Minimum Vector Ranks and Complement-Critical Graphs

SMC Author

Xiaowei Li, Michael Nathanson, Rachel Phillips

SMC Affiliated Work

1

Status

Faculty

School

School of Science

Department

Math/Computer Science

Document Type

Article

Publication Date

2-2014

Publication / Conference / Sponsorship

Electronic Journal of Linear Algebra

Description/Abstract

Given a graph G, a real orthogonal representation of G is a function from its set of vertices to R d such that two vertices are mapped to orthogonal unit vectors if and only if they are not neighbors. The minimum vector rank of a graph is the smallest dimension d for which such a representation exists. This quantity is closely related to the minimum semidefinite rank of G, which has been widely studied. Considering the minimum vector rank as an analogue of the chromatic number, this work defines critical graphs as those for which the removal of any vertex decreases the minimum vector rank; and complement critical graphs as those for which the removal of any vertex decreases the minimum vector rank of either the graph or its complement. It establishes necessary and sufficient conditions for certain classes of graphs to be complement critical, in the process calculating their minimum vector rank. In addition, this work demonstrates that complement critical graphs form a sufficient set to prove the Graph Complement Conjecture, which remains open.

Keywords

Minimum vector rank, Minimum semidefinite rank, Orthogonal representation, Complement critical graph, Graph complement conjecture

Scholarly

yes

DOI

10.13001/1081-3810.1606

Volume

27

First Page

100

Last Page

123

Disciplines

Computer Sciences | Mathematics

Rights

Open Access journal

Comments

Note: Xiaowei Li and Rachel Phillips are Saint Mary’s students.

Original Citation

Michael Nathanson (Mathematics and Computer Science): "Minimum Vector Ranks and Complement-Critical Graphs," with Xiaowei Li and Rachel Phillips, in the Electronic Journal of Linear Algebra (2014), v.27 pp.100-123. doi:10.13001/1081-3810.1606

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