Brazilian Journal of Probability and Statistics
Given a simple random walk on an undirected connected graph, the commute time of the vertices x and y is defined as C(x,y) = ExTy + EyTx. We give a new proof, based on the optional sampling theorem for martingales, of the formula C(x,y) = wyWy x) i" terms of the escape probability e(y,x ) (the probability that once the random walk leaves x, it hits y before it returns to x) and the stationary distribution TT(-). We use this formula for C(x,y) to show that the maximum commute time among all barbell-type graphs in N vertices is attained for the lollipop graph and its value is O(~j-).
Northshield, S., & Palacios, J. (1995). On the Commute Time of Random Walks on Graphs. Brazilian Journal of Probability and Statistics, 9 (2), 169-175.