero-nonzero pattern is that of the adjacency matrix of $G$. Here $G$ is taken to be a circulant graph, and only circulant matrices are considered. The resulting graph parameter is termed the minimum circulant rank of the graph. This value is determined for every circulant graph in which a vertex neighborhood forms a consecutive set, and in this case is shown to coincide with the usual minimum rank. Under the additional restriction to positive semidefinite matrices, the resulting parameter is shown to be equal to the smallest number of dimensions in which the graph has an orthogonal representation with a certain symmetry property, and also to the smallest number of terms appearing among a certain family of polynomials determined by the graph. This value is then determined when the number of vertices is prime. The analogous parameter over the reals is also investigated.Comment: 27 pages, 3 figures; to appear in Linear Algebra and its Application

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On the minimum semidefinite rank of a simple graph

Cited by 25 publications

References 17 publications

A new lower bound for the positive semidefinite minimum rank of a graph

A new lower bound for the positive semidefinite minimum rank of a graph

Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions

The minimum rank problem for circulants

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