The Colin de Verdière Graph Parameter for Threshold Graphs

We consider Schrödinger operators on threshold graphs and prove a formula for the Colin de Verdière parameter in terms of the building sequence. We construct an optimal Colin de Verdière matrix for each connected threshold graph $G$ of $n$ vertices. For a large subclass of threshold graphs we construct an alternative Colin de Verdière matrix depending on a large parameter. As a corollary to this last construction, we give estimates on the size of the non-zero eigenvalues of this matrix.

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