The Logic of Random Regular Graphs

The first order language of graphs is a formal language in which one can express many properties of graphs — known as first order properties. The classic Zero-One law for random graphs states that if p is some constant probability then for every first order property the limiting probability of the binomial random graph G(n, p) having this property is either zero or one. The case of sparse random graphs has also been studied in detail for the binomial random graph model. We obtain results for random regular graphs that match the main results for G(n, p). In particular we prove that if the degree d is linear in the number of vertices n, or if d = n for 0 < α < 1 irrational, then the random d-regular graph Gn,d obeys the Zero-One law. On the contrary, if d = n for rational 0 < α < 1, then there is a (theoretically explicit) first order property with no limiting probability in Gn,d.