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Oct
9
comment Minimum word length for an unusual set of generators of the symmetric group
By the way, I knew about this problem previously but did not recognise it immediately. I computed a few of the values of the diameter in magma and then searched the OEIS for the corresponding sequence together with the word "diameter". The only hit was the right one. This is a good methodology to follow for this type of question.
Oct
9
answered Minimum word length for an unusual set of generators of the symmetric group
Oct
8
comment Minimum word length for an unusual set of generators of the symmetric group
Isn't this just prefix reversal? In other words, you are asking for the diameter of the "pancake graph". This is open but see oeis.org/A058986 for lots of information.
Oct
3
answered When is Aut(G) the symmetric group of an Aut(G)-invariant generating set?
Oct
3
comment When is Aut(G) the symmetric group of an Aut(G)-invariant generating set?
The cyclic group of order 6 is also an example.
Sep
25
comment What do we know about isospectral Cayley graphs?
@user6818 I already gave an example. The complete graph is a Cayley graph for any group of the appropriate cardinality (finite or not). Thus, in some extreme cases, it is impossible to recover any information about a group from a Cayley graph except its order. If you want more, then you need to tell us something about what you are assuming about the Cayley graphs.
Sep
25
comment What do we know about isospectral Cayley graphs?
@Paul Siegel Claiming that "the Cayley graph of a finite group is completely uninteresting" is inflammatory and, in fact, completely wrong. I'm assuming this is a troll. Moreover, my comment did not even assume finiteness...
Sep
25
comment What do we know about isospectral Cayley graphs?
I apologise in advance for this trivial comment: the complete graph is a Cayley graph for every group of the appropriate cardinality. Hence, even knowning that two groups have isomorphic Cayley graphs doesn't tell you much about the groups. You'll probably need to make your question more precise.
Sep
20
comment Existence of neighborhood inclusion for 4-chordal graphs
I think you mean "Up to symmetry" rather than "By symmetry".
Sep
10
comment What are smallest finite images of triangle groups?
Rather than using the SmallGroup library, you might want to use the lowindexnormalsubgroup command in magma on the triangle groups, you'll probably be able to get much further.
Sep
4
comment Are there number-theoretic graphs that are far from being isomorphic
For example, take the star on n vertices and the path on n vertices (both with n-1 edges). In this case, $\epsilon$ will be close to 1 no? It shouldn't be hard to generalise this example for different number of edges...
Sep
4
comment Are there number-theoretic graphs that are far from being isomorphic
It seems easier to simply take graphs with extremely different degree sequences.
Aug
27
comment Cubic Cayley (undirected) graphs
I'm a bit confused about your last paragraph. Clearly, graphs of type A and B are not ``Cayley isomorphic'', but what is meant by the first sentence? Certainly the same group can give rise to graphs of type A and B, even (graph) isomorphic ones.
Aug
27
answered Cubic Cayley (undirected) graphs
May
15
awarded  Revival
May
15
answered simple graphs of degree 16 with a semiregular normal subgroup isomorphic to the quaternion group $Q_8$
May
10
comment Triple Transitive Graphs
I only knew that the groups of rank 3 were classified and assumed that the graphs of rank 3 also were. After failing to find a reference and asking around a bit, it seems that they havn't really been explicitly listed anywhere, although how to do it is clear, at least in principle. It may be that the extra information you have in your situation (for example the girth) would allow one to finish case iv) without findind all the rank 3 graphs.
May
1
comment Triple Transitive Graphs
More specifically, aren't graphs in (iv) rank 3 graphs, which have been classified?
May
1
comment Triple Transitive Graphs
On the other hand, this paper predates the classification of finite simple groups and this seems like the kind of question where the classification could make a big difference (for example, the Higman-Sims graph involves a sporadic simple group, and, as noted in the sketch of the proof in the paper under question, a vertex-stabiliser acts 3-transitively on its neighbours, and 3-transitive groups were only classified post-classification, which might play a role.) Do you have access to reference [1] from the paper? That might be helpful, to see how the Higman-Sims graph comes up.
Apr
23
awarded  Nice Question