Asymmetric graphs

Question

I am working from the book, "Invitation to Discrete Math." On page 117 there's a question that asks to construct an Asymmetric graph. I thought I did with a graph of 5 vertices. However, immediately after the question, there's another one which asks to prove that no asymmetric graph of less than or equal to 5 vertices exist. (Image to be attached). Could you show me why the graph I constructed is not asymmetric? I have highlighted the vertices by circling in green. Thank you!

Answer 1

Let me name the vertices. The top left is A, the top right is B, the middle is C, the bottom left is D and the bottom right is E.

Then you can use this permutation: $\{A,B,C,D,E\}\to\{E,B,D,C,A\}$

Answer 2

In your first posting, the graph consisted of two disjoint triangles on vertices a,b,c and c,d,e, say, where c is the common vertex. A symmetry of the graph is the transposition (a,b) and another is to interchange d and e. One can also permute the two sets $\{a,b\}$ and $\{d,e\}$.

In your second posting, the graph again had five vertices, where there was a unique vertex of degree 4. Any symmetry must therefore fix this vertex. If this vertex is removed, then the remaining graph is the path graph $P_4$ on four vertices, which does have a nontrivial symmetry: one can interchange the first and last vertex of the path and interchange the second and third vertex of the path. Hence, the given graph has a nontrivial symmetry.