A finite, discrete-time, time-homogeneous Markov chain is a type of mathematical model used to describe dynamical systems which transition between a finite number of possible states in discrete time increments. A Markov chain model may be used in a wide range of applications, such as urban road traffic, computational drug design, and the spread of disease. In many such applications, there are existing constraints on the structure of the underlying network dictating which transitions are possible, and which are not. In this thesis, the influence of these combinatorial constraints on the behaviour of a Markov chain is explored.