The Eigenvalue Region of a Family of Markov Chains with Randomness Confined to Two States

We determine the complete eigenvalue region of the following Markov chains on $2q$ states, for $q\ge2$. The states are $A_0,\ldots,A_{q-1}$ and $B_0,\ldots,B_{q-1}$. Each transition $A_j\to A_{j+1}$ and $B_j\to B_{j+1}$, for $0\le j<q-1$, has probability one. From $A_{q-1}$, the chain moves to $A_0$