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 ) .
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