Reset Thresholds of Automata with Two Cycle Lengths

We present several series of synchronizing automata with multiple parameters, generalizing previously known results. Let p and q be two arbitrary co-prime positive integers, q > p. We establish reset thresholds of the colorings of primitive digraphs with exactly one simple cycle of length p and one simple cycle of length q. Also, we study reset thresholds of the colorings of primitive digraphs with exactly one simple cycle of length q and two simple cycles of length p. We obtain the reset thresholds of the form (q-1)(p-1)+ (p,q), where (p,q) is a linear function of p and q. © 2015 World Scientific Publishing Company