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In the context of directed or undirected graphs, matrices such as adjacency and Laplacian matrices are commonly used. The eigenbasis of these matrices addresses some practical implications, such as centrality measures (like PageRank), and it is also crucial for clustering (community detection).

Q. What other practical implications have been identified in the literature by analyzing the eigenbasis?

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  • $\begingroup$ Cross posted. $\endgroup$
    – Jochen Glueck
    Commented May 2 at 13:51
  • $\begingroup$ In any case, I think this question would benefit from removing some of the buzzwords from the title. $\endgroup$
    – Jochen Glueck
    Commented May 2 at 13:53

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A place to find a wealth of examples is in graph signal processing, where spectral decomposition of the Laplacian is known as the graph Fourier transform.

  • A. Ortega, Introduction to Graph Signal Processing. Cambridge (2022), chapter 3 for the GFT and chapter 7 for examples.

Spectral graph theoretic examples are discussed in various places for example:

This finds applications in graph and convolution based machine learning, and neuroimaging/brain connectivity, and perhaps as a illustrative example on road traffic data.

There is also work in TDA and Applied Topology/Persistence along these lines such as persistent laplacians:

There are applications in protein engineering, covid variant forcasting and genomics.

  • Wei, Xiaoqi, and Guo-Wei Wei. "Persistent Topological Laplacians--a Survey." arXiv preprint arXiv:2312.07563 (2023).
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