Question: How to simulate random walks on graphs?

This week I am participating in 19th Ising lectures (see https://drive.google.com/folderview?id=0B0uPwoK-03XgSEZpYWljYnpXN0U&usp=sharing). The Serguei Nechaev's talk inspired me to ask the question:
"How to simulate a random walk on an undirected and unweighted (and, of course, connected) graph
(All the paths from a vertex of degree k have the same probability 1/k.)?"
A Maple procedure to this end is welcome.

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