Hitting time quasi-metric and its forest representation

Let (m) over cap (ij) be the weighted digraph whose vertex set coincides with the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. It holds that (m) over cap (ij) = q(j)(-1) center dot {f(ij), if i not equal j, q, if i= j, where f(ij) is the total weight of 2-tree spanning converging forests in Gamma that have one tree containing i and the other tree converging to j, q(j) is the total weight of spanning trees converging to j in Gamma, and q= Sigma(n)(=1)qj is the total weight of all spanning trees in Gamma. Moreover, f(ij) and q(j) can be calculated by an algebraic recurrent procedure. A forest expression for Kemeny's constant is an immediate consequence of this result. Further, we discuss the properties of the hitting time quasi-metric m on the set of vertices of Gamma: m(i,j) = (m) over cap (ij), i not equal j, and m(i, i)=0. We also consider a number of other metric structures on the set of graph vertices related to the hitting time quasi-metric m-along with various connections between them. The notions and relationships under study are illustrated by two examples.