Let $n\geq4$. What is the maximum possible number of edges in a graph
with $n$ vertices and $n-2$ connected components? Prove your answer.

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

Let $n \geq 4$. What is the maximum possible number of edges in a graph with $n$ vertices and $n-2$ connected components? Prove your answer.

This problem explores the maximum number of edges in a graph with n vertices and n-2 connected components. It involves understanding graph structure and applying combinatorial principles to maximize the number of edges, suitable for students in Grades 11-12.

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