A note on the multiplicities of the eigenvalues of a tree

Abstract

Let T be a tree of order n with μ as an eigenvalue of multiplicity mT(μ). Wong et al. showed that mT(μ)≤n−43 when n>6 and μ2 is an integer at least 2. In this paper, we generalize this result by first showing that if T is a tree of order n and t is a positve integer, then (i) if n≤2t+2, then mT(t)≤1; (ii) If n≥2t+3, then mT(t)≤n−(t+2)t+1, with equality if and only if T has a vertex v such that T−v=kK1,t for some integer k≥2.