A cubic graph has no eigenvalues in (-2,0) if and only if it is the 3-prism, K3,3, the Petersen graph, the dodecahedron, Tutte's 8-cage, or one of the graphs X(n) in a single infinite family.
Cubic Ramanujan graphs.Combinatorica, 12(3):275–285, 1992
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Cubic graphs with no eigenvalues in the interval (-2,0)
A cubic graph has no eigenvalues in (-2,0) if and only if it is the 3-prism, K3,3, the Petersen graph, the dodecahedron, Tutte's 8-cage, or one of the graphs X(n) in a single infinite family.