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For each prime $p$, we study the eigenvalues of a 3-regular graph on roughly$p^2$ vertices constructed from the Markoff surface. We show theyasymptotically follow the Kesten-McKay law, which also describes theeigenvalues of a random regular graph. The proof is based on the method ofmoments and takes advantage of a natural group action on the Markoff surface.

Kesten-McKay law for the Markoff surface mod p