  Affinity has proven to be a useful tool for quantifying the non-equilibrium
character of time continuous Markov processes since it serves as a measure for
the breaking of time reversal symmetry. It has recently been conjectured that
the number of coherent oscillations, which is given by the ratio of imaginary
and real part of the first non-trivial eigenvalue of the corresponding master
matrix, is constrained by the maximum cycle affinity present in the network. In
this paper, we conjecture a bound on the whole spectrum of these master
matrices that constrains all eigenvalues in a fashion similar to the well known
Perron-Frobenius theorem that is valid for any stochastic matrix. As in other
studies that are based on affinity-dependent bounds, the limiting process that
saturates the bound is given by the asymmetric random walk. For unicyclic
networks, we prove that it is not possible to violate the bound by small
perturbation of the asymmetric random walk and provide numerical evidence for
its validity in randomly generated networks. The results are extended to
multicyclic networks, backed up by numerical evidence provided by networks with
randomly constructed topology and transition rates.
