I'm talking about eigenvalues/eigenvectors of the adjacency matrix, though for regular graphs, the adjacency and Laplacian matrices are interchangeable.
Yes, the 1-skeleton of the 7-prism (2 faces of length 7 and 7 faces of length 4) has eigenvalues that are roots of a cubic. Its minimal polynomial is:
(x - 3)(x - 1)(x^3 - 2x^2 - x + 1)^2 (x^3 + 4x^2 + 3x - 1)^2.
The graph is the Cartesian product of the 7-cycle with an edge. The 9-prism also has eigenvalues that are roots of a cubic.
In general, the n-prism "lifts" the eigenvalues of the n-cycle, whose characteristic polynomial (of the adj matrix) by given the Chebyshev polynomial of the 1st kind. This is a good way to make graphs with interesting Galois automorphisms, because the eigenvalues are then (functions of) cosines of rational multiples of pi.
Krystal Guo
npub14p…40m69
2026-05-17 12:48:29 UTC
in reply to nevent1q…ef6w
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2026-05-17 12:48:29 UTCEvent JSON
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"content": "I'm talking about eigenvalues/eigenvectors of the adjacency matrix, though for regular graphs, the adjacency and Laplacian matrices are interchangeable. \n\nYes, the 1-skeleton of the 7-prism (2 faces of length 7 and 7 faces of length 4) has eigenvalues that are roots of a cubic. Its minimal polynomial is:\n(x - 3)(x - 1)(x^3 - 2x^2 - x + 1)^2 (x^3 + 4x^2 + 3x - 1)^2.\nThe graph is the Cartesian product of the 7-cycle with an edge. The 9-prism also has eigenvalues that are roots of a cubic. \n\nIn general, the n-prism \"lifts\" the eigenvalues of the n-cycle, whose characteristic polynomial (of the adj matrix) by given the Chebyshev polynomial of the 1st kind. This is a good way to make graphs with interesting Galois automorphisms, because the eigenvalues are then (functions of) cosines of rational multiples of pi.",
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