Determine the number of symmetries of the graph attached.
The Attempt at a Solution
I know the answer, I would just like someone to look over my argument to make sure I haven't missed out anything important. I also have some questions along the way.
Vertex 1 is the only vertex with valency 4 and so by defintion, symmetries preserve valancy, therefore any symmetry must leave vertex 1 in place.
Given that 1 is fixed, 2,3,4 must be attached to 1, but they need not be in the same position, so there are 3! permutations of these three vertices. (First question: How can I be sure that it is definitely 3!? I know the symmetries of a graph are independent on the way you draw it, so would there not be a chance that two of such permutations might actually correspond to the same permutation. The answer is probably no, but it is not that obvious to me.)
Similarly vertex 5 has valence 3 so it must be fixed and by same reasoning above 2! permutations of 6 and 7. Total number of symmetries = 3! . 2! = 12.
(2nd Question: Is the distribution of vertices 2,3,4 independent of that of 6,7. I.e if I send 2 to 3 then does that mean anything for what happens to 6 on the other side, for example? I feel the answer is no, but this is completely against what a symmetry would mean in a non-mathematical context. Is it correct to say that a symmetry in mathematics is not the same as a the notion of symmetry I would have had before I took this class?
(Last Question: Is it possible to say what these symmetries are? I.e one is the identity, one I think reflection, and the rest just rotations)