How did I produce these coordinates? The answer is eigenmagicks! The eigenvalues of the icosahedral graph are 5, -1, sqrt(5) and -sqrt(5). The multiplicity of the irrational eigenvalues are each 3. So I found an eigenbasis for the -sqrt(5) eigenspace, which is 3 vectors. Then I just used the v-th entry of these 3 vectors to obtain (x,y,z) coordinates for vertex v.
The least eigenvalue of a graph appears to be a rigid graph invariant, particularly when its absolute value is small. Graphs with least eigenvalue at least -2 or at least -3, admit geometric realizations in lattices (excepting finitely many sporadic graphs).
Krystal Guo
npub14p…40m69
2026-05-16 16:45:14 UTC
in reply to nevent1q…0te0
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2026-05-16 16:45:14 UTCEvent JSON
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"content": "How did I produce these coordinates? The answer is eigenmagicks! The eigenvalues of the icosahedral graph are 5, -1, sqrt(5) and -sqrt(5). The multiplicity of the irrational eigenvalues are each 3. So I found an eigenbasis for the -sqrt(5) eigenspace, which is 3 vectors. Then I just used the v-th entry of these 3 vectors to obtain (x,y,z) coordinates for vertex v. \n\nThe least eigenvalue of a graph appears to be a rigid graph invariant, particularly when its absolute value is small. Graphs with least eigenvalue at least -2 or at least -3, admit geometric realizations in lattices (excepting finitely many sporadic graphs).",
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