To find the total number of regular graphs of degree 2 on 12 vertices, we need to understand what a regular graph of degree 2 means.

A regular graph of degree 2 means that every vertex has exactly 2 edges connected to it. This type of graph is essentially a collection of disjoint cycles.

For 12 vertices:

• We can have cycles of different lengths, but they must sum up to 12. The possible configurations of cycles are:

  1. One cycle of length 12: There is only 1 way to arrange all 12 vertices in a single cycle.
  2. Two cycles of length 6: The number of ways to arrange 6 vertices in a cycle is $(6-1)!/2 = 5!$, but we have 2 identical cycles, so we divide by 2. So the number of ways is $\frac{(5!)(5!)}{2}$.
  3. Three cycles of length 4: The number of ways to arrange 4 vertices in a cycle is $(4-1)!/2 = 3!$. So the number of ways is $\frac{(3!)(3!)(3!)}{3!}$.
  4. Four cycles of length 3: The number of ways to arrange 3 vertices in a cycle is $(3-1)!/2 = 2!$. So the number of ways is $\frac{(2!)(2!)(2!)(2!)}{4!}$.
  5. Six cycles of length 2: This is just pairing up vertices, and there is only 1 way to do this.

Now, calculating the number of distinct graphs:

1. One cycle of length 12: There is only 1 such graph.
2. Two cycles of length 6: $\frac{(5!)(5!)}{2} = \frac{120 \times 120}{2} = 7200$.
3. Three cycles of length 4: $\frac{(3!)(3!)(3!)}{3!} = 6^2 = 36$.
4. Four cycles of length 3: $\frac{(2!)(2!)(2!)(2!)}{4!} = 1$ (since all cycles are identical).
5. Six cycles of length 2: There is only 1 such graph.

After counting all possibilities, considering that different cyclic configurations lead to distinct graphs, it turns out that:

• The number of distinct configurations considering is 6.

Hence, the correct answer is: (a) 6

Would you like more details or have any questions? Here are some related questions:

1. What is a regular graph of degree 3 on 12 vertices?
2. How does the cycle decomposition theorem apply to regular graphs?
3. Can a regular graph of degree 2 on 12 vertices be connected?
4. How do the properties of cycles affect the number of regular graphs?
5. How would the answer change for a different number of vertices?

Tip: When calculating possible graphs, consider symmetry and identical cycles to avoid over-counting.