Bounds on graph eigenvalues II

We prove three results about the spectral radius (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 (G) < (Tr (n)) unless G = Tr (n): (b) For most irregular graphs G of order n and size m; (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 (G) min (G)