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Spectral geometry of differential operators

We are primarily interested in the spectral geometry of differential operators: how the shape and structure of an object affect qualitative and quantitative properties of solutions of certain differential equations such as the heat and the wave equation defined on that object. An area of particular focus is the study of spectral geometry on graphs and networks, more precisely so-called quantum graphs: for example, how can knowledge about the rate and nature of diffusion within a network of thin pipes (or transmission lines, or ... ) allow one to identify the presence of bottlenecks or clusters within the network?