I am currently struggling with understanding isomorphisms and automorphisms.
Can somebody in more "layman's terms" explain these terms for me and help me with this practice problem?
How many automorphisms are there for each one of the graphs A,B,C,D and E?
Two graphs are isomorphic if one is essentially a relabeling of the other. That is, there is a one-to-correspondence of the vertices such that two vertices in one graph are adjacent if and only if the corresponding vertices in the other graph are adjacent. The one-to-one correspondence itself is called an "isomorphism."
When the two graphs are the same graph, the isomorphism is called an "automorphism." In this case, the automorphism is just a permutation of the vertices.
In example A, there are two automorphisms. Note that an isomorphism must preserve vertex degrees. Let $f$ be an automorphism. Then $f(1)$ can only be $1$ or $5$. If $f(1)=1$, then $f(2)$ must be adjacent to $f(1)$ so $f(2)=2$. Also, $f(3)$ is adjacent to $f(2)$, so $f(3)$ is $1$ or $2$. But $f(1)=1$ and $f$ is a permutation, so $f(3)=3$. Continuing in this manner, we find that $f$ is the identity.
The other choice is $f(1)=5$. Reasoning similar to the above shows that $f(2)=4, f(3)=3,f(4)=2,f(5)=1.$
The other examples are a bit more complicated than this one, but not much.
Please ask me if there is anything here you don't understand. Try the others for yourself, and if you run into trouble, ask another question. I think it's best if you ask about one graph at a time, showing how far you've gotten, and pinpointing your difficulty as much as possible.
Good luck.
By the way, the set of all automorphisms of a graph, together with the operation of function composition form a structure called a group, so the tag "automorphism group" that you've applied is relevant, if somewhat marginally. On the other hand, the tag "group isomorphism" has nothing at all to do with this. It's best not to use tags unless you're sue you know what they mean.