The Colin de Verdi\`ere Graph Parameter for Threshold Graphs

We consider Schr\"odinger operators on threshold graphs and prove a formula for the Colin de Verdi\`ere parameter in terms of the building sequence. We construct an optimal Colin de Verdi\`ere matrix for each connected threshold graph $G$ of $n$ vertices. For a large subclass of threshold graphs we construct an alternative Colin de Verdi\`ere 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.