To find the maximum possible number of edges in a graph with $n$ vertices and $n - 2$ connected components, we start by analyzing the structure of the graph.

1. Understanding Components:

   • If a graph has $n - 2$ connected components, this means that one component will have 2 vertices and the remaining $n - 2$ components will each have 1 vertex (since $n$ total vertices must be accounted for).

2. Edges in Components:

   • The component with 2 vertices can have a maximum of 1 edge (since a complete graph with 2 vertices, $K_2$, has exactly 1 edge).
   • Each of the $n - 2$ components consisting of a single vertex has no edges.

3. Calculating the Maximum Number of Edges:

   • The only edges present in the graph will come from the component with 2 vertices. Therefore, the total number of edges $E$ in the graph is: $E = 1 \text{ (from the component with 2 vertices)}$

4. Conclusion:

   • The maximum possible number of edges in a graph with $n$ vertices and $n - 2$ connected components is 1.

Proof:

To formalize this:

• Let $C_1, C_2, \ldots, C_{n-2}$ be the $n - 2$ connected components of the graph.
• Without loss of generality, we can assume:
  • $|C_1| = 2$ (the component with 2 vertices),
  • $|C_2| = |C_3| = \ldots = |C_{n-2}| = 1$ (the components with 1 vertex each).
• Thus: $|C_1| + |C_2| + |C_3| + \ldots + |C_{n-2}| = 2 + (n - 2) \cdot 1 = n$
• The component $C_1$ can have at most $\binom{2}{2} = 1$ edge, while the others contribute 0 edges.

Final Result:

Thus, the maximum number of edges in a graph with $n$ vertices and $n - 2$ connected components is: $\boxed{1}$

Would you like more details on this topic or have any questions? Here are some related questions you might consider:

1. What happens to the number of edges if we increase the number of components?
2. How does the structure change if we allow more than one edge between components?
3. Can we achieve more edges with different configurations of components?
4. What is the maximum number of edges in a connected graph with $n$ vertices?
5. How does this result relate to other graph theory concepts, such as trees and forests?

Tip: When studying graph theory, it's helpful to draw diagrams to visualize the relationships between vertices and edges.