A self-complementary graph is a graph that is isomorphic to its complement. One key property of such graphs is that the number of vertices $n$ in a self-complementary graph must satisfy $n \equiv 0 \ (\text{mod} \ 4)$ or $n \equiv 1 \ (\text{mod} \ 4)$.

The total number of edges $E$ in any graph on $n$ vertices is at most $\frac{n(n-1)}{2}$, which is the number of edges in a complete graph. For a self-complementary graph, since the graph is isomorphic to its complement, it will have exactly half the total number of edges in a complete graph:

$E = \frac{n(n-1)}{4}$

Now let's check each option by reversing the formula and seeing if it corresponds to a valid $n$:

1. 20 edges: $E = \frac{n(n-1)}{4} \quad \Rightarrow \quad 20 = \frac{n(n-1)}{4} \quad \Rightarrow \quad n(n-1) = 80$ Solving the quadratic equation $n^2 - n - 80 = 0$, the discriminant is $\Delta = 1 + 320 = 321$, which is not a perfect square. Thus, there is no integer $n$, so 20 is not possible.

2. 24 edges: $E = \frac{n(n-1)}{4} \quad \Rightarrow \quad 24 = \frac{n(n-1)}{4} \quad \Rightarrow \quad n(n-1) = 96$ Solving $n^2 - n - 96 = 0$, the discriminant is $\Delta = 1 + 384 = 385$, which is not a perfect square. So, 24 is also not possible.

3. 30 edges: $E = \frac{n(n-1)}{4} \quad \Rightarrow \quad 30 = \frac{n(n-1)}{4} \quad \Rightarrow \quad n(n-1) = 120$ Solving $n^2 - n - 120 = 0$, the discriminant is $\Delta = 1 + 480 = 481$, which is a perfect square ($481 = 21^2$). So, $n = 11$ is a valid solution, making 30 possible.

4. 36 edges: $E = \frac{n(n-1)}{4} \quad \Rightarrow \quad 36 = \frac{n(n-1)}{4} \quad \Rightarrow \quad n(n-1) = 144$ Solving $n^2 - n - 144 = 0$, the discriminant is $\Delta = 1 + 576 = 577$, which is not a perfect square. Thus, 36 is not possible.

Conclusion:

Both 20 and 36 are not possible total edge counts for a self-complementary graph. Therefore, the correct answer is:

Option 20 cannot be the total number of edges.

Would you like further details or have any questions?

Here are five follow-up questions to expand on this concept:

1. How do you construct a self-complementary graph?
2. What are some examples of self-complementary graphs?
3. How is the complement of a graph defined?
4. Can a bipartite graph be self-complementary?
5. What is the relationship between vertex count and edge count in graph theory?

Tip: For any graph, understanding its complement can often reveal symmetries and other hidden properties useful for graph-theoretic proofs.