Spanning tree invariants, loop systems and doubly stochastic matrices (Q1044548)

\textit{D. Lind} and \textit{S. Tuncel} [Codes, systems, and graphical models. IMA workshop, Minneapolis, MN, USA, August 2--13, 1999. New York, NY: Springer. IMA Vol. Math. Appl. 123, 487--497 (2001; Zbl 1030.37005)] introduced an invariant of block isomorphism for Markov shifts. The invariant is called a spanning tree invariant and is obtained by taking the weight of all spanning trees of a presentation of the Markov shift. This invariant is observed in the context of loop systems of Markov chains. For \(n=1,2,3\) the spanning tree invariants of the loop systems of a Markov chain determined by an irreducible stochastic matrix \(P\) of degree \(n\) coincide if and only if \(P\) is doubly stochastic; and in this case, the common value of the spanning tree invariants of the loop systems is \(n\).