On the structure of random graphs with constant $r$-balls

Abstract

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 \in \mathbb{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 \in \mathbb{N} , we let G_n = 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 \to \infty , 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 = G_n(F,r) has largest component of order at most n^{5/6} with high probability, and has at least \exp(n^{\delta}) automorphisms with high probability, where \delta>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 \mathbb{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.