The multiplicity of eigenvalues of trees

Abstract

Let T be a tree on n vertices. Denote by mT,θ the multiplicity of θ as an eigenvalue of the adjacency matrix of T. From two previous publications of Rowlinson and Wong et al., it is shown that mT,0≤n−2, mT,θ≤n−32 if θ≠0, and mT,θ≤n−43 if θ≠0,±1 and n≥7. In this note, we exploit the Parter-Wiener Theorem (different from the star complement technique used in previous publications) to obtain an upper bound for mT,θ for any fixed real number θ, thereby generalizing the results of Rowlinson and Wong et al., as well as two new papers from Tian and Wang, and Zhang et al.