What are multiple isomorphisms?

Question

For example, this graph has "multiple isomorphisms." What does that mean?

And could you list them? I don't understand how there can be more than one.

Answer 1

There can be lots of isomorphisms between the two. One isomorphism is obvious. Just take the image on the left and set it on top of the image on the right. Map each edge and vertex to the ones that overlap.

Now you can get another isomorphism by doing the same thing as before, but flipping the image on the left over as you place it on the image on the right.

You may be getting confused because isomorphism between objects say that the objects are essentially the same, that they aren't unique relative to each other. However, that doesn't mean that the isomorphism is unique.

In fact, I'm sure there's even more isomorphisms than the two we created above. It's just that those are the easiest ones to see. Try finding any more for yourself as an exercise!

Answer 2

Call these graphs $G$ and $H$ (from left to right in your image). Define an isomorphism $f: V(G)\to V(H)$ by setting $f(1) = a$, $f(2) = b$, $f(3) = c$, and so on. In other words, note $H$ is just a relabeling of the same underlying polygons (those formed by the vertices and edges when viewed as subsets of the plane) as $G$.

However, $f$ isn't the only isomorphism between these graphs. (Recall that a graph isomorphism $g$ is just a bijection $g: V(G) \to V(H)$ which "preserves edges" in the sense that $v_1$ and $v_2$ are adjacent in $G$ $\iff$ $f(v_1)$ and $f(v_2)$ are adjacent in $H$.) Indeed, define $g: V(G) \to V(H)$ by $g(1) = b$, $g(2) = a$, and $g(n) = f(n)$ for $3 \leq n \leq 6$. Then $g$ is another bijection that preserves edges and therefore an isomorphism. If you think about the assignments being made, $g$ is just another formal way to describe how $G$ and $H$ are the same underlying graph with different labels.