We are tasked with finding the maximum possible number of edges in a graph with $n$ vertices and $n-2$ connected components, where $n \geq 4$.

Step 1: Understand the Structure of the Graph

Let the graph have $n$ vertices and $n-2$ connected components. This means that the graph consists of $n-2$ disjoint connected subgraphs, and the total number of vertices in these subgraphs must sum to $n$.

Step 2: Maximize the Number of Edges

To maximize the number of edges, each connected component should have as many edges as possible. The maximum number of edges in a connected component with $k$ vertices is achieved when the component is a complete graph, denoted $K_k$. The number of edges in $K_k$ is given by:

$\binom{k}{2} = \frac{k(k-1)}{2}.$

Thus, to maximize the number of edges in the entire graph, each connected component should be as large as possible while still maintaining $n-2$ components.

Step 3: Distribute the Vertices

Let the sizes of the connected components be $k_1, k_2, \ldots, k_{n-2}$, where $k_1 + k_2 + \cdots + k_{n-2} = n$. To maximize the number of edges, we want to distribute the vertices in such a way that the sizes of the components are as large as possible. This is achieved by making the sizes of the components as equal as possible.

Case 1: All components are of size 1 or 2

• If all components are of size 1, then there are no edges at all, which is not optimal.
• If there are $n-2$ components of size 1, then the remaining 2 vertices must form a single component. The maximum number of edges is the maximum size that they can form, yielding a total 2 edges at most