 
Summary: Spanning tree invariants, loop systems and doubly
stochastic matrices
Ricardo G´omez
, Jos´e Miguel Salazar
Instituto de Matem´aticas de la Universidad Nacional Aut´onoma de M´exico.
Circuito Exterior, Ciudad Universitaria C.P. 04510, M´exico D.F. M´exico.
Abstract
The spanning tree invariant of Lind and Tuncel [12] 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
(n×n)matrix P 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.
Key words: spanning tree, loop system, doubly stochastic, Markov chain
2000 MSC: 15A18, 15A51, 37B10
1. Introduction
Lind and Tuncel introduce the spanning tree invariant as an invariant of
block isomorphism of Markov chains [12]. Its definition is as follows. Let P
be a (finite and irreducible) stochastic matrix. For every subdigraph H of the
underlying digraph D(P) = (V (P), E(P)) induced by P, let
wtP (H) =
