A co-preLie structure from chronological loop erasure in graph walks

TL;DR

This paper introduces a novel preLie co-algebra structure derived from Lawler's loop-erasing process on graph walks, revealing new algebraic frameworks and sub-Hopf algebras related to specific walk types.

Contribution

It establishes a new algebraic structure from cycle removal in graph walks and connects it with existing algebraic frameworks like Hopf algebras and brace coalgebras.

Findings

PreLie co-algebra from cycle erasure on graph walks

Explicit antipodes for tensor and symmetric algebras

Identification of sub-Hopf algebras for specific walk types

Abstract

We show that the chronological removal of cycles from a walk on a graph, known as Lawler's loop-erasing procedure, generates a preLie co-algebra on the vector space spanned by the walks. In addition, we prove that the tensor and symmetric algebras of graph walks are graded Hopf algebras, provide their antipodes explicitly and recover the preLie co-algebra from a brace coalgebra on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras associated to particular types of walks.