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1d
comment Vertex-primitive graphs with two vertices having almost the same neighbourhood
EDIT: I've added a proof in the affine case, in the hope that somebody sees how to generalise it.
1d
revised Vertex-primitive graphs with two vertices having almost the same neighbourhood
added 3846 characters in body
Nov
18
comment 4-regular graph with every edge lying in a unique 4-cycle
@BenBarber What about the graph on {1,2,3,4,5,6} with the two cycles 1234 and 2536? They are edge-disjoint but share two vertices, and there are no other 4-cycles...
Nov
12
comment Vertex-primitive graphs with two vertices having almost the same neighbourhood
Right. More generally, let $\Gamma'$ be the graph with the same vertex-set but two vertices adjacent if they had almost the same neighbourhood in $\Gamma$. Then $Aut(\Gamma)$ acts on $\Gamma'$ hence it is also vertex-primitive. In particular, if it is two-valent, it has prime order. In other words, the critical case is if there are at least three vertices with almost the same neighbourhood as a given vertex $v$. (I don't know any non-complete examples where this happens.)
Nov
11
awarded  Fanatic
Nov
10
comment Vertex-primitive graphs with two vertices having almost the same neighbourhood
Right, any cycle satisfies the second part of the hypothesis, but the only cycles that are vertex-primitive are the ones of prime order.
Nov
10
revised Vertex-primitive graphs with two vertices having almost the same neighbourhood
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Nov
10
asked Vertex-primitive graphs with two vertices having almost the same neighbourhood
Nov
3
awarded  Enlightened
Nov
3
awarded  Nice Answer
Oct
30
revised The number of Sylow subgroups of a group
added 45 characters in body
Oct
30
comment The number of Sylow subgroups of a group
Thanks for the correction!
Oct
27
answered The number of Sylow subgroups of a group
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...