An Elementary Proof of the Markov Chain Tree Theorem

The Markov Chain Tree Theorem is a classical result which expresses the stable distribution of an irreducible Markov matrix in terms of directed spanning trees of its associated graph. In this article, we present what we believe to be an original elementary proof of the theorem (Theorem 5.1). Our proof uses only linear algebra and graph theory, and in particular, it does not rely on probability theory. For this reason, this article could serve as a pedagogical tool or a gentle introduction to the theory of Markov matrices for undergraduate computer science and mathematics students. A version of our proof of the Markov Chain Tree Theorem appeared in John Wicks’ PhD thesis [4]. Other proofs of the Markov Chain Tree Theorem which use probability theory can be found in Broder [2, Theorem 1], or in more general form in Anantharam and Tsoucas [1]. The interested reader can find more information on Markov chains, matrices, and graphs in Kemeny and Snell [3]. In Section 2, we introduce basic facts and terminology that we will need when working with graphs. In Section 3, we define Markov matrices and provide an algebraic formula for the stable distribution of a unichain Markov matrix. In Section 4, we discuss directed trees and prove the existence of directed spanning trees of unichain graphs. In Section 5, we prove the Markov Chain Tree Theorem by rewriting the algebraic formula for the stable distribution provided in Section 3 as a sum of weights of directed spanning trees.