2-regular (simple) graphs on n vertices are determined (up to isomorphism) by the partition of n induced by the size of its connected components. This is because connected components of 2-regular graphs must be cycles.
Conversely, given a partition $d_1,d_2,\ldots,d_k$ of n into parts of size at least 3 (you can't have cycles of length 1 or 2, which would induce loops and multiedges) then you can construct a 2-regular graph consisting of a $d_i$-cycle for each $d_i$.
In the above case, the partition is 7 (since the graph is a 7-cycle).
0 0 1 0 0 0 1
0 0 0 1 0 1 0
1 0 0 1 0 0 0
0 1 1 0 0 0 0
0 0 0 0 0 1 1
0 1 0 0 1 0 0
1 0 0 0 1 0 0
There exist an automorphism of the graph that maps vertex u to vertex v provided u and v both belong to cycles of the same length. To illustrate, you can fix everything outside of the components containing u and v, then
- if u and v are in the same cycle, just rotate the cycle, or otherwise,
- swap the cycles containing u and v, then rotate each appropriately.