Talk:Hoffman–Singleton graph

Not a Cayley graph

I removed the assertion that this graph is a Cayley graph. To prove it isn't a Cayley graph, find generators for its automorphism group (using nauty, for example) and note they are all even permutations. This means every automorphism is even. However, if it was a Cayley graph there would be an automorphism with 25 2-cycles, which is an odd permutation. You can call this original research if you like, but since there was no citation for the claim removing it is allowed. McKay (talk) 02:13, 5 June 2012 (UTC)

That was an error, I miss-read data from Mathematica (via Wolfram Alpha). Koko90 (talk) 07:51, 5 June 2012 (UTC)

Thank you Koko90 for correcting this up also in Slovene WP. --xJaM (talk) 08:49, 5 June 2012 (UTC)

Fano plane construction

What obvious "permutation" (?) of the vertices of the Fano plane gives 30 variants? Presumably the special middle vertex remains fixed, or the number of permutations would be a multiple of 7, but what are the other constraints down from 6! permutations? It must be clarified. — Preceding unsigned comment added by 89.97.187.68 (talk) 10:29, 21 November 2016 (UTC)

This paragraph is unsourced and also makes no sense to me. The Fano plane has 168 symmetries; 30 is not a divisor of 168. If it can't be clarified it should be deleted. —David Eppstein (talk) 18:11, 21 November 2016 (UTC)

I found a very likely source for the mangled explanation. About half length, after the pretty pictures: "The symmetry group of the Fano plane has 168 elements. So, the number of different Fano plane structures on a 7-element set is 7!/168=30. Call each of these a Fano." http://blogs.ams.org/visualinsight/2016/02/01/hoffman-singleton-graph/

Symmetry group of the Hoffman Singleton graph

According to reference [5] in the article and also according to the [ATLAS], the symmetry group of the Hoffman–Singleton graph is not PSU(3,5²).2, but PSU(3,5).2. The latter means an extension of PSU(3,5) by a group of order 2. This appears to be a question of notation: PSU(3,5²) in reference [7] of the article is the same thing as PSU(3,5) in [5] and in the [ATLAS].

[ATLAS] Conway et. al., ATLAS of finite groups

--Martin Seysen (talk) 22:08, 4 March 2017 (UTC)

Subgraph isomorphism for dodecahedron

After looking fruitlessly for some card-carrying mathematician to have written a note about whether the HS graph contains a dodecahedral subgraph or not, I finally installed a subgraph isomorphism solver called Glasgow https://github.com/ciaranm/glasgow-subgraph-solver and did the search myself. It instantly found such a dodecahedron in HS and printed the mapping. It took 12 seconds to count all of them: 5040000, and 2 minutes to print them. I verified the results independently. Apparently no card-carrying mathematician published it because it was beneath their dignity to print actual numbers. Anyway, given how easily the program found the mappings and verified the 252000 automorphisms, doubtless someone must have known the result already, so I'll add it to the article. If any of you matherati find a reason why it is 5040000, please add that. I observe that 5040000 == the 252000 HS automorphisms * the 20 dodecahedron vertices, but I have no idea what that means and am not adding it. -- Quack5quack (talk) 01:43, 18 August 2023 (UTC)

I found today that the subgraph program gives the number of distinct mappings, which must be divided by the 120 automorphisms of the dodecahedron to give the number of distinct dodecahedra, otherwise the same set of vertices is counted 120 ways. Thus the reduced answer is 5.04 million / 120 = 42000 dodecahedra. I will correct it. -- Quack5quack (talk) 14:39, 18 August 2023 (UTC)