On Lyndon-Word Representable Graphs

TL;DR

This paper introduces Lyndon graphs derived from words over ordered alphabets, defines Lyndon-word representable graphs, and explores the generalized Stirling cycle number, concluding with open questions and conjectures.

Contribution

It proposes a new graph construction called Lyndon graphs, defines Lyndon-word representable graphs, and introduces the generalized Stirling cycle number, expanding the theoretical framework of Lyndon words.

Findings

Defined Lyndon graphs from words over ordered alphabets

Introduced Lyndon-word representable graphs as isomorphic to Lyndon graphs

Presented the generalized Stirling cycle number $S(N;n,k)$ and its combinatorial significance

Abstract

In this short note, we first associate a new simple undirected graph with a given word over an ordered alphabet of $n$-letters. We will call it the Lyndon graph of that word. Then, we introduce the concept of the Lyndon-word representable graph as a graph isomorphic to a Lyndon graph of some word. Then, we introduce the generalized Stirling cycle number $S(N;n,k)$ as the number words of length $N$ with $k$ distinct Lydon words in their Lyndon factorization over an ordered alphabet of $n$-letters . Finally, we conclude the paper with several interesting open questions and conjectures for interested audiences.