About a weighted sum of hitting times for random walks on graphs Consider a random walk on an undirected, non-bipartite graph. Let $\pi$ be the stationary distribution of this process, and let the hitting time $H(s,t)$ be the expected time until a walk beginning at node $s$ reaches node $t$. 
I learned from Random walks on graphs: a survey, by L. Lovasz, that the quantity 
$$ \sum_{t} \pi(t) H(s,t)$$
is independent of $s$. In the Lovasz survey, this falls out as a byproduct of a lengthy calculation. I have two questions:
(i) Is there a simple (calculation-free, combinatorial) proof of this statement?
(ii) In what generality does this statement hold? Random walks on undirected graphs are a very particular kind of Markov process. What conditions on the probability transition matrix of the process do you need for the above to hold?

To answer (ii), I think it needs to be ergodic, i.e., aperiodic and recurrent.
