Linearly independent vertices and minimum semidefinite rank

Abstract. Tree-width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these parameters, including several Colin de Verdière type parameters, and introduce numerous variations, including the minor monotone floors and ceilings of some of these parameters. This leads to new graph parameters and to new characterizations of existing graph parameters. In particular, tree-width, largeur d'arborescence, path-width, and proper path-width are each characterized in terms of a minor monotone floor of a certain zero forcing parameter defined by a color change rule.

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“…19. But H is a subgraph of the 11-tree obtained from K 14 by deleting the three edges {1, 9}, {1, 10}, {2, 10}, so tw(H) ≤ 11.…”

Section: A1 Examples For Non-equality Between Parametersmentioning

confidence: 99%

Parameters Related to Tree‐Width, Zero Forcing, and Maximum Nullity of a Graph

Barioli

Barrett

Fallat

et al. 2012

Journal of Graph Theory

111

158

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“…By studying the support of vectors in the null space of any matrix representation of G one can show that |G| − Z(G) is a lower bound for mr(G). Hackney et al [10] introduced the notion of an OS-set to construct a lower bound for mr K + (G). A set S ⊂ V is a OS-set if it satisfies a combinatorial condition involving connectivity in various induced subgraphs.…”

Section: S(g) = {A ∈ M N (R) : a T = A G(a) = G}mentioning

confidence: 99%

“…By considering vector representations of a graph G one can show that OS(G) is a lower bound for mr K + (G). Theorem 1.2 [10,Proposition 3.3]. OS(G) mr K + (G).…”

Section: S(g) = {A ∈ M N (R) : a T = A G(a) = G}mentioning

confidence: 99%

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

Zimmer

2013

Linear Algebra and its Applications

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“…Minimum rank problems seek to find the minimum rank over matrices in a given subset of C(G, F) for a specified G and F. For positive semidefinite (psd) matrices, this is the minimum semidefinite rank, mr + (G) for F = R and msr(G) for F = C. This problem has been previously studied both for multigraphs as we have presented it [1,4,7,19] and when the graph G is required to be simple [3,11].…”

mentioning

confidence: 99%

“…Thus, isolated and singly-isolated vertices do not influence the msr of a graph. In certain situations, such as computing the msr of the join of two graphs [7], it has been beneficial to consider only non-degenerate vector representations that do not include zero vectors. This led to the definition of the minimum vector rank (mvr)…”

mentioning

confidence: 99%

On the minimum vector rank of multigraphs

Mitchell

Narayan

Rogers³

2010

ELA

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Abstract. The minimum vector rank (mvr) of a graph over a field F is the smallest d for which a faithful vector representation of G exists in F d . For simple graphs, minimum semidefinite rank (msr) and minimum vector rank differ by exactly the number of isolated vertices. We explore the relationship between msr and mvr for multigraphs and show that a result linking the msr of chordal graphs to clique cover number also holds for the mvr of multigraphs. We study the difference between msr and mvr in the removal of duplicate vertices in multigraphs, and relate mvr to certain coloring problems.

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Linearly independent vertices and minimum semidefinite rank

Cited by 46 publications

References 14 publications

Parameters Related to Tree‐Width, Zero Forcing, and Maximum Nullity of a Graph

Parameters Related to Tree‐Width, Zero Forcing, and Maximum Nullity of a Graph

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

On the minimum vector rank of multigraphs

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