The distribution of eigenvalues of graphs Original Research Article

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 ≥ 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) λ3(G) < −1 iff G congruent with P3; (2) λ3(G) = −1 iff Gc is isomorphic to a complete bipartite graph plus isolated vertices: (3) there exist no graphs such that −1 < λ3(G) < (1 − √5)/2. Consequently, if Gc is not a complete bipartite graph plus isolated vertices, λ3(G) ≥ λ3(D*n), where D*n is the complement of the double star S(1, n − 3).