On the number of Hamilton cycles in pseudo-random graphs

TL;DR

This paper establishes a formula for counting Hamilton cycles in certain pseudo-random graphs based on spectral properties, under specific regularity and eigenvalue conditions.

Contribution

It provides a new asymptotic count of Hamilton cycles in (n,d,lambda)-graphs with spectral conditions, extending understanding of pseudo-random graph structures.

Findings

Number of Hamilton cycles approximated by n!(d/n)^n with high accuracy.

Conditions on d and lambda ensure the pseudo-randomness necessary for the count.

Results apply to a broad class of regular graphs with spectral gap constraints.

Abstract

We prove that if G is an (n,d,lambda)-graph (a d-regular graph on n vertices, all of whose non-trivial eigenvalues are at most lambda) and the following conditions are satisfied: 1. d/lambda >= (log n)^{1+epsilon} for some constant epsilon>0; 2.log d * lod (d/lambda) >> log n, then the number of Hamilton cycles in G is n!(d/n)^n(1+o(1))^n.