#### Document Type

Article

#### Publication Date

11-1995

#### Publication Title

Brazilian Journal of Probability and Statistics

#### Abstract

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-).

#### Recommended Citation

Northshield, S.,
&
Palacios, J.
(1995).
On the Commute Time of Random Walks on Graphs.
*Brazilian Journal of Probability and Statistics, 9* (2), 169-175.