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.