Algebraic combinatorics on trace monoids: extending number theory to walks on graphs - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2016

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

Abstract

Partially commutative monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy 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 direct correspondence 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 zeta function.
Fichier principal
Vignette du fichier
AlgComb_revision3.pdf (453.71 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-01249007 , version 1 (29-12-2015)
hal-01249007 , version 2 (05-01-2016)
hal-01249007 , version 3 (14-10-2016)

Identifiers

Cite

P.-L Giscard, P Rochet. Algebraic combinatorics on trace monoids: extending number theory to walks on graphs. 2016. ⟨hal-01249007v3⟩
191 View
417 Download

Altmetric

Share

Gmail Facebook X LinkedIn More