Klein four-group as automorphism group of a graph. Every finite abstract group is the automorphism group of some graph.
Can someone show an example of a graph whose automorphism group is isomorphic to the Klein four-group?
 A: Sure, take two (nonisomorphic) graphs $G_1, G_2$ with trivial symmetry groups (easy), and then take the disjoint union of two copies of $G_1$ and two copies of $G_2.$
A: Try a simple graph with $4$ vertices and $5$ edges. In other words, draw the complete graph $K_4$ and remove one edge.
Or the complementary graph, with $4$ vertices and $1$ edge.
A: You can get infinitely many such graphs by starting with the (colored) Cayley graph $C$ of $G = (\mathbb{Z}/2\mathbb{Z})\times (\mathbb{Z}/2\mathbb{Z})$ and replacing colored edges with (uncolored) graphs having trivial symmetry and isomorphic to each other if and only if they replace edges of the same color. This is (one way) how to prove Frucht's theorem in the first place.
A: Try this. Start with a Schlegel diagram for a 3-dimensional cube 

Now put X's through the "upper" and "lower" faces (i.e., edges {0,3} and {1,2} for the "upper X", in this picture). That should create two perpendicular axes of mirror symmetry, and I don't see any other symmetries aside from the $180^\circ$ rotation.
