On a family of Schreier graphs of intermediate growth associated with a self-similar group

For every infinite sequence ω=x1x2…, with xi∈{0,1}, we construct an infinite 4-regular graph Xω. These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space {0,1}∞. We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs Xω have intermediate growth.