rewrite the last term of \[e\].mtt’/ as (I-P) &= \_ (1 - ) = - \_ \_[k 1]{} \^k/k &= - \_[k 1]{} \_ \^k/k = - \_[k 1]{} [1 k]{} (P\^k - 1) , where in the last step, we have used the fact that the eigenvalue $1$ of $P$ has multiplicity 1 since $\gh$ is connected. Since $\tr P^k = \sum_{x \in \verts(\gh)} p_k(x;\gh)$, the desired formula now follows from this and \[e\].mtt’/.

r.mc There is an extension of \[p\].finite/ to any irreducible Markov chain with transition matrix $P$. In this case, a spanning tree of the associated directed graph, with all edges leading towards a single vertex, is often called a “spanning arborescence". Let $\cp'(P)$ be the sum over all spanning trees (with all possible roots) of the product of $P(e)$ over all $e$ in the tree. The analogue of \[p\].finite/ states that $$\log \cp'(P)
=
- \sum_{k \ge 1}
{1 \over k} \left(\sum_{x \in \verts(\gh)} p_k(x;\gh) - 1 \right)
\,.$$ This is reminiscent of a formula that appears in \[b\].LindTun/. In fact, combining their formula with ours, one gets an expression for the derivative at $1$ of the reciprocal of the so-called stochastic zeta function of 