On the structure of random graphs with constant r-balls

We continue the study of the properties of graphs in which the ball of radius r around each vertex induces a graph isomorphic to the ball of radius r in some fixed vertex-transitive graph F, for various choices of F and r. This is a natural extension of the study of regular graphs. More precisely, if F is a vertex-transitive graph and r ε ℕ, we say a graph G is r -locally F if the ball of radius r around each vertex of G induces a graph isomorphic to the graph induced by the ball of radius r around any vertex of F. We consider the following random graph model: for each n ε ℕ, we let G_n D G_n.F; r/be a graph chosen uniformly at random from the set of all unlabelled, n-vertex graphs that are r -locally F. We investigate the properties possessed by the random graph G_n with high probability, i.e. with probability tending to 1 as n → 1, for various natural choices of F and r. We prove that if F is a Cayley graph of a torsion-free group of polynomial growth, and r is sufficiently large depending on F, then the random graph G_n D G_n.F; r/has largest component of order at most n⁵⁼⁶ with high probability, and has at least exp.n^ı/automorphisms with high probability, where ı > 0 depends upon F alone. Both properties are in stark contrast to random d -regular graphs, which correspond to the case where F is the infinite d -regular tree. We also show that, under the same hypotheses, the number of unlabelled, n-vertex graphs that are r -locally F grows like a stretched exponential in n, again in contrast with d -regular graphs. In the case where F is the standard Cayley graph of ℤ^d, we obtain a much more precise enumeration result, and more precise results on the properties of the random graph G_n.F; r/. Our proofs use a mixture of results and techniques from geometry, group theory and combinatorics. We make several conjectures regarding what happens for Cayley graphs of other finitely generated groups.