On mean recurrence times of Markov chains and spanning tree invariants

We show that the mean recurrence times of (countable state) irreducible and positively recurrent Markov chains are the spanning tree invariants of the first return loop systems. Then, by the Perron–Frobenius theorem, the spanning tree invariants of the first return loop systems of a finite state Markov chain are all equal if and only if the process is doubly stochastic, settling a conjecture on a question in [1] where it was verified for matrices of size at most three.