Broder and Karlin's formula for hitting times and the Kirchhoff Index

We give an elementary proof of an extension of Broder and Karlin's formula for the hitting times of an arbitrary ergodic Markov chain. Using this formula in the particular case of random walks on graphs, we give upper and tight lower bounds for the Kirchhoff index of any N- vertex graph in terms of N and its maximal and minimal degrees. We also apply the formula to a closely related index that takes into account the degrees of the vertices between which the effective resistances are computed. We give an upper bound for this alternative index and show that the bound is attained-up to a constant-for the barbell graph.