  As key subjects in spectral geometry and combinatorial graph theory
respectively, the (continuous) Hodge Laplacian and the combinatorial Laplacian
share similarities in revealing the topological dimension and geometric shape
of data and in their realization of diffusion and minimization of harmonic
measures. It is believed that they also both associate with vector calculus,
through the gradient, curl, and divergence, as argued in the popular usage of
"Hodge Laplacians on graphs" in the literature. Nevertheless, these Laplacians
are intrinsically different in their domains of definitions and applicability
to specific data formats, hindering any in-depth comparison of the two
approaches.
  To facilitate the comparison and bridge the gap between the combinatorial
Laplacian and Hodge Laplacian for the discretization of continuous manifolds
with boundary, we further introduce Boundary-Induced Graph (BIG) Laplacians
using tools from Discrete Exterior Calculus (DEC). BIG Laplacians are defined
on discrete domains with appropriate boundary conditions to characterize the
topology and shape of data. The similarities and differences of the
combinatorial Laplacian, BIG Laplacian, and Hodge Laplacian are then examined.
Through an Eulerian representation of 3D domains as level-set functions on
regular grids, we show experimentally the conditions for the convergence of BIG
Laplacian eigenvalues to those of the Hodge Laplacian for elementary shapes.

Differential Geometry