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Vertexprimitive 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 
Vertexprimitive graphs with two vertices having almost the same neighbourhood
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Nov 18 
comment 
4regular graph with every edge lying in a unique 4cycle
@BenBarber What about the graph on {1,2,3,4,5,6} with the two cycles 1234 and 2536? They are edgedisjoint but share two vertices, and there are no other 4cycles... 
Nov 12 
comment 
Vertexprimitive graphs with two vertices having almost the same neighbourhood
Right. More generally, let $\Gamma'$ be the graph with the same vertexset but two vertices adjacent if they had almost the same neighbourhood in $\Gamma$. Then $Aut(\Gamma)$ acts on $\Gamma'$ hence it is also vertexprimitive. In particular, if it is twovalent, 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 noncomplete examples where this happens.) 
Nov 11 
awarded  Fanatic 
Nov 10 
comment 
Vertexprimitive 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 vertexprimitive are the ones of prime order. 
Nov 10 
revised 
Vertexprimitive graphs with two vertices having almost the same neighbourhood
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Nov 10 
asked  Vertexprimitive 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 
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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 
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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 
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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 
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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 
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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... 