Spectral analogues of Erdős’ and Moon–Moser’s theorems on Hamilton cycles.

In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number and minimum degree of graphs which generalized Ore’s theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper, we present the spectral analogues of Erdős’ theorem and Moon–Moser’s theorem, respectively. Letbe the class of non-Hamiltonian graphs of ordernand minimum degree at leastk. We determine the maximum (signless Laplacian) spectral radius of graphs in(for large enoughn), and the minimum (signless Laplacian) spectral radius of the complements of graphs in. All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements. [ABSTRACT FROM PUBLISHER]