  We consider Catalan-pair graphs, a family of graphs that can be viewed as
representing certain interactions between pairs of objects which are enumerated
by the Catalan numbers. In this paper we study random Catalan-pair graphs and
deduce various properties of these random graphs. In particular, we
asymptotically determine the expected number of edges and isolated vertices,
and more generally we determine the expected number of (induced) subgraphs
isomorphic to a given connected graph.
