TY - JOUR

T1 - Forbidden minors for the class of graphs G with $\xi (G) \leq 2$

AU - Hogben, L.

AU - Holst, van der, H.

PY - 2007

Y1 - 2007

N2 - For a given simple graph 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.], ¿(G) is defined to be the maximum corank (i.e., nullity) among having the Strong Arnold Property; ¿ is used to study the minimum rank/maximum eigenvalue multiplicity problem for G. Since ¿ is minor monotone, the graphs G such that ¿(G)k can be described by a finite set of forbidden minors. We determine the forbidden minors for ¿(G)2 and present an application of this characterization to computation of minimum rank among matrices in .

AB - For a given simple graph 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.], ¿(G) is defined to be the maximum corank (i.e., nullity) among having the Strong Arnold Property; ¿ is used to study the minimum rank/maximum eigenvalue multiplicity problem for G. Since ¿ is minor monotone, the graphs G such that ¿(G)k can be described by a finite set of forbidden minors. We determine the forbidden minors for ¿(G)2 and present an application of this characterization to computation of minimum rank among matrices in .

U2 - 10.1016/j.laa.2006.08.003

DO - 10.1016/j.laa.2006.08.003

M3 - Article

VL - 423

SP - 42

EP - 52

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1

ER -