ON THE NUMBER OF LAPLACIAN EIGENVALUES OF TREES SMALLER THAN TWO

Let m(T) [0, 2) be the number of Laplacian eigenvalues of a tree T in [0, 2), multiplicities included. We give best possible upper bounds for m(T) [0, 2) using the parameters such as the number of pendant vertices, diameter, matching number, and domination number, and characterize the trees T of order n with m(T) [0, 2) = n - 1, n - 2, and [n/2], respectively, and in particular, show that m(T) [0, 2) = [n/2] if and only if the matching number of T is [n/2].