Bounds on graph eigenvalues I

We improve some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n >= 2, maximum degree d, and girth at least 5, then mu(G) <= min {Delta, root n-1}, where mu(G) is the largest eigenvalue of the adjacency matrix of G. Also, if G is a graph of order n >= 2 with dominating number gamma(G) = gamma, then lambda 2(G) <= {(n if gamma =1,)(n - gamma if gamma >= 2,) lambda(n)(G) >= [n/gamma], where 0 = lambda(1) (G) <= lambda(2)(G) <= ... <= lambda(n) (G) are the eigenvalues of the Laplacian of G. We also determine all cases of equality in the above inequalities. (c) 2006 Elsevier Inc. All rights reserved.