Bounds on graph eigenvalues II

We prove three results about the spectral radius μ(G) of a graph G:(a)Let Tr(n)Tr(n) be the r  -partite Turán graph of order nn. If G   is a Kr+1Kr+1-free graph of order n, thenμ(G)<μ(Tr(n))μ(G)<μ(Tr(n))unless G=Tr(n)G=Tr(n).(b)For most irregular graphs G of order n and size m,μ(G)-2m/n>1/(2m+2n).μ(G)-2m/n>1/(2m+2n).(c)Let 0⩽k⩽l0⩽k⩽l. If G is a graph of order n   with no K2+K¯k+1 and no K2,l+1K2,l+1, thenμ(G)⩽minG),(k-l+1+(k-l+1)2+4l(n-1))/2}.