Eigenvalue multiplicity in quartic graphs

Let G be a connected quartic graph of order n with μ as an eigenvalue of multiplicity k. We show that if μ∉{−1,0} then k≤(2n−5)/3 when n≤22, and k≤(3n−1)/5 when n≥23. If μ∈{−1,0} then k≤(2n+2)/3, with equality if and only if G=K5 (with μ=−1) or G=K4,4 (with μ=0).