Bounds on graph eigenvalues II

We prove three results about the spectral radius μ(G) of a graph G: (a) Let Tr (n) be the r-partite Turán graph of order n. If G is a Kr+1-free graph of order n, then μ(G) < μ(Tr (n)) unless G = Tr (n). (b) For most irregular graphs G of order n and size m, μ(G) − 2m/n > 1/(2m + 2n). (c) Let 0 k l. If G is a graph of order n with no K2 + Kk+1 and no K2,l+1, then μ(G) min { (G), ( k − l + 1 + √ (k − l + 1)2 + 4l(n − 1) )/ 2 } . © 2007 Elsevier Inc. All rights reserved. AMS classification: Primary 05C50; Secondary 05C35