SOME EIGENVALUE PROPERTIES IN GRAPHS (CONJECTURES OF GRAFFITI .2.)

In this paper we improve some classical bounds on the greatest eigenvalue of the adjuacency matrix of a graph. We also give inequalities between the eigenvalues and some other parameters. These results allow us to prove some conjectures of the program Graffiti written by Fajtlowicz. Moreover, the study of the spectrum of graphs obtained by some simple constructions yields infinite families of counterexamples for other conjectures of this program.