  For every infinite sequence $\omega=x_1,x_2,...$, with $x_i\in\{0,1\}$, we
construct an infinite 4-regular graph $X_{\omega}$. These graphs are precisely
the Schreier graphs of the action of a certain self-similar group on the space
$\{0,1\}^{\infty}$. We solve the isomorphism and local isomorphism problems for
these graphs, and determine their automorphism groups. Finally, we prove that
all graphs $X_\omega$ have intermediate growth.
