Sparse pseudo-random graphs are hamiltonian

In this article we study Hamilton cycles in sparse pseudorandom graphs. We prove that if the second largest absolute value λ of an eigenvalue of a d-regular graph G on n vertices satisfies λ ≤ (log log n)²/1000a log n (log log n) d and n is large enough, then G is Hamiltonian. We also show how out main result can be used to prove that for every c > 0 and large enough n a Cayley graph X(G,S), formed by choosing a set S of c log⁵ n random generators in a group G of order n, is almost surely Hamiltonian.