17) In a complete graph with 720 distinct Hamilton circuits, there is a total of B) 9 vertices. A) 6 vertices. C) 7 vertices. D) 5 vertices.
17) In a complete graph with 720 distinct Hamilton circuits, there is a total of B) 9 vertices. A) 6 vertices. C) 7 vertices. D) 5 vertices.
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:**
In a complete graph with 720 distinct Hamilton circuits, there is a total of
A) 6 vertices.
B) 9 vertices.
C) 7 vertices.
D) 5 vertices.
**Explanation:**
A Hamilton circuit in a complete graph involves visiting every vertex exactly once and returning to the starting point. The number of Hamilton circuits in a complete graph with \( n \) vertices is given by the formula:
\[
\frac{(n-1)!}{2}
\]
Here, we know there are 720 distinct Hamilton circuits. Thus, we set up the equation:
\[
\frac{(n-1)!}{2} = 720
\]
Solving this would give us the number of vertices. Note that as a complete graph grows in the number of vertices, the number of possible Hamilton circuits grows factorially.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F284f9aca-91c7-4c35-8043-db359fa93315%2F80188327-210e-43b5-94f3-7820648865e2%2Fwfe4sif_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
In a complete graph with 720 distinct Hamilton circuits, there is a total of
A) 6 vertices.
B) 9 vertices.
C) 7 vertices.
D) 5 vertices.
**Explanation:**
A Hamilton circuit in a complete graph involves visiting every vertex exactly once and returning to the starting point. The number of Hamilton circuits in a complete graph with \( n \) vertices is given by the formula:
\[
\frac{(n-1)!}{2}
\]
Here, we know there are 720 distinct Hamilton circuits. Thus, we set up the equation:
\[
\frac{(n-1)!}{2} = 720
\]
Solving this would give us the number of vertices. Note that as a complete graph grows in the number of vertices, the number of possible Hamilton circuits grows factorially.
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