DIAMETER VS. LAPLACIAN EIGENVALUE DISTRIBUTION

Let G be a simple graph of order n. It is known that any Laplacian eigenvalue of G belongs to the interval [0, n]. For an interval I subset of [0, n], denote by m(G)I the number of Laplacian eigenvalues of G in I, counted with multiplicities. Let d be the diameter of G. If 2 <= d <= n-4, we show that m(G)[n-d, n] <= n-d+ 2, and it may be improved into m(G)[n-d, n] <= n-d+ 1 when d = 2, 3, 4. We also show that m(G)[n - 2d + 4, n] <= n - 2 if d = 2, [n+3/2], and m(G)[n - 2d + 4, n] <= n - 3 if 3 <= d <= [n+1/2]. The diameter constraint provides an insightful approach to understand how the Laplacian eigenvalues are distributed.