Allostery is the regulation of a protein's activity through a perturbation at a location distant from its active site. Such regulation is central to many biochemical processes. Targeting allosteric sites with drugs promises to allow fine-tuning of protein
activity. However, proteins are complex systems composed of thousands of atoms interacting over multiple temporal and spatial scales. Direct observation of the non-equilibrium response of proteins to allosteric perturbations is still a major challenge. This limits our understanding of how the signal induced by the perturbation propagates
across the protein and hampers our ability to predict the location of novel allosteric sites. Graph theory provides a way of representing proteins in a reduced form that still captures the full complexity of their underlying physico-chemical interactions. In this thesis, we develop a number of
novel graph-theoretic methods for analysing allosteric behaviour. We start by constructing an atomistic, energy-weighted graph representation of a protein. We then use the behaviour of dynamic
processes on this graph to explore how signals propagate within the protein. We use three distinct, but related methods. Markov stability identifies hierarchical community structure in the graph; Markov transients identifies anisotropic pathways of flow; and our bond-bond propensity measure quantifies the effect of instantaneous bond
fluctuations propagating through the protein.

These methods are applied to a number of biologically important allosteric proteins. Markov stability identifies dynamic coupling between the active and allosteric sites in caspase-1. The pathways involved in this coupling are revealed by combining a Markov transients analysis with computational mutagenesis. In caspase-1,
CheY and h-Ras, the bond-bond propensity correctly predicts the location of the allosteric site and identifies key allosteric interactions. Evaluating the Markov transients and bond-bond propensity methods against a larger set of allosteric proteins, we demonstrate that these measures are good predictors of a site's allosteric propensity.