In this paper, we investigate the relation between the -spectrum and the structure of in terms of the circumference of . Exploiting this relation, we give a novel necessary condition for a graph to be Hamiltonian by means of its -spectrum. We also determine the graphs with exactly one or two -eigenvalues greater than or equal to and obtain all minimal forbidden subgraphs and maximal graphs, as induced subgraphs, with respect to the latter property.
