Determinants, their applications to Markov processes, and a random walk proof of Kirchhoff's matrix tree theorem

TL;DR

This paper presents a new proof of Kirchhoff's matrix tree theorem using Markov process techniques, connecting graph theory, linear algebra, and stochastic processes to simplify understanding and extend applications.

Contribution

It introduces a novel proof of Kirchhoff's theorem based on Markov process methods and demonstrates their applicability to related computations in finite Markov chains.

Findings

Proof of Kirchhoff's theorem via Markov processes

Extension of methods to other Markov chain computations

Simplification of existing graph-theoretic proofs

Abstract

Kirchhoff's matrix tree theorem is a well-known result that gives a formula for the number of spanning trees in a finite, connected graph in terms of the graph Laplacian matrix. A closely related result is Wilson's algorithm for putting the uniform distribution on the set of spanning trees. We will show that when one follows Greg Lawler's strategy for proving Wilson's algorithm, Kirchhoff's theorem follows almost immediately after one applies some elementary linear algebra. We also show that the same ideas can be applied to other computations related to general Markov chains and processes on a finite state space.