The growing emphasis on data collection and machine learning has renewed the contributions of the ubiquitous Laplace operator in shape and data analysis. Variants and simplifications of the differential geometry de Rham-Hodge Laplacian have emerged as fast and concise topological and geometric shape descriptors for complex data sets. However, choosing the appropriate type of Laplace operator depends on the application and discretization scheme, especially in the context of volumes with 2-manifold boundary where treatment of boundary conditions is crucial.In this dissertation, we present the Boundary-Induced Graph (BIG) Laplacian, introduced using tools from Discrete Exterior Calculus (DEC), to bring the graph Laplacian and Hodge Laplacian on an equal footing for manifolds with boundary. BIG Laplacians are defined on discrete domains, accounting for appropriate normal or tangential boundary conditions. We examine the similarities and differences of the graph Laplacian, BIG Laplacian, and Hodge Laplacian through an in-depth comparison.Furthermore, we demonstrate experimentally the conditions for convergence of BIG Laplacian eigenvalues to those of the Hodge Laplacian for elementary shapes using an Eulerian representation of 3D domains as level-set functions on regular grids. Additionally, we show that similar schemes for defining Laplacians can be used as the kinetic energy component for the Hamiltonian operator of the density of small biological molecules. The spectra of such Hamiltonians serve as useful features for machine learning tasks in drug design and density function theory advancements, offering potential implications for practical applications.
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