Linearly independent vertices and minimum semidefinite rank

Philip Hackney, Benjamin Harris, Margaret Lay, Lon H. Mitchell, Sivaram K. Narayan*, Amanda Pascoe

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

34 Citations (Scopus)

Abstract

We study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs. We present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph, and conjecture that the resulting graph invariant is equal to minimum semidefinite rank. Rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs. We also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex. (C) 2009 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)1105-1115
Number of pages11
JournalLinear Algebra and Its Applications
Volume431
Issue number8
DOIs
Publication statusPublished - 1 Sep 2009
Externally publishedYes

Keywords

  • Minimum semidefinite rank
  • Join
  • Linearly independent vertices
  • GRAPHS
  • REPRESENTATIONS
  • MATRICES

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