SOME RESULTS ON THE LARGEST AND LEAST EIGENVALUES OF GRAPHS

Let G = (V,E) be a simple graph with vertex set V(G) = {v(1), v(2), . . . , v(n)} and edge set E(G). In this paper, first some sharp upper and lower bounds on the largest and least eigenvalues of graphs are given when vertices are removed. Some conjectures in [M. Aouchiche. Comparaison Automatisee dInvariants en Theorie des Graphes. Ph.D. Thesis, Ecole Polytechnique de Montreal, February 2006.] and [M. Aouchiche, G. Caporossi, and P. Hansen. Variable neighborhood search for extremal graphs, 20. Automated comparison of graph invariants. MATCH Commun. Math. Comput. Chem., 58:365384, 2007.] involving the spectral radius, diameter and matching number are also proved. Furthermore, the extremal graph which attains the minimum least eigenvalue among all quasi-tree graphs is characterized.