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Determine whether the following pair of graphs are isomorphic, if they are give the isomorphism, if they are not, state why not. V₁ V3 V5 (a) (b) G: V2 G1 SO V4 H: u2 V5 V4 V1 G2 V3 u4 V2

Determine whether the following pair of graphs are isomorphic, if they are give the isomorphism, if they are not, state why not. V₁ V3 V5 (a) (b) G: V2 G1 SO V4 H: u2 V5 V4 V1 G2 V3 u4 V2

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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### Educational Content on Graph Isomorphism **Problem Statement:** Determine whether the following pair of graphs are isomorphic. If they are, provide the isomorphism. If they are not, explain why they are not. --- #### Graph Pair (a) - **Graph \( G_1 \):** - Vertices: \( V_1, V_2, V_3, V_4, V_5 \) - Edges: \( (V_1, V_2), (V_1, V_4), (V_3, V_2), (V_3, V_5), (V_5, V_4) \) - Structure: This graph forms an hourglass shape where the vertices are strategically interconnected. - **Graph \( G_2 \):** - Vertices: \( V_1, V_2, V_3, V_4, V_5 \) - Edges: \( (V_1, V_2), (V_2, V_3), (V_3, V_4), (V_4, V_5), (V_5, V_1) \) - Structure: This graph is a simple pentagon shape, with vertices connected in a loop. **Analysis:** Graph \( G_1 \) and \( G_2 \) have the same number of vertices and edges. However, \( G_1 \) has intersecting edges forming two triangles, whereas \( G_2 \) is a simple cycle. Hence, they are not isomorphic since their structures do not match despite having the same vertex and edge count. --- #### Graph Pair (b) - **Graph \( G \):** - Vertices: \( V_1, V_2, V_3, V_4, V_5 \) - Shape: Resembles a pyramid or star, with \( V_1 \) connected to every other vertex. - **Graph \( H \):** - Vertices: \( U_1, U_2, U_3, U_4, U_5 \) - Shape: It appears as a typical star graph with intersecting edges, with \( U_1 \) as the central vertex like an hourglass shape. **Analysis:** Graphs \( G \) and \( H \) have
Transcribed Image Text:### Educational Content on Graph Isomorphism **Problem Statement:** Determine whether the following pair of graphs are isomorphic. If they are, provide the isomorphism. If they are not, explain why they are not. --- #### Graph Pair (a) - **Graph \( G_1 \):** - Vertices: \( V_1, V_2, V_3, V_4, V_5 \) - Edges: \( (V_1, V_2), (V_1, V_4), (V_3, V_2), (V_3, V_5), (V_5, V_4) \) - Structure: This graph forms an hourglass shape where the vertices are strategically interconnected. - **Graph \( G_2 \):** - Vertices: \( V_1, V_2, V_3, V_4, V_5 \) - Edges: \( (V_1, V_2), (V_2, V_3), (V_3, V_4), (V_4, V_5), (V_5, V_1) \) - Structure: This graph is a simple pentagon shape, with vertices connected in a loop. **Analysis:** Graph \( G_1 \) and \( G_2 \) have the same number of vertices and edges. However, \( G_1 \) has intersecting edges forming two triangles, whereas \( G_2 \) is a simple cycle. Hence, they are not isomorphic since their structures do not match despite having the same vertex and edge count. --- #### Graph Pair (b) - **Graph \( G \):** - Vertices: \( V_1, V_2, V_3, V_4, V_5 \) - Shape: Resembles a pyramid or star, with \( V_1 \) connected to every other vertex. - **Graph \( H \):** - Vertices: \( U_1, U_2, U_3, U_4, U_5 \) - Shape: It appears as a typical star graph with intersecting edges, with \( U_1 \) as the central vertex like an hourglass shape. **Analysis:** Graphs \( G \) and \( H \) have
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