Synchronizing automata and principal eigenvectors of the underlying digraphs

TL;DR

This paper explores the relationship between spectral properties of digraphs and the synchronization behavior of automata derived from them, revealing conditions linking eigenvectors, stationary distributions, and synchronizing words.

Contribution

It establishes a novel connection between the principal eigenvectors of digraphs and the synchronization properties of their colorings, extending spectral analysis to automata theory.

Findings

Non-synchronizing colorings correspond to stationary distributions with equal-sum partitions.

Existence of such partitions implies non-synchronizing automata.

Shortest synchronizing word length is bounded by a quadratic function of the eigenvector sum.

Abstract

A coloring of a digraph with a fixed out-degree k is a distribution of k labels over the edges resulting in a deterministic finite automaton. An automaton is called synchronizing if there exists a word which sends all states of the automaton to a single state. In the present paper we study connections between spectral and synchronizing properties of digraphs. We show that if a coloring of a digraph is not synchronizing, then the stationary distribution of an associated Markov chain has a partition of coordinates into blocks of equal sum. Moreover, if there exists such a partition, then there exists a non-synchronizing automaton with such stationary distribution. We extend these results to bound the number of non-synchronizing colorings for digraphs with particular eigenvectors. We also demonstrate that the length of the shortest synchronizing word of any coloring is at most $w^2 - 3w + 3$, where $w$ is the sum of the coordinates of the integer principal eigenvector of the digraph.