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Partially commutative monoids provide a powerful tool to study graphs,viewingwalks as words whose letters, the edges of the graph, obey a specificcommutation rule. A particularclass of traces emerges from this framework, thehikes, whose alphabet is the set of simple cycleson the graph. We show thathikes characterize undirected graphs uniquely, up to isomorphism, andsatisfyremarkable algebraic properties such as the existence and uniqueness of a primefactorization.Because of this, the set of hikes partially ordered bydivisibility hosts a plethora of relations in directcorrespondence with thosefound in number theory. Some applications of these results arepresented,including a permanantal extension to MacMahon's master theorem and aderivation of the Ihara zetafunction.

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