The Colin de Verdière Graph Parameter for Threshold Graphs

Details The Colin de Verdière Graph Parameter for Threshold Graphs The Colin de Verdière Graph Parameter for Threshold Graphs

2006-12-22

arxiv.org/abs/math/0612720v2

Mathematics

Combinatorics

19 pages, 1 figure. Added sharp result with F. Goldberg

Scientific paper

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.

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