Cerny's conjecture, synchronizing automata, group representation theory

TL;DR

This paper introduces new infinite families of Cerny Cayley graphs by applying group representation theory over the rationals to analyze synchronizing automata.

Contribution

It leverages group representation theory to identify and construct new classes of Cerny Cayley graphs, advancing understanding of Cerny's conjecture.

Findings

Established new infinite families of Cerny Cayley graphs

Connected group representation theory to automata synchronization

Provided theoretical tools for analyzing synchronizing automata

Abstract

Let us say that a Cayley graph $\Gamma$ of a group $G$ of order $n$ is a Cerny Cayley graph if every synchronizing automaton containing $\Gamma$ as a subgraph with the same vertex set admits a synchronizing word of length at most $(n-1)^2$. In this paper we use the representation theory of groups over the rational numbers to obtain a number of new infinite families of {\v{C}}ern{\'y} Cayley graphs.