Primitive digraphs with large exponents and slowly synchronizing automata

TL;DR

This paper explores infinite series of synchronizing automata linked to primitive digraphs with large exponents, demonstrating that their reset words are nearly quadratic in length relative to the number of states.

Contribution

It introduces new classes of automata with near-quadratic reset lengths, connecting automata synchronization properties with primitive digraph exponents.

Findings

Reset word lengths are close to the square of the number of states.

Automata are closely related to primitive digraphs with large exponents.

Provides infinite series of such automata with specific properties.

Abstract

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.