Show that it is not possible to creak graph the" degree with g vertices such that every a of vertex is 3.
Show that it is not possible to creak graph the" degree with g vertices such that every a of vertex is 3.
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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Question
![**Problem 3**
Show that it is not possible to create a graph with 9 vertices such that the degree of every vertex is 3.
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*Explanation for Educational Context:*
In this problem, we are asked to demonstrate that it is impossible to construct a graph with 9 vertices where each vertex has a degree of 3. A graph is a set of vertices (or nodes) connected by edges. The degree of a vertex refers to the number of edges connected to it.
*Key concept:*
- **Degree Sum Theorem**: This theorem states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges (because each edge contributes to the degree of two vertices).
Using this principle, if each of the 9 vertices has a degree of 3, the total sum of the degrees would be:
\[
9 \times 3 = 27
\]
However, since the degree sum must be even (as it is twice the number of edges), 27 being an odd number indicates that it's not possible to construct such a graph.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9de804fd-1a97-452a-9cf6-b7d2ceae8a21%2Fbed62970-33b8-43cd-b811-dbf970617bef%2F8cl8keh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3**
Show that it is not possible to create a graph with 9 vertices such that the degree of every vertex is 3.
---
*Explanation for Educational Context:*
In this problem, we are asked to demonstrate that it is impossible to construct a graph with 9 vertices where each vertex has a degree of 3. A graph is a set of vertices (or nodes) connected by edges. The degree of a vertex refers to the number of edges connected to it.
*Key concept:*
- **Degree Sum Theorem**: This theorem states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges (because each edge contributes to the degree of two vertices).
Using this principle, if each of the 9 vertices has a degree of 3, the total sum of the degrees would be:
\[
9 \times 3 = 27
\]
However, since the degree sum must be even (as it is twice the number of edges), 27 being an odd number indicates that it's not possible to construct such a graph.
Expert Solution
Step 1
Given that
The no of vertices = 9
degree of each vertices =3
To show such graph is not possible.
Step by step
Solved in 2 steps
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