Algebraic combinatorics on trace monoids: extending number theory to walks on graphs

TL;DR

This paper explores how algebraic combinatorics on trace monoids can be used to analyze walks on graphs, revealing unique graph characterizations and algebraic properties that extend classical number theory concepts.

Contribution

It introduces hikes as a new class of traces that uniquely characterize undirected graphs and exhibit prime factorization, linking graph theory with number theory.

Findings

Hikes uniquely identify undirected graphs up to isomorphism.

Hikes satisfy a prime factorization property similar to integers.

Applications include extending MacMahon's theorem and deriving the Ihara zeta function.

Abstract

Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycleson the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, andsatisfy remarkable algebraic properties such as the existence and uniqueness of a prime factorization.Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in directcorrespondence with those found in number theory. Some applications of these results are presented,including a permanantal extension to MacMahon's master theorem and a derivation of the Ihara zetafunction.