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For the following graph, explain, giving reason for your answer, why an Euler circuit exists. Indicate the circuit. a g b h

For the following graph, explain, giving reason for your answer, why an Euler circuit exists. Indicate the circuit. a g b h

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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**Graph Analysis and Euler Circuit Explanation** For the following graph, explain, giving reason for your answer, why an Euler circuit exists. Indicate the circuit. **Graph Details:** The graph consists of 10 vertices labeled from a to j. The connections (edges) between these vertices are shown in a planar representation, with some vertices having multiple connections. To determine if an Euler circuit exists, we must check if the graph is connected and if all vertices have an even degree (the number of edges connected to each vertex). **Vertices and Their Degrees:** - Vertex a: connected to d, g, e, b (degree 4) - Vertex b: connected to a, e, c, h (degree 4) - Vertex c: connected to b, e, f, i (degree 4) - Vertex d: connected to a, f, g (degree 3) - Vertex e: connected to a, b, c, h (degree 4) - Vertex f: connected to c, d, i, j (degree 4) - Vertex g: connected to a, d, h, i (degree 4) - Vertex h: connected to b, e, g, i (degree 4) - Vertex i: connected to c, f, g, h, j (degree 5) - Vertex j: connected to f, i (degree 2) **Reasoning:** For an Euler circuit to exist, all vertices must have even degrees. The above inspection shows that vertex d has an odd degree (3) and vertex i also has an odd degree (5), indicating that an Euler circuit does not exist in this graph. Thus, not all conditions for an Euler circuit are met in the given graph.
Transcribed Image Text:**Graph Analysis and Euler Circuit Explanation** For the following graph, explain, giving reason for your answer, why an Euler circuit exists. Indicate the circuit. **Graph Details:** The graph consists of 10 vertices labeled from a to j. The connections (edges) between these vertices are shown in a planar representation, with some vertices having multiple connections. To determine if an Euler circuit exists, we must check if the graph is connected and if all vertices have an even degree (the number of edges connected to each vertex). **Vertices and Their Degrees:** - Vertex a: connected to d, g, e, b (degree 4) - Vertex b: connected to a, e, c, h (degree 4) - Vertex c: connected to b, e, f, i (degree 4) - Vertex d: connected to a, f, g (degree 3) - Vertex e: connected to a, b, c, h (degree 4) - Vertex f: connected to c, d, i, j (degree 4) - Vertex g: connected to a, d, h, i (degree 4) - Vertex h: connected to b, e, g, i (degree 4) - Vertex i: connected to c, f, g, h, j (degree 5) - Vertex j: connected to f, i (degree 2) **Reasoning:** For an Euler circuit to exist, all vertices must have even degrees. The above inspection shows that vertex d has an odd degree (3) and vertex i also has an odd degree (5), indicating that an Euler circuit does not exist in this graph. Thus, not all conditions for an Euler circuit are met in the given graph.
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