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.