  In this paper we generalise the results on eigenvalues and eigenvectors of
unnormalized (combinatorial) Laplacian of two-dimensional grid presented by
Edwards:2013 first to a grid graph of any dimension, and second also to other
types of Laplacians, that is unoriented Laplacians, normalized Laplacians, and
random walk Laplacians. While the closed-form or nearly closed form solutions
to the eigenproblem of multidimensional grid graphs constitute a good test suit
for spectral clustering algorithms for the case of no structure in the data,
the multidimensional weighted grid graphs, presented also in this paper can
serve as testbeds for these algorithms as graphs with some predefined cluster
structure. The weights permit to simulate node clusters not perfectly separated
from each other. This fact opens new possibilities for exploitation of
closed-form or nearly closed form solutions eigenvectors and eigenvalues of
graphs while testing and/or developing such algorithms and exploring their
theoretical properties. Besides, the differences between the weighted and
unweighted case allow for new insights into the nature of normalized and
unnormalized Laplacians.
