The existence of a graph with certain strange-looking properties will imply a counterexample to something I am playing with. I'm stuck on figuring out whether such a graph can exist and so I thought I'd post it here as question.
Specifically, for any integer $n$, is it always possible to construct an undirected graph on $n$ nodes such that:
(i) the simple random walk on it has constant spectral gap?
(ii) there exist two vertices $i,j$ such that the stationary distribution at both of them is lower bounded by a constant and the hitting time from $i$ to $j$ is at least linear in $n$?
More formally, the question is as follows. Given an undirected graph $G$, the simple random walk is the Markov chain which transitions to a uniformly random out-neighbor at each step. Let $P$ be its probability transition matrix and $\pi$ be its stationary distribution, i.e, $\pi^T P = \pi^T$.
Do there exist positive constants $\alpha, \beta, c$ such that:
i) For each $n$, there exists an undirected graph on $n$ vertices for which the corresponding matrix $P$ satisfies $\lambda_2(P) \leq 1-\alpha$.
ii) In the same graph, there exist nodes $i,j$ such that $\pi_i \geq \beta$, $\pi_j \geq \beta$ and the hitting time from $i$ to $j$ is at least $cn$?
I suspect that this is not possible.