A spectral characterization for concentration of the cover time

We prove that for a sequence of finite vertex-transitive graphs of increasing sizes, the cover times are asymptotically concentrated if and only if the product of the spectral-gap and the expected cover time diverges. In fact, we prove this for general reversible Markov chains under the much weaker assumption (than transitivity) that the maximal hitting time of a state is of the same order as the average hitting time.