  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.
