Dynamical systems in which the orbit of a point may escape from the state space through a hole are classified as open dynamical systems. The escape rate represents the average rate at which the orbits escape into the hole. This depends on the size and the position of the hole in the state space. The larger the escape rate, the faster the orbits terminate. In this talk, we will consider a particular class of open dynamical systems namely subshifts of finite type with a hole. Using the concept of escape rate, we describe the relationship between the Perron root and Perron eigenvectors of an irreducible subshift of finite type with the correlation between the forbidden words in the subshift. As an application, we obtain expressions for the Perron eigenvectors for irreducible non-negative matrices which are adjacency matrices for directed graphs. Moreover, we will present an alternate definition of the Parry measure where the explicit role of forbidden words in the subshift is highlighted.