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Prove that the two graphs below are isomorphic.

Prove that the two graphs below are isomorphic.

Advanced Engineering Mathematics
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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**Problem Statement:** (c) Prove that the two graphs below are isomorphic. **Graph Representation:** 1. **Left Graph:** - **Vertices:** 1, 2, 3, 4, 5, 6 - **Edges:** The graph includes the following connections: - 1 is connected to 2, 3, 5, and 6. - 2 is connected to 3, 5, and 6. - 3 is connected to 4. - 4 is connected to 5 and 6. - 5 is connected to 6. 2. **Right Graph:** - **Vertices:** a, b, c, d, e, f - **Edges:** The graph includes the following connections: - a is connected to b, e, and f. - b is connected to c, e, and f. - c is connected to d. - d is connected to e and f. - e is connected to f. **Explanation of Isomorphism:** To prove that the two graphs are isomorphic, one needs to find a one-to-one correspondence (bijection) between the vertex sets of the two graphs such that the adjacency is preserved. This means that if two vertices are connected by an edge in the first graph, their corresponding vertices in the second graph should also be connected by an edge, and vice versa. By comparing the connections, we see that: - Vertex 1 in the left graph corresponds to vertex a in the right graph. - Vertex 2 corresponds to vertex b. - Vertex 3 corresponds to vertex c. - Vertex 4 corresponds to vertex d. - Vertex 5 corresponds to vertex e. - Vertex 6 corresponds to vertex f. This mapping preserves the adjacency relations, thus proving that the two graphs are isomorphic.
Transcribed Image Text:**Problem Statement:** (c) Prove that the two graphs below are isomorphic. **Graph Representation:** 1. **Left Graph:** - **Vertices:** 1, 2, 3, 4, 5, 6 - **Edges:** The graph includes the following connections: - 1 is connected to 2, 3, 5, and 6. - 2 is connected to 3, 5, and 6. - 3 is connected to 4. - 4 is connected to 5 and 6. - 5 is connected to 6. 2. **Right Graph:** - **Vertices:** a, b, c, d, e, f - **Edges:** The graph includes the following connections: - a is connected to b, e, and f. - b is connected to c, e, and f. - c is connected to d. - d is connected to e and f. - e is connected to f. **Explanation of Isomorphism:** To prove that the two graphs are isomorphic, one needs to find a one-to-one correspondence (bijection) between the vertex sets of the two graphs such that the adjacency is preserved. This means that if two vertices are connected by an edge in the first graph, their corresponding vertices in the second graph should also be connected by an edge, and vice versa. By comparing the connections, we see that: - Vertex 1 in the left graph corresponds to vertex a in the right graph. - Vertex 2 corresponds to vertex b. - Vertex 3 corresponds to vertex c. - Vertex 4 corresponds to vertex d. - Vertex 5 corresponds to vertex e. - Vertex 6 corresponds to vertex f. This mapping preserves the adjacency relations, thus proving that the two graphs are isomorphic.
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