Kn\"odel walks in a B\"ohm-Hornik environment

TL;DR

This paper combines ideas from Kn"odel and B"ohm-Hornik regarding walks in specific graphs, providing explicit generating functions that describe their behavior, akin to classical symmetric random walks.

Contribution

It unifies and extends previous concepts of walks in graphs by explicitly deriving generating functions for these processes.

Findings

Explicit generating functions for walks in Kn"odel and B"ohm-Hornik environments

Unified framework for analyzing walks similar to symmetric random walks

Enhanced understanding of walk behaviors in complex graph structures

Abstract

Ideas of Kn\"odel and B\"ohm-Hornik about walks in certain graphs, resembling the classical symmetric random walk on the integers, are combined. All the relevant generating functions (although occasionally quite involved) are made fully explicit.