A linear bound for the Colin de Verdière parameter $\mu$ for graphs embedded on surfaces
Résumé
The Colin de Verdière graph parameter $\mu(G)$ was introduced in 1990 by Y. Colin de Verdière. It is defined via spectral properties of a certain type of matrices, called Schrödinger operators, associated to a graph $G$. We provide a combinatorial and self-contained proof that for all graphs $G$ embedded on a surface $S$, the Colin de Verdière parameter $\mu(G)$ is upper bounded by $7 − 2\chi(S)$, where $\chi(S)$ is the Euler characteristic of $S$.
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