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The application of graph Laplacian eigenvectors has been quite popular in the graph signal processing field: one can use them as ingredients to design smooth multiscale basis. Our long-term goal is to study and understand the dual geometry of graph Laplacian eigenvectors. In order to do that, it is necessary to define a certain metric to measure the behavioral differences between each pair of the eigenvectors. Saito (2018) considered the ramified optimal transportation (ROT) cost between the square of the eigenvectors as such a metric. Clonginger and Steinerberger (2018) proposed a way to measure the affinity (or `similarity') between the eigenvectors based on their Hadamard (HAD) product. In this article, we propose a simplified ROT metric that is more computational efficient and introduce two more ways to define the distance between the eigenvectors, i.e., the time-stepping diffusion (TSD) metric and the difference of absolute gradient (DAG) pseudometric. The TSD metric measures the cost of "flattening" the initial graph signal via diffusion process up to certain time, hence it can be viewed as a time-dependent version of the ROT metric. The DAG pseudometric is the l2-distance between the feature vectors derived from the eigenvectors, in particular, the absolute gradients of the eigenvectors. We then compare the performance of ROT, HAD and the two new "metrics: on different kinds of graphs. Finally, we investigate their relationship as well as their pros and cons. Keywords: Graph Laplacian eigenvectors, metrics between orthonormal vectors, dual geometry of graph Laplacian eigenvectors, multiscale basis dictionaries on graphs, heat diffusion on graphs, Wasserstein distance, optimal transport
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Hadamard Diagonalizable Graphs of Order at Most 36
If the Laplacian matrix of a graph has a full set of orthogonal eigenvectors with entries $$\pm1$$, then the matrix formed by taking the columns as the eigenvectors is a Hadamard matrix and the graph is said to be Hadamard diagonalizable. In this article, we prove that if $n=8k+4$ the only possible Hadamard diagonalizable graphs are $$K_n$$, $$K_{n/2,n/2}$$, $$2K_{n/2}$$, and $$nK_1$$, and we develop a computational method for determining all graphs diagonalized by a given Hadamard matrix of any order. Using these two tools, we determine and present all Hadamard diagonalizable graphs up to order 36. Note that it is not even known how many Hadamard matrices there are of order 36.
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- PAR ID:
- 10382202
- Date Published:
- Journal Name:
- The Electronic Journal of Combinatorics
- Volume:
- 29
- Issue:
- 2
- ISSN:
- 1077-8926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.37236/9725
