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.