Forbidden minors for the class of graphs G with ξ(G) ⩽ 2

Date
2007-05-01
Authors
Hogben, Leslie
van der Holst, Hein
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Altmetrics
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Research Projects
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Mathematics
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Abstract

For a given simple graph G, S(G) is defined to be the set of real symmetric matrices A whose (i,j)th entry is nonzero whenever i≠j and ij is an edge in G. In [F. Barioli, S. Fallat, L. Hogben, A variant on the graph parameters of Colin de Verdière: Implications to the minimum rank of graphs, Electron. J. Linear Algebra 13 (2005) 387–404.], ξ(G) is defined to be the maximum corank (i.e., nullity) among A∈S(G) having the Strong Arnold Property; ξ is used to study the minimum rank/maximum eigenvalue multiplicity problem for G. Since ξ is minor monotone, the graphs G such that ξ(G)⩽k can be described by a finite set of forbidden minors. We determine the forbidden minors for ξ(G)⩽2 and present an application of this characterization to computation of minimum rank among matrices in S(G).

Description

This is a manuscript of an article from Linear Algebra and its Applications 423 (2007): 42, doi:10.1016/j.laa.2006.08.003. Posted with permission.

Keywords
Minimum rank, Graph minor, Corank, Strong Arnold property, Symmetric matrix
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