Orthogonal representations, minimum rank, and graph complements

Date
2008-06-01
Authors
Hogben, Leslie
Hogben, Leslie
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Mathematics
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Abstract

Orthogonal representations are used to show that complements of certain sparse graphs have (positive semidefinite) minimum rank at most 4. This bound applies to the complement of a 2-tree and to the complement of a unicyclic graph. Hence for such graphs, the sum of the minimum rank of the graph and the minimum rank of its complement is at most two more than the order of the graph. The minimum rank of the complement of a 2-tree is determined exactly.

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<p>This is a manuscript of an article from <em>Linear Algebra and its Applications </em>428 (2008); 2560, doi:<a href="http://dx.doi.org/10.1016/j.laa.2007.12.004" target="_blank">10.1016/j.laa.2007.12.004</a>. Posted with permission.</p>
Keywords
Minimum rank, Orthogonal representation, 2-Tree, Unicyclic graph, Graph complement
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