THE DISTRIBUTION OF EIGENVALUES OF GRAPHS

We show that every limit point of the kth largest eigenvalues of graphs is a limit point of the (k + 1)th largest eigenvalues, and we find out the smallest limit point of the kth largest eigenvalues and an upper bound of the limit points of the kth smallest eigenvalues. For k greater than or equal to 4, we prove that there exists a gap beyond the smallest limit point in which no point is the limit point of the kth largest eigenvalues. For the third largest eigenvalues of a graph G with at least three vertices, we obtain that (1) lambda(3)(G) < -1 iff G congruent to P-3; (2) lambda(3)(G) = -1 iff G(c) is isomorphic to a complete bipartite graph plus isolated vertices: (3) there exist no graphs such that -1 < lambda(3)(G) < (1 - root 5)/2. Consequently, if G(c) is not a complete bipartite graph plus isolated vertices, lambda(3)(G) greater than or equal to lambda(3)(D-n*), where D-n* is the complement of the double star S(1, n - 3).