Eigenvalue multiplicity in cubic graphs

Let G be a connected cubic graph of order n with μ as an eigenvalue of multiplicity k. We show that (i) if μ∉{−1,0} then k⩽12n, with equality if and only if μ=1 and G is the Petersen graph; (ii) if μ=−1 then k⩽12n+1, with equality if and only if G=K4; (iii) if μ=0 then k⩽12n+1, with equality if and only if G=2K3¯.