$5$-vertex graphs with vertices of degree $2$ I'm trying to show that all graphs with $5$ vertices, each of degree $2,$ are isomorphic to each other.  Is there a more clever way than simply listing them all out?

Consider this simple, powerful theorem: all vertices of a graph have even degree if and only if the set of its edges can be partitioned in cycles.
The number of edges of your graph $G$ is $e(G) = \frac 1 2 \sum_{x\in V(G)} d(x) = 5$. Observe that there are not isolated vertices and that every cycle has at least three edges. Therefore the only possible partition in cycles is $E(G)$ itself, i.e. all the graphs with the required property are 5-cycles and thus they are isomorphic.
