On the distribution of Laplacian eigenvalues of a graph

This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of 0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result (Theorem 13), Grone and Merris’ conjecture The Laplacian spectrum of graph II. SIAM J. Discrete Math., 7, 221–229 (1994)] is partially proved.