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Let G be a complete undirected graph of 6 vertices. If vertices of G are labelled, then the number of distinct cycles of length 4 in G is equal to (Give your answer up to 2 digit whole numbers)
Correct answer is '45'. Can you explain this answer?
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Let G be a complete undirected graph of 6 vertices. If vertices of G a...
Explanation:
Complete Undirected Graph:
- A complete undirected graph of 6 vertices has all possible edges between the vertices.
- In this case, there are a total of 6 vertices, and each vertex is connected to every other vertex.
Distinct Cycles of Length 4:
- To find the number of distinct cycles of length 4 in a complete undirected graph of 6 vertices, we can use the formula:
- Number of distinct cycles of length 4 = nC4 * (4-1)! / 2
- Where n is the total number of vertices in the graph (in this case, n=6)
Calculation:
- Substituting n=6 in the formula:
- Number of distinct cycles of length 4 = 6C4 * 3! / 2
- 6C4 = 6! / (4! * (6-4)!) = 15
- 3! = 6
- Number of distinct cycles of length 4 = 15 * 6 / 2 = 45
Therefore, the number of distinct cycles of length 4 in a complete undirected graph of 6 vertices is 45.
Complete Undirected Graph:
- A complete undirected graph of 6 vertices has all possible edges between the vertices.
- In this case, there are a total of 6 vertices, and each vertex is connected to every other vertex.
Distinct Cycles of Length 4:
- To find the number of distinct cycles of length 4 in a complete undirected graph of 6 vertices, we can use the formula:
- Number of distinct cycles of length 4 = nC4 * (4-1)! / 2
- Where n is the total number of vertices in the graph (in this case, n=6)
Calculation:
- Substituting n=6 in the formula:
- Number of distinct cycles of length 4 = 6C4 * 3! / 2
- 6C4 = 6! / (4! * (6-4)!) = 15
- 3! = 6
- Number of distinct cycles of length 4 = 15 * 6 / 2 = 45
Therefore, the number of distinct cycles of length 4 in a complete undirected graph of 6 vertices is 45.
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Community Answer
Let G be a complete undirected graph of 6 vertices. If vertices of G a...
There can be total 6C4 ways to pick 4 vertices from 6. The value of 6C4 is 15.
Note that the given graph is complete, so any 4 vertices can form a cycle.
There can be 6 different cycle with 4 vertices. For example, consider 4 vertices as a, b, c and d.
Cycles should be like this:
(a, b, c, d, a)
(a, b, d, c, a)
(a, c, b, d, a)
(a, c, d, b, a)
(a, d, b, c, a)
(a, d, c, b, a)
and
(a, b, c, d, a) and (a, d, c, b, a)
(a, b, d, c, a) and (a, c, d, b, a)
(a, c, b, d, a) and (a, d, b, c, a)
are same cycles.
So, total number of distinct cycles are (15 × 3) = 45.
Note that the given graph is complete, so any 4 vertices can form a cycle.
There can be 6 different cycle with 4 vertices. For example, consider 4 vertices as a, b, c and d.
Cycles should be like this:
(a, b, c, d, a)
(a, b, d, c, a)
(a, c, b, d, a)
(a, c, d, b, a)
(a, d, b, c, a)
(a, d, c, b, a)
and
(a, b, c, d, a) and (a, d, c, b, a)
(a, b, d, c, a) and (a, c, d, b, a)
(a, c, b, d, a) and (a, d, b, c, a)
are same cycles.
So, total number of distinct cycles are (15 × 3) = 45.
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Let G be a complete undirected graph of 6 vertices. If vertices of G are labelled, then the number of distinct cycles of length 4 in G is equal to (Give your answer up to 2 digit whole numbers)Correct answer is '45'. Can you explain this answer?
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