https://file.aiquickdraw.com/mathbot/file/17308838426968dop1w30.jpg

Let’s go through each part of the question based on the network diagram provided.

### Part (a): State the number of nodes and edges.
- **Nodes**: The network contains 7 nodes, labeled as \( A, B, C, D, E, F, G \).
- **Edges**: Counting the lines connecting each pair of nodes, we find there are 9 edges in total.

### Part (b): Name a path starting at \( A \) and ending at \( F \).
- One possible path from \( A \) to \( F \) is:
  \( A \to D \to G \to F \).

### Part (c): Name a circuit which contains 4 edges.
- A circuit (a closed loop where you return to the starting point without retracing any edge) containing 4 edges is:
  \( A \to B \to E \to D \to A \).

### Part (d): Draw a spanning tree for the network.
- A **spanning tree** includes all nodes with the minimum number of edges required to keep the graph connected (no cycles). A possible spanning tree could include the edges:
  - \( A \to B \)
  - \( A \to C \)
  - \( A \to D \)
  - \( B \to E \)
  - \( D \to G \)
  - \( F \to G \)

This spanning tree has 6 edges and includes all nodes, avoiding any cycles.

Would you like me to provide further details on any part of this solution, or do you have any additional questions?

### Follow-up Questions
1. What is the difference between a path and a circuit in graph theory?
2. How many different spanning trees can you draw for this network?
3. Can you explain why the spanning tree doesn’t have cycles?
4. What would be an example of a path from \( C \) to \( E \)?
5. How would removing one edge affect the connectivity of a spanning tree?

### Tip
In a network, a **spanning tree** is a subgraph that connects all the nodes together without any cycles and uses the minimum number of edges, making it a fundamental concept in optimizing connected graphs.

Consider the network alongside.

a. State the number of nodes and edges.
b. Name a path starting at A and ending at F.
c. Name a circuit which contains 4 edges.
d. Draw a spanning tree for the network.

Explore graph theory concepts with this problem involving a network of nodes and edges. Learn to identify paths, circuits, and spanning trees, while practicing essential graph analysis skills. Suitable for Grades 9-12.

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