Relations between connected and self-avoiding walks in a digraph

TL;DR

This paper explores the mathematical relationships between walks and self-avoiding hikes in directed graphs using incidence algebra and characteristic polynomials, revealing new connections and enumeration methods.

Contribution

It introduces novel relations between walks and self-avoiding hikes through incidence algebra and polynomial truncations, providing new enumeration techniques.

Findings

Derived relations between walks and self-avoiding hikes

Introduced matrices counting self-avoiding hikes of length  from vertex to vertex

Connected polynomial truncations to hike enumeration

Abstract

Walks in a directed graph can be given a partially ordered structure that extends to possibly unconnected objects, called hikes. Studying the incidence algebra on this poset reveals unsuspected relations between walks and self-avoiding hikes. These relations are derived by considering truncated versions of the characteristic polynomial of the weighted adjacency matrix, resulting in a collection of matrices whose entries enumerate the self-avoiding hikes of length $\ell$ from one vertex to another.