Why walks lead us astray in the study of graphs

Jean Fromentin (1) , P.-L Giscard (1) , T Karaboghossian (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

We show that the commonly held assumption that walks can be used to infer properties of digraphs is highly problematic. Since in particular closed walks are describable from combinations of simple cycles, we study the trace monoid formed by these cycles on a digraph under the rule that two such cycles commute if and only if they are vertex disjoint. We show that most graph properties can be lost while maintaining the monoidal structure of cycles and thus cannot be inferred from it, including vertex-transitivity, regularity, planarity, Hamiltonicity, graph spectra, degree distribution and more. Conversely we find that even allowing for multidigraphs, many arrangements of simple cycles are not possible at all. The problem of determining whether a certain arrangement of simple cycles is realizable is highly non-trivial. We show at least that it is decidable and equivalent to the existence of integer solutions to systems of polynomial equations.

Mots clés: walk; simple cycle; trace monoid; hike monoid

Domaines: Théorie des groupes [math.GR]; Combinatoire [math.CO]

Jean Fromentin, P.-L Giscard, T Karaboghossian. Why walks lead us astray in the study of graphs. 2021. ⟨hal-03408041v1⟩