Three Ways to Count Walks in a Digraph

TL;DR

This paper explores three distinct methods—combinatorics, linear algebra, and symbolic dynamics—to count walks in directed graphs, providing a comprehensive understanding of the problem from multiple mathematical perspectives.

Contribution

It introduces a unified framework for counting walks in digraphs using three different mathematical approaches, highlighting their connections and differences.

Findings

Enumerative formulas for walk counts

Linear algebraic methods for counting walks

Symbolic dynamics approach to walk enumeration

Abstract

We approach the problem of counting the number of walks in a digraph from three different perspectives: enumerative combinatorics, linear algebra, and symbolic dynamics.