This talk will focus on how graph-structures can be compactly characterised using measurements motivated by diffusion processes and random walks. It will commence by explaining the relationship between the heat equation on a graph, the spectrum of the Laplacian matrix (the degree matrix minus the weighted adjacency matrix) and the steady-state random walk. The talk will then focus in some depth on how the heat kernel, i.e. the solution of the heat equation, can be used to characterize graph structure in a compact way. One of the important steps here is to show that the zeta function is the moment generating functions of the heat kernel trace, and that the zeta function is determined by the distribution of paths and the number os spanning trees in a graph. We will then explore a number of applications of these ideas in image analysis and computer vision.