https://file.aiquickdraw.com/mathbot/file/1729043318767dtnx7p0o.jpg

The question asks to classify the sequence \( (E, Q, H, V, E) \) in the provided graph by checking all applicable terms: Walk, Trail, Path, Circuit, and Cycle.

Let’s break down each concept:

1. **Walk**: A sequence of vertices where each consecutive pair is connected by an edge. The sequence \( (E, Q, H, V, E) \) is a walk if all edges exist in the graph. From the image, it appears that there are edges between each consecutive pair, so it qualifies as a walk.

2. **Trail**: A walk in which no edge is repeated. In the sequence \( (E, Q, H, V, E) \), the edges between vertices are not repeated, so it qualifies as a trail.

3. **Path**: A trail where no vertices (except possibly the first and last) are repeated. In this sequence, the vertex \( E \) appears at the beginning and end, so it does **not** qualify as a path.

4. **Circuit**: A closed trail, meaning it starts and ends at the same vertex, and no edge is repeated. Since the sequence starts and ends at \( E \) and does not repeat any edge, it qualifies as a circuit.

5. **Cycle**: A circuit in which no vertices (except the first and last) are repeated. Since the vertices \( Q, H, V \) are all distinct, and the sequence starts and ends at \( E \), it qualifies as a cycle.

### Conclusion:
- The sequence \( (E, Q, H, V, E) \) is a **Walk**, **Trail**, **Circuit**, and **Cycle**, but **not a Path**.

Would you like further details or clarification on any of these concepts? Here are 5 related questions to expand on this:

1. How do you determine if a sequence in a graph is a trail versus a path?
2. What is the difference between an Eulerian path and a Hamiltonian cycle in graph theory?
3. Can a walk contain repeated vertices and still be a cycle? Why or why not?
4. What are the key characteristics of a graph that can contain circuits and cycles?
5. How would the classification change if the sequence was \( (E, Q, H, V) \) without returning to \( E \)?

**Tip**: When analyzing sequences in a graph, always check both vertex and edge repetition to classify paths, trails, and circuits correctly.

Select all terms that apply to the sequence (E, Q, H, V, E) in the graph.

Learn how to classify sequences in graph theory. This problem focuses on determining if a sequence (E, Q, H, V, E) is a Walk, Trail, Path, Circuit, or Cycle based on their definitions. Suitable for college-level students.

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