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To determine which option is *not* a Hamilton circuit, let's first define a Hamilton circuit:

A Hamilton circuit is a path that visits every vertex of a graph exactly once and returns to the starting vertex.

Given this graph with vertices \( A, B, C, E \), each Hamilton circuit should:
1. Start at \( A \),
2. Visit each of the other vertices \( B \), \( C \), and \( E \) exactly once,
3. Return to \( A \) at the end.

Let's analyze each option:

1. **\( A, E, B, C, A \)**: This path starts at \( A \), visits all vertices once, and returns to \( A \). This is a Hamilton circuit.
2. **\( A, C, B, E, A \)**: This path also starts at \( A \), visits each vertex exactly once, and returns to \( A \). This is a Hamilton circuit.
3. **\( A, B, C, E, A \)**: This path meets the criteria of starting and ending at \( A \), and visiting each vertex exactly once. It’s a Hamilton circuit.
4. **\( A, B, C, E, B, A \)**: This path visits vertex \( B \) twice, meaning it does not visit each vertex exactly once. Therefore, this is *not* a Hamilton circuit.

### Answer:
The correct answer is **\( A, B, C, E, B, A \)**, as it is *not* a Hamilton circuit.

Would you like more details or have any questions?

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### Related Questions:
1. What is the difference between a Hamilton path and a Hamilton circuit?
2. Can a Hamilton circuit exist in any type of graph?
3. How does a Hamilton circuit differ from an Euler circuit?
4. Is there a way to algorithmically find Hamilton circuits in larger graphs?
5. What are some real-world applications of Hamiltonian circuits?

### Tip:
When checking for Hamilton circuits, ensure each vertex is visited exactly once before returning to the start, without repeating any vertices.

What is not a Hamilton circuit when starting and ending at vertex A?

Determine which path is not a Hamilton circuit in a given graph with vertices A, B, C, and E. A Hamilton circuit must visit each vertex exactly once before returning to the starting point. Suitable for students in Grades 9-12.

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