Let's examine each statement in terms of the definitions of walks, trails, paths, circuits, and closed walks in graph theory.

Definitions:

• Walk: A sequence of vertices where each consecutive pair is connected by an edge. Repetition of vertices and edges is allowed.
• Trail: A walk where no edge is repeated (vertices can be repeated).
• Path: A walk with no repeated vertices (and, consequently, no repeated edges).
• Circuit: A closed trail (i.e., a trail that starts and ends at the same vertex).
• Closed Walk: A walk that starts and ends at the same vertex.

Analyzing Each Statement:

1. $C, G, F, B, E$ is a walk from $C$ to $E$:
   This is true. It is a sequence of vertices with each consecutive pair connected by an edge, and it starts at $C$ and ends at $E$.

2. $A, C, E, B, F, E, C, G$ is a trail from $A$ to $G$:
   This is false. Although it is a walk from $A$ to $G$, it repeats the edge $E$-$C$, which disqualifies it as a trail.

3. $A, C, E, A$ is a closed walk:
   This is true. It starts and ends at $A$, forming a closed sequence.

4. $A, D, C, G, F, B, E, C, A$ is a circuit:
   This is false. Although it starts and ends at $A$, it repeats the edge $C$-$A$, which disqualifies it as a circuit.

5. $A, C, A, C, A, C, A, C, E$ is a walk from $A$ to $E$:
   This is true. It is a valid sequence from $A$ to $E$, although it repeats vertices and edges.

6. $A, C, F, G, D$ is a path from $A$ to $G$:
   This is false. Although it is a walk, it does not end at $G$, so it’s not a path from $A$ to $G$.

7. $A, C, F, G, D$ is a trail from $A$ to $G$:
   This is false. Same reasoning as above; it doesn't end at $G$, so it’s not a trail from $A$ to $G$.

8. $D, C, F, G, C$ is a closed walk:
   This is true. It starts and ends at $C$, forming a closed walk.

Summary of Answers:

1. True
2. False
3. True
4. False
5. True
6. False
7. False
8. True

Would you like further details on any specific explanation or have any questions?

Related Questions:

1. How does a circuit differ from a closed walk?
2. Can a path also be a trail? Under what conditions?
3. What makes a walk a trail in graph theory?
4. How do you determine if a sequence of vertices forms a path?
5. What is the significance of edge repetition in trails and paths?

Tip:

Remember, a trail allows repeated vertices but not repeated edges, while a path allows neither repeated vertices nor repeated edges.