The logic of random regular graphs

In this paper we discuss the behavior of sets of properties in random regular graphs or, more generally, in random graphs with a given degree sequence. The properties under investigation are those describable in the standard first order language of graphs (to be defined in a few paragraphs). This blend of combinatorics and logic shed light on the connection between random regular graphs and binomial random graphs, and also on the descriptive power of the first order language of graphs.