Lyndon word decompositions and pseudo orbits on q-nary graphs

TL;DR

This paper extends Lyndon word decompositions to count certain factorizations and applies these results to analyze primitive pseudo orbits on q-nary graphs, impacting quantum graph theory.

Contribution

It generalizes the Chen-Fox-Lyndon theorem and connects Lyndon word decompositions to counting primitive pseudo orbits on q-nary graphs.

Findings

Derived the proportion of decreasing Lyndon decompositions.

Counted primitive pseudo orbits on q-nary graphs.

Provided a diagonal approximation for quantum graph variance.

Abstract

A foundational result in the theory of Lyndon words (words that are strictly earlier in lexicographic order than their cyclic permutations) is the Chen-Fox-Lyndon theorem which states that every word has a unique non-increasing decomposition into Lyndon words. This article extends this factorization theorem, obtaining the proportion of these decompositions that are strictly decreasing. This result is then used to count primitive pseudo orbits (sets of primitive periodic orbits) on q-nary graphs. As an application we obtain a diagonal approximation to the variance of the characteristic polynomial coefficients q-nary quantum graphs.