Bounds on graph eigenvalues II

We prove three results about the spectral radius mu(G) of a graph G: (a) Let T-r (n) be the r-partite Turan graph of order n. If G is a Kr+ 1 -free graph of order n, then mu(G) < mu(T-r(n)) unless G = T-r(n). (b) For most irregular graphs G of order n and size m, mu(G) - 2m/n > 1/(2m + 2n). (c) Let 0 <= k <= l. If G is a graph of order n with no K-2 + (K) over bar (k+1) and no K-2,K-l+1, then mu(G) <= min { Delta(G), (k - l + 1 + root(k - 1 + 1)(2) + 4l (n - 1) / 2 }. (C) 2007 Elsevier Inc. All rights reserved.