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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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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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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