Lifts, derandomization, and diameters of Schreier graphs of Mealy   automata
  It is known that random 2-lifts of graphs give rise to expander graphs. We present a new conjectured derandomization of this construction based on certain Mealy automata. We verify that these graphs have polylogarithmic diameter, and present a class of automata for which the same is true. However, we also show that some automata in this class do not give rise to expander graphs. 
Anton Malyshev, Igor Pak
math.CO