Abstract
Word-representable graphs are vital in the Combinatorics of Words and Graph Theory.We define one such graph, the \(\ell\) -Rauzy graph, and study its structural properties for a few infinite words. The \(\ell\) -Rauzy graph of order \(k\) for an infinite word \(w\) is a directed graph in which each vertex is a subword of length \(k\) of \(w\) , where any two vertices \(u,v\) form an arc \(uv\) iff the prefix of \(v\) of length \(\ell\) is the same as the suffix of \(u\) of length \(\ell\) and the concatenated word formed by the arc \(uv\) of length \(2k-\ell\) is a subword of \(w\) . As main results, we show that the \(\ell\) -Rauzy graph of order \(k\) for the infinite Fibonacci word \(f\) is strongly connected, and we explicitly find the graph structure of the \(\ell\) -Rauzy graph of order \(k\) for an infinite periodic word, by knowing the locations of subwords of \(f\) and infinite periodic words. As locating the subwords in an infinite word isnot easy, we show that the \(\ell\) -Rauzy graph of order \(k\) for an Arnoux-Rauzy word is strongly connected using a different approach. \(\ell\)
