The graph G₁ has 6 vertices, all of degree 4. How many edges does G₁ have? The graph G₂ has 4 vertices, all of degree k. Also, G₂ has 6 edges. What is k? E The graph G3 has v vertices, all of degree 4. Also, G3 has 18 edges. What is v?
The graph G₁ has 6 vertices, all of degree 4. How many edges does G₁ have? The graph G₂ has 4 vertices, all of degree k. Also, G₂ has 6 edges. What is k? E The graph G3 has v vertices, all of degree 4. Also, G3 has 18 edges. What is v?
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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![**Transcription for Educational Website**
1. **The graph \( G_1 \) has 6 vertices, all of degree 4. How many edges does \( G_1 \) have?**
[Input Box]
2. **The graph \( G_2 \) has 4 vertices, all of degree \( k \). Also, \( G_2 \) has 6 edges. What is \( k \)?**
[Input Box]
3. **The graph \( G_3 \) has \( v \) vertices, all of degree 4. Also, \( G_3 \) has 18 edges. What is \( v \)?**
[Input Box]
**Explanation for Graph and Edge Concepts:**
- In these problems, we are dealing with concepts of graph theory, particularly concerning vertices, edges, and degrees of a graph.
- The degree of a vertex in a graph is the number of edges connected to it.
- The sum of the degrees of all vertices in a graph is twice the number of edges, according to the Handshaking Lemma. This concept is beneficial for solving problems related to the structure of graphs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0db8c0f3-7e1f-4cf8-bdbd-38ddd8711042%2Fccd0fdd4-7b17-41ae-8b3a-c9e00b75b864%2Febto16v_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
1. **The graph \( G_1 \) has 6 vertices, all of degree 4. How many edges does \( G_1 \) have?**
[Input Box]
2. **The graph \( G_2 \) has 4 vertices, all of degree \( k \). Also, \( G_2 \) has 6 edges. What is \( k \)?**
[Input Box]
3. **The graph \( G_3 \) has \( v \) vertices, all of degree 4. Also, \( G_3 \) has 18 edges. What is \( v \)?**
[Input Box]
**Explanation for Graph and Edge Concepts:**
- In these problems, we are dealing with concepts of graph theory, particularly concerning vertices, edges, and degrees of a graph.
- The degree of a vertex in a graph is the number of edges connected to it.
- The sum of the degrees of all vertices in a graph is twice the number of edges, according to the Handshaking Lemma. This concept is beneficial for solving problems related to the structure of graphs.
Transcribed Image Text:**Question:**
Which of the following degree sequences are possible for a graph?
**Options:**
- A. (5, 5, 5, 5, 4, 4)
- B. (5, 3, 2, 2, 2, 1)
- C. (3, 3, 3, 3, 2, 1)
- D. (9, 8, 6, 6, 6, 6, 4, 4, 3)
**Explanation:**
This question focuses on determining whether each degree sequence presented is graphical, meaning whether a simple graph can be constructed with these vertex degrees. Each sequence must satisfy the Handshaking Lemma, which states that the sum of the degrees must be even, among other conditions.
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