Zeta functions and topological entropy of the Markov-Dyck shifts

The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in Keller (G. Keller, {\it Circular codes, loop counting, and zeta-functions}, J. Combinatorial Theory {\bf 56} (1991), pp. 75–83). For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topological entropy ae determined.