Let’s go through each part of the question:

(a) State the number of nodes and edges.

From the image, we observe the following nodes and edges:

• Nodes: A, B, C, D, E, F, G (7 nodes in total)
• Edges: The lines connecting nodes represent edges. Counting these, we have 9 edges.

So, the answer is:

• Number of nodes: 7
• Number of edges: 9

(b) Name a path starting at A and ending at F.

A path from A to F could be as follows:

• Path: $A \to D \to G \to F$

(c) Name a circuit which contains 4 edges.

A circuit is a closed path that starts and ends at the same node without repeating any edges. One such circuit with 4 edges is:

• Circuit: $A \to B \to E \to D \to A$

(d) Draw a spanning tree for the network.

A spanning tree is a subgraph that includes all the nodes and enough edges to connect them without any cycles. One possible spanning tree includes these edges:

• $A \to B$
• $A \to C$
• $A \to D$
• $D \to G$
• $G \to F$
• $D \to E$

This choice of edges connects all nodes without forming any cycles.

Would you like me to further explain any of these answers, or do you have additional questions?

Here are five related questions to deepen understanding:

1. What distinguishes a path from a circuit in graph theory?
2. How many spanning trees can a graph have?
3. Why is it important that a spanning tree has no cycles?
4. Can a graph have more than one possible path between two nodes?
5. What is the difference between an edge and a vertex in graph theory?

Tip: In a spanning tree, the number of edges is always the number of nodes minus one (i.e., $n - 1$).