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Consider a random walk on an undirected graph which, at each step, moves to a uniformly random neighbor. Define $T(u,v)$ to be the expected time until such a walk, starting from $u$, arrives at $v$, and let $T = \max_{u,v} T(u,v)$. Define $G(u)$ to be the expected time until such a walk, starting from $u$, visits every vertex and let $G = \max_u G(u)$. Is it true that $$G \leq cT$$ for some constant $c$ which does not depend on the graph and any of its parameters (e.g., number of nodes)?

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btw, I believe the "standard" notation for $T(u,v)$ is actually $H(u,v)$ (or $H_{uv}$). And $C$ instead of $G$. – Aryabhata May 10 '12 at 1:46
up vote 3 down vote accepted

That seems false.

For the complete graph $K_n$, $T = T(u,v) = \mathcal{O}(n)$, but $G = \Theta(n \log n)$.

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This is a bit simpler than my example :-) – joriki May 9 '12 at 22:11

No. Take a graph with one central vertex and $n$ vertices connected only to the central vertex. The expected time to visit every vertex is determined by the coupon collector's problem and goes as $n\log n$, whereas the expected time to visit any vertex from any other vertex only goes as $n$.

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