**Graph Theory Exercise****9. Consider the following graph.***Instructions:*- Label all the vertices and edges.- Determine whether the graph has a Hamiltonian path. - Determine whether the graph has a Hamiltonian circuit. Justify your answers.*Graph Description:*The graph depicted consists of five vertices forming an irregular pentagon. The vertices are connected in a combination of triangular and quadrilateral fashion, resulting in multiple intersecting lines.**Tasks:**1. **Labeling:** - Assign a label to each vertex, such as V1, V2, V3, V4, and V5. - Label each edge connecting the vertices.2. **Hamiltonian Path:** - A Hamiltonian path visits every vertex exactly once. Analyze the graph to see if such a path exists, and list the sequence if it does.3. **Hamiltonian Circuit:** - A Hamiltonian circuit visits every vertex exactly once and returns to the starting point. Determine if this is possible with the given graph structure. Please analyze the graph structure and respond with your findings regarding Hamiltonian paths and circuits.

Consider the given graph and label it.

Consider the following graph. Label all the vertices and edges. Determine whether the graph has a Hamilton path. Determine whether the graph has a Hamilton circuit. Justify your answers. Hamiltonian path. Then

Consider the following graph. Label all the vertices and edges. Determine whether the graph has a Hamilton path. Determine whether the graph has a Hamilton circuit. Justify your answers. Hamiltonian path. Then

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Graph Theory Exercise**

**9. Consider the following graph.**

*Instructions:*

- Label all the vertices and edges.
- Determine whether the graph has a Hamiltonian path.
- Determine whether the graph has a Hamiltonian circuit. Justify your answers.

*Graph Description:*

The graph depicted consists of five vertices forming an irregular pentagon. The vertices are connected in a combination of triangular and quadrilateral fashion, resulting in multiple intersecting lines.

**Tasks:**

1. **Labeling:**
- Assign a label to each vertex, such as V1, V2, V3, V4, and V5.
- Label each edge connecting the vertices.

2. **Hamiltonian Path:**
- A Hamiltonian path visits every vertex exactly once. Analyze the graph to see if such a path exists, and list the sequence if it does.

3. **Hamiltonian Circuit:**
- A Hamiltonian circuit visits every vertex exactly once and returns to the starting point. Determine if this is possible with the given graph structure.

Please analyze the graph structure and respond with your findings regarding Hamiltonian paths and circuits.

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