Classification of trees by Laplacian eigenvalue distribution and edge covering number

For a connected graph G of order n and an interval I, denote by mGI the number of Laplacian eigenvalues of G in I. In this paper, we present bounds for mGI in terms of the structural parameter ,9'(G), the edge covering number of G. We prove a known result that m(G) [1, n] = ,9'(G). We also show that all graphs G ? C-3, C(7 )with minimum degree at least two, m(G) [1, n] = ,9'(G) + 1. We present a short proof of the known result that m(G) (n - 1, n] = ?(G), where ?(G) is the vertex connectivity of G. Additionally, we classify all trees T such that mT(n - i, n] = j, for 1 = i, j = 2.