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(c) Prove that the two graphs below are isomorphic.

(c) 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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### (c) Prove that the two graphs below are isomorphic. #### Figure 4 Explanation: - **Description**: Two undirected graphs, each with 6 vertices. - **Arrangement and Vertices**: - **First Graph**: - Vertices are arranged in two rows and three columns. - Top row, from left to right: Vertices 1, 2, 3. - Bottom row, from left to right: Vertices 6, 5, 4. - Edges connect the following vertex pairs: (1,2), (2,3), (1,5), (2,5), (5,3), (2,4), (3,6), (6,4), (5,4). - **Second Graph**: - Vertices labeled a through f. - Vertices d, a, c (vertically inline). - Vertices e, f, b (horizontally to the right of vertices d, a, c, respectively). - Edges connect the following vertex pairs: (a,d), (a,c), (c,e), (e,a), (b,d), (b,a), (f,e), (c,f), (b,f). ### (d) Show that the pair of graphs are not isomorphic by showing that there is a property that is preserved under isomorphism which one graph has and the other does not. #### Figure 5 Explanation: - **Description**: Two undirected graphs. - **First Graph**: - Pentagonal arrangement with 5 vertices. - Vertices labeled 1, 2, 3, 4, 5 (top vertex to bottom, clockwise). - Edges connect: (1,2), (2,3), (3,4), (4,5), (5,1). - **Second Graph**: - 4 vertices labeled a, b, c, d. - Vertices d, c (horizontally inline, d is left of c). - Vertex a above d and c. - Vertex b right of a but above other vertices. - Edges connect: (a,b), (b,c), (a,d), (c,d), (d,b).
Transcribed Image Text:### (c) Prove that the two graphs below are isomorphic. #### Figure 4 Explanation: - **Description**: Two undirected graphs, each with 6 vertices. - **Arrangement and Vertices**: - **First Graph**: - Vertices are arranged in two rows and three columns. - Top row, from left to right: Vertices 1, 2, 3. - Bottom row, from left to right: Vertices 6, 5, 4. - Edges connect the following vertex pairs: (1,2), (2,3), (1,5), (2,5), (5,3), (2,4), (3,6), (6,4), (5,4). - **Second Graph**: - Vertices labeled a through f. - Vertices d, a, c (vertically inline). - Vertices e, f, b (horizontally to the right of vertices d, a, c, respectively). - Edges connect the following vertex pairs: (a,d), (a,c), (c,e), (e,a), (b,d), (b,a), (f,e), (c,f), (b,f). ### (d) Show that the pair of graphs are not isomorphic by showing that there is a property that is preserved under isomorphism which one graph has and the other does not. #### Figure 5 Explanation: - **Description**: Two undirected graphs. - **First Graph**: - Pentagonal arrangement with 5 vertices. - Vertices labeled 1, 2, 3, 4, 5 (top vertex to bottom, clockwise). - Edges connect: (1,2), (2,3), (3,4), (4,5), (5,1). - **Second Graph**: - 4 vertices labeled a, b, c, d. - Vertices d, c (horizontally inline, d is left of c). - Vertex a above d and c. - Vertex b right of a but above other vertices. - Edges connect: (a,b), (b,c), (a,d), (c,d), (d,b).
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