Probing the influence of topological and geometric disorder on the spectrum of the Laplacian operator on networks

Metric graphs are network-shaped structures on which one can solve partial differential equations to simulate a wide range of physical systems including conjugated molecules, photonic crystals, quantum mechanics in waveguide networks, and acoustic metamaterials. In particular, we consider metric graphs, which can be considered as a complex of edges, on which we define function spaces and differential operators (a combination sometimes called a quantum graph). Recent software advancements have made it feasible to analyze partial differential equations on large, compact metric graphs with a vast array of structures. Here, we generate compact metric graph structures using the spatial tessellations of hyperuniform point patterns, which have suppressed large-scale density fluctuations. This choice of structure is inspired by the exotic physical properties of network materials with these structures in other contexts. Then, we characterize the eigenspectrum structure of the Laplace operator on these graphs. In particular, we examine how disorder in the edge lengths and local graph topology affects the size and location of gaps in the eigenvalue spectrum. Importantly, many of the structures we consider are realizable in Euclidean space, meaning they are well-suited for practical applications in, e.g., metamaterial design. This work can thus be used to inform the design of metric graph-based systems with desired spectral properties.