We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. All these automata are tightly related to primitive digraphs with large exponent. * A preliminary version of a part of the results of this paper was published in [3].

We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. These automata are closely related to primitive digraphs with large exponent.

In an earlier paper, we have studied reset words for synchronizing automata whose states admit a stable linear order. Here we show that the same bound on the length of the shortest reset word persists for synchronizing automata satisfying much weaker stability restriction. This result supports our conjecture concerning the length of reset words for synchronizing automata accepting only star-free languages.

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We present two infinite series of synchronizing automata with a letter of deficiency 2 whose shortest reset words are longer than those for synchronizing automata obtained by a straightforward modification ofČerný's construction.