Walks and the spectral radius of graphs

Given a graph G, write μ(G) for the largest eigenvalue of its adjacency matrix, ω(G) for its clique number, and wk(G) for the number of its k-walks. We prove that the inequalities hold for all r > 0 and odd q > 0. We also generalize a number of other bounds on μ(G) and characterize semiregular and pseudo-regular graphs in spectral terms.