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Show that G1 and G2 are not isomorphic. G₁ G2

Show that G1 and G2 are not isomorphic. G₁ G2

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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**Title: Understanding Non-Isomorphic Graphs** **Task: Show that \( G_1 \) and \( G_2 \) are not isomorphic.** **Graph Explanation:** - **Graph \( G_1 \):** - Structure: This graph is a complete bipartite graph with a pair of edges crossing each other, forming an "X" shape within a quadrilateral. - Vertices: 5 - Edges: 6 - Notable Characteristics: The graph has a central vertex connected to four vertices, effectively dividing the graph into multiple triangular sections. - **Graph \( G_2 \):** - Structure: This is a pentagon-shaped graph. - Vertices: 5 - Edges: 5 - Notable Characteristics: The graph forms a simple cycle with no crossed edges and a consistent path formed by connecting each vertex to the next in a circular manner. **Key Differences:** 1. **Number of Edges:** - \( G_1 \) has 6 edges, whereas \( G_2 \) has 5 edges. 2. **Graph Structure:** - \( G_1 \) includes intersecting edges, creating more complex connectivity within its structure. - \( G_2 \) is a simple cycle graph with a clean pentagonal shape and no intersections. **Conclusion:** These differences in edge connections and structures demonstrate that \( G_1 \) and \( G_2 \) are not isomorphic. An isomorphic graph must have the same number of vertices connected in the same way, which is not the case here. This exercise illustrates the principles of graph isomorphism by contrasting complete bipartite and cycle graphs.
Transcribed Image Text:**Title: Understanding Non-Isomorphic Graphs** **Task: Show that \( G_1 \) and \( G_2 \) are not isomorphic.** **Graph Explanation:** - **Graph \( G_1 \):** - Structure: This graph is a complete bipartite graph with a pair of edges crossing each other, forming an "X" shape within a quadrilateral. - Vertices: 5 - Edges: 6 - Notable Characteristics: The graph has a central vertex connected to four vertices, effectively dividing the graph into multiple triangular sections. - **Graph \( G_2 \):** - Structure: This is a pentagon-shaped graph. - Vertices: 5 - Edges: 5 - Notable Characteristics: The graph forms a simple cycle with no crossed edges and a consistent path formed by connecting each vertex to the next in a circular manner. **Key Differences:** 1. **Number of Edges:** - \( G_1 \) has 6 edges, whereas \( G_2 \) has 5 edges. 2. **Graph Structure:** - \( G_1 \) includes intersecting edges, creating more complex connectivity within its structure. - \( G_2 \) is a simple cycle graph with a clean pentagonal shape and no intersections. **Conclusion:** These differences in edge connections and structures demonstrate that \( G_1 \) and \( G_2 \) are not isomorphic. An isomorphic graph must have the same number of vertices connected in the same way, which is not the case here. This exercise illustrates the principles of graph isomorphism by contrasting complete bipartite and cycle graphs.
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