  The rate of convergence of the Gibbs sampler for the generalized
one-dimensional Ising model is determined by the second largest eigenvalue of
its transition matrix in absolute value denoted by $\beta^*$. In this paper we
generalize a result from Shiu and Chen $(2015)$ for the one-dimensional Ising
model with two states which gives a bound for $\beta^*$ . The method is based
on Diaconis and Stroock bound for reversible Markov processes. The new bound
presented in this paper improves Ingrassia's $(1994)$ result.
