Timeline for Factorization of the characteristic polynomial of the adjacency matrix of a graph
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| May 5, 2022 at 12:59 | answer | added | baronbrixius | timeline score: 0 | |
| Sep 6, 2020 at 23:01 | vote | accept | Joe Silverman | ||
| Sep 6, 2020 at 22:50 | comment | added | Joe Silverman | @RichardStanley Thanks. I think that my graph does have a large automophism group. I'm not sure I can compute it exactly, but pulling out a large piece should give something. It's also helpful to know that although symmetry causes factorization, the converse isn't necessarily true; I'll keep that in mind. | |
| Sep 6, 2020 at 20:24 | answer | added | Timothy Chow | timeline score: 4 | |
| Sep 6, 2020 at 14:55 | answer | added | Tsemo Aristide | timeline score: 5 | |
| Sep 6, 2020 at 14:49 | comment | added | Richard Stanley | Let me add that there seem to be quite a few examples of graphs for which $P_G(X)$ factors more than is explained by symmetry. An example is mathoverflow.net/questions/369515. For a different kind of example of this phenomenon of "extra factorization," see Exercise 10.9(d) of my book Algebraic Combinatorics, second ed. | |
| Sep 6, 2020 at 12:03 | answer | added | joro | timeline score: 2 | |
| Sep 6, 2020 at 11:32 | comment | added | M. Winter | Since the graphs seems to have many eigenspaces of dimension three, I would probably try to get some intiution for its structure by plotting the 3-dimensional spectral graph realizations to these eigenvalues. Maybe some of these are of a quite recognizable shape (a Platonic solid etc). See here for what I mean by "spectral realization". | |
| Sep 6, 2020 at 10:39 | answer | added | M. Winter | timeline score: 10 | |
| Sep 6, 2020 at 10:23 | history | became hot network question | |||
| Sep 6, 2020 at 9:05 | comment | added | M. Winter | Large eigenspaces of the adjacency matrix often point to either large symmetry, or more generally, to large regularity of the graph (whatever that means precisely). You could check whether your graph is distance-regular or at least $k$-walk-regular for some small $k\ge1$ as this often comes with large eigenspaces. | |
| Sep 6, 2020 at 9:00 | history | edited | M. Winter | CC BY-SA 4.0 |
edited tags; edited title
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| Sep 6, 2020 at 8:26 | answer | added | Qiaochu Yuan | timeline score: 27 | |
| Sep 6, 2020 at 2:33 | comment | added | Richard Stanley | Do your graphs have a lot of automorphisms? This will induce a factorization of $P_G(X)$. | |
| Sep 6, 2020 at 2:17 | history | asked | Joe Silverman | CC BY-SA 4.0 |