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Characterize graphs with exactly three normalized Laplacian eigenvalues

Determine all finite simple graphs whose normalized Laplacian has exactly three distinct eigenvalues.

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Background

Understanding the number of distinct normalized Laplacian eigenvalues ties directly to structural graph characterization. It is known that graphs with exactly two normalized Laplacian eigenvalues are precisely the complete graphs, but the three-eigenvalue case remains unresolved.

A full characterization would contribute to spectral graph theory’s classification program, complementing known results on multiplicities and on spectra determined by graph operations or structure (e.g., complete multipartite and petal graphs), and would parallel existing results for other matrices such as adjacency and signless Laplacian.

References

Some other problems involving the normalized Laplacian regard multiplicities, for example: Which graphs have two normalized Laplacian eigenvalues, and which graphs have three normalized Laplacian eigenvalues ? The answer to the first question is: only complete graphs, while the answer to the second question is not known.

— At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian (2402.09160 - Beers et al., 14 Feb 2024) in Introduction, Subsection 1.2 (At the end of the Laplacian spectrum)