Which cubic graphs have an eigenvalue of $\sqrt{6}$?

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Which cubic graphs have an eigenvalue of $\sqrt{6}$?

Can these graphs be constructed?

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After a quick look at GraphData[] in Mathematica, here are some simple graphs with eigenvalues of $\sqrt{6}$. That value is also in their spectrum.

some graphs with eigenvalue sqrt(6)

Ah, you're asking for cubic graphs. Same process:

cubic graphs with eigenvalue sqrt(6)