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In this paper we generalise the results on eigenvalues and eigenvectors ofunnormalized (combinatorial) Laplacian of two-dimensional grid presented byEdwards:2013 first to a grid graph of any dimension, and second also to othertypes of Laplacians, that is unoriented Laplacians, normalized Laplacians, andrandom walk Laplacians. While the closed-form or nearly closed form solutionsto the eigenproblem of multidimensional grid graphs constitute a good test suitfor spectral clustering algorithms for the case of no structure in the data,the multidimensional weighted grid graphs, presented also in this paper canserve as testbeds for these algorithms as graphs with some predefined clusterstructure. The weights permit to simulate node clusters not perfectly separatedfrom each other. This fact opens new possibilities for exploitation ofclosed-form or nearly closed form solutions eigenvectors and eigenvalues ofgraphs while testing and/or developing such algorithms and exploring theirtheoretical properties. Besides, the differences between the weighted andunweighted case allow for new insights into the nature of normalized andunnormalized Laplacians.

Spectral Analysis of Laplacians of an Unweighted and Weighted  Multidimensional Grid Graph -- Combinatorial versus Normalized and Random  Walk Laplacians