Hamilton cycles in dense vertex-transitive graphs

A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.Comment: 26 pages, 3 figures; referees' comments incorporated; accepted for publication in Journal of Combinatorial Theory, series

Independence ratio and random eigenvectors in transitive graphs

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $\lambda_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a $3$-regular transitive graph is at least $$q=\frac{1}{2}-\frac{3}{4\pi}\arccos\biggl(\frac{1-\lambda _{\min}}{4}\biggr).$$ The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least $q-o(1)$. We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.Comment: Published at http://dx.doi.org/10.1214/14-AOP952 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

All finite transitive graphs admit self-adjoint free semigroupoid algebras

In this paper we show that every non-cycle finite transitive directed graph has a Cuntz-Krieger family whose WOT-closed algebra is $B(\mathcal{H})$. This is accomplished through a new construction that reduces this problem to in-degree $2$-regular graphs, which is then treated by applying the periodic Road Coloring Theorem of B\'eal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.Comment: Added missing reference. 16 pages 2 figure