7. If you want to draw a simple graph that has 6 vertices and two components, then: (a) What is the minimum number of edges possible in the graph? Draw such a graph. (b) What is the maximum number of edges possible in the graph? Draw the graph.

Let's tackle the two parts of this problem one at a time:

### Part (a) - Minimum Number of Edges in a Graph with 6 Vertices and 2 Components

To find the minimum number of edges, we need to consider the graph as consisting of two separate components. A component must have at least one edge to be connected. To minimize the number of edges:

- One option is to have one component be a single vertex (which does not require any edges), and the other component be a connected graph (the simplest being a tree with \(n-1\) edges).
  
- The simplest connected graph is a tree, where the number of edges is \(n-1\), where \(n\) is the number of vertices.

To minimize the number of edges, we can have the following configuration:
- One component is a tree with 5 vertices, which has \(5-1 = 4\) edges.
- The other component is a single isolated vertex, which adds 0 edges.

Thus, the **minimum number of edges** is 4.

### Simple Graph Example for Part (a):
- Component 1: A tree with 5 vertices connected by 4 edges.
- Component 2: A single isolated vertex.

### Part (b) - Maximum Number of Edges in a Graph with 6 Vertices and 2 Components

To maximize the number of edges, we need to consider that the graph still has two components. A component is defined as a maximal connected subgraph. To maximize the number of edges:

- Each component can be a **complete graph**. The maximum number of edges in a complete graph \(K_n\) with \(n\) vertices is given by the formula:

\[
\text{Max edges} = \frac{n(n-1)}{2}
\]

To maximize the number of edges, we split the vertices as evenly as possible between the two components.

- One possible configuration is to have:
  - Component 1 with 3 vertices (maximum number of edges = \( \frac{3(3-1)}{2} = 3 \))
  - Component 2 with 3 vertices (maximum number of edges = \( \frac{3(3-1)}{2} = 3 \))

Thus, the **maximum number of edges** is \(3 + 3 = 6\).

### Simple Graph Example for Part (b):
- Component 1: A complete graph \(K_3\) with 3 vertices.
- Component 2: A complete graph \(K_3\) with 3 vertices.

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Let me know if you'd like me to draw the graphs for both parts or if you have any questions.

### Relative Questions:
1. What are the key differences between connected and disconnected graphs?
2. How does the number of edges in a tree compare to a complete graph?
3. What is the significance of the number of components in a graph?
4. How would the problem change if we required the graph to be connected?
5. How do you calculate the maximum number of edges in a complete graph?

### Tip:
When calculating the minimum number of edges in a disconnected graph, consider if some vertices can be isolated to reduce edge count.

Learn how to calculate the minimum and maximum number of edges in a simple graph with 6 vertices and two components. This problem uses concepts from graph theory, including trees, connected components, and complete graphs. Suitable for students in Grades 10-12.

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