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Random walks on G

Markov process with state graph G

Take a Markov chain with probability matrix $P_{u, v} = \frac{1}{deg(u)}$. $\pi_{v}=\frac{d(v)}{2|E|}$. \chk

Hitting and cover times

Hitting time $h_{u,v}$: expected time to get to v from u. Commute time: $C_{u,v} = h_{u,v} + h_{v, u}$. $h_{u,v}<2|E|$ \why.

Cover time C(G) = $\max_{v}$ E[time to hit all nodes starting from v]. $C(G) < 4|V||E|$ \why.

Connection with resistive electric networks

Let every $e\in E$ have resistance 1, thence get n/w N(G). $R_{u,v}$: effective resistance between u, v. $C_{u, v} = 2m R_{u, v}$ \why.