A Note On $\ell$-Rauzy Graphs for the Infinite Fibonacci Word

TL;DR

This paper investigates the properties of $ ext{ell}$-Rauzy graphs associated with the infinite Fibonacci word, establishing their strong connectivity and fundamental characteristics.

Contribution

It provides new insights into the structure and connectivity of $ ext{ell}$-Rauzy graphs specifically for the infinite Fibonacci word.

Findings

Proved basic properties of $ ext{ell}$-Rauzy graphs for the Fibonacci word.

Established that these graphs are strongly connected.

Analyzed the structural features of these graphs.

Abstract

The $\ell$-Rauzy graph of order $k$ for any infinite word is a directed graph in which an arc $(v_1,v_2)$ is formed if the concatenation of the word $v_1$ and the suffix of $v_2$ of length $k-\ell$ is a subword of the infinite word. In this paper, we consider one of the important aperiodic recurrent words, the infinite Fibonacci word for discussion. We prove a few basic properties of the $\ell$-Rauzy graph of the infinite Fibonacci word. We also prove that the $\ell$-Rauzy graphs for the infinite Fibonacci word are strongly connected.