On the relation between reversibility and monotonicity of fluctuation spectra for discrete time finite state Markov chains

For the irreducible aperiodic discrete time finite state Markov chain, (i) if it is reversible, we provide a necessary and sufficient condition for all fluctuation spectra to be monotonic on [0,π] and (ii) if it is irreversible, then there exist some nonmonotonic fluctuation spectra.