Total number of regular graphs of degree 2 on 12 vertices.
(a) 6 (b) 7 (c) 8 (d) 9

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.

Discover how to calculate the total number of regular graphs of degree 2 on 12 vertices. Understand the concept of regular graphs, cycle decomposition, and specific configurations of cycles. Explore detailed solutions and the application of mathematical theorems in graph theory.

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