The spectral radius of graphs without paths and cycles of specified length

Let G be a graph with n vertices and mu(G) be the largest eigenvalue of the adjacency matrix of G. We study how large mu(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give right bounds on the spectral radius of graphs without given even cycles. We also raise a number of open problems. (C) 2009 Elsevier Inc. All rights reserved.