We are tasked with finding the number of vertices $n$ in the graph $G$. Here's the solution:

Key Definitions and Facts:

1. $G$ is a simple graph with no self-loops, so it has $n$ vertices.
2. The total number of possible edges in a complete graph $K_n$ with $n$ vertices is: $\binom{n}{2} = \frac{n(n-1)}{2}$
3. The complement graph $\overline{G}$ has all the edges that are not in $G$. Thus: $\text{Edges in } G + \text{Edges in } \overline{G} = \binom{n}{2}$

Information Provided:

• $G$ has 21 edges.
• $\overline{G}$ has 15 edges.

Equating the Total Number of Edges:

$21 + 15 = \binom{n}{2}$ $36 = \binom{n}{2}$ Using the formula for $\binom{n}{2}$, we have: $\frac{n(n-1)}{2} = 36$ Multiply through by 2: $n(n-1) = 72$

Solving the Quadratic Equation:

Rearrange: $n^2 - n - 72 = 0$ Factorize: $(n - 9)(n + 8) = 0$ Thus: n = 9 \quad \text{(since n > 0)}.

Verification:

• The total number of possible edges in $G$ and $\overline{G}$ is: $\binom{9}{2} = \frac{9 \cdot 8}{2} = 36.$
• $G$ has 21 edges, and $\overline{G}$ has $36 - 21 = 15$, which matches the problem statement.

Final Answer:

The graph $G$ has 9 vertices.

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Would you like a breakdown of related concepts or further details? Here are 5 questions to explore:

1. How do we derive the formula for $\binom{n}{2}$?
2. What are the properties of simple graphs and their complements?
3. Can $G$ and $\overline{G}$ have the same number of edges?
4. How does the complement operation affect connectedness in graphs?
5. How does the solution change if we include self-loops?

Tip: Always verify the complement of a graph by ensuring the total number of edges sums to $\binom{n}{2}$.