Eigenvalue multiplicity in regular graphs

Abstract Let G be a connected r -regular graph of order n with μ as an eigenvalue of multiplicity k , where r > 2 and μ ≠ − 1 , 0 . We show that k ∕ n ≤ ( r − 1 ) ∕ ( r + 1 ) , with equality if and only if r = 3 , μ = 1 and G is the Petersen graph. We observe that whenever r > 2 there exists an r -regular graph with an eigenvalue μ ≠ − 1 , 0 for which k ∕ n > ( r − 2 ) ∕ ( r + 2 ) . Lastly we find an improved upper bound for k when r > 3 and G has a tree as a star complement for μ .