To every convex $d$-polytope with the dual graph $G$ a matrix is associated.
The matrix is shown to be a discrete Schr\”odinger operator on $G$ with the
second least eigenvalue of multiplicity $d$. This implies that the Colin de
Verdi\`ere parameter of $G$ is greater or equal $d$. The construction
generalizes the one given by Lov\’asz in the case $d = 3$.
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