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Let G be a complete undirected graph of six vertices. If the vertices of G are labelled, then the number of distinct cycles of length 4 in G is
Correct answer is '45'. Can you explain this answer?
Most Upvoted Answer
Let G be a complete undirected graph of six vertices. If the vertices ...
There can be total 6C4 ways to pick four vertices from six. The value of 6C4 is 15.
Note that the given graph is complete, so any four vertices can form a cycle.
There can be six different cycles with four vertices. For example, consider four vertices a, b, c and d.
(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); and (a, c, b, d, a) and (a, d, b, c, a) are same cycles.
So, total number of distinct cycles = (15 x 3) = 45
Note that the given graph is complete, so any four vertices can form a cycle.
There can be six different cycles with four vertices. For example, consider four vertices a, b, c and d.
(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); and (a, c, b, d, a) and (a, d, b, c, a) are same cycles.
So, total number of distinct cycles = (15 x 3) = 45
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Let G be a complete undirected graph of six vertices. If the vertices ...
Understanding Complete Graphs
A complete undirected graph, denoted as K_n, is a graph where every pair of distinct vertices is connected by a unique edge. For K_6, which has 6 vertices, the properties of this graph are essential for counting cycles.
Defining Cycles
A cycle in a graph is a path that starts and ends at the same vertex, with no other vertices repeated. In this case, we are interested in cycles of length 4.
Choosing Vertices for the Cycle
1. Select 4 Vertices:
- To form a cycle of length 4, you need to choose 4 vertices from the 6 available.
- The number of ways to choose 4 vertices from 6 is calculated using combinations:
- C(6, 4) = 15.
Arranging the Vertices in Cycles
2. Arranging the Selected Vertices:
- Once 4 vertices are chosen, we must arrange them to form a cycle.
- The number of distinct arrangements (cycles) for 4 vertices is (4-1)! because rotations of the cycle are considered the same.
- This gives us: 3! = 6 arrangements per selection of 4 vertices.
Calculating Total Distinct Cycles
3. Total Count of Cycles:
- Multiply the number of ways to choose the vertices by the number of arrangements:
- Total cycles = C(6, 4) * (4-1)! = 15 * 6 = 90.
4. Accounting for Symmetry:
- Each cycle can be traversed in two directions (clockwise and counterclockwise), so we need to divide by 2:
- Final total = 90 / 2 = 45.
Conclusion
Thus, the number of distinct cycles of length 4 in the complete graph K_6 is 45.
A complete undirected graph, denoted as K_n, is a graph where every pair of distinct vertices is connected by a unique edge. For K_6, which has 6 vertices, the properties of this graph are essential for counting cycles.
Defining Cycles
A cycle in a graph is a path that starts and ends at the same vertex, with no other vertices repeated. In this case, we are interested in cycles of length 4.
Choosing Vertices for the Cycle
1. Select 4 Vertices:
- To form a cycle of length 4, you need to choose 4 vertices from the 6 available.
- The number of ways to choose 4 vertices from 6 is calculated using combinations:
- C(6, 4) = 15.
Arranging the Vertices in Cycles
2. Arranging the Selected Vertices:
- Once 4 vertices are chosen, we must arrange them to form a cycle.
- The number of distinct arrangements (cycles) for 4 vertices is (4-1)! because rotations of the cycle are considered the same.
- This gives us: 3! = 6 arrangements per selection of 4 vertices.
Calculating Total Distinct Cycles
3. Total Count of Cycles:
- Multiply the number of ways to choose the vertices by the number of arrangements:
- Total cycles = C(6, 4) * (4-1)! = 15 * 6 = 90.
4. Accounting for Symmetry:
- Each cycle can be traversed in two directions (clockwise and counterclockwise), so we need to divide by 2:
- Final total = 90 / 2 = 45.
Conclusion
Thus, the number of distinct cycles of length 4 in the complete graph K_6 is 45.
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Let G be a complete undirected graph of six vertices. If the vertices of G are labelled, then the number of distinct cycles of length 4 in G isCorrect answer is '45'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Let G be a complete undirected graph of six vertices. If the vertices of G are labelled, then the number of distinct cycles of length 4 in G isCorrect answer is '45'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a complete undirected graph of six vertices. If the vertices of G are labelled, then the number of distinct cycles of length 4 in G isCorrect answer is '45'. Can you explain this answer?.
Let G be a complete undirected graph of six vertices. If the vertices of G are labelled, then the number of distinct cycles of length 4 in G isCorrect answer is '45'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Let G be a complete undirected graph of six vertices. If the vertices of G are labelled, then the number of distinct cycles of length 4 in G isCorrect answer is '45'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a complete undirected graph of six vertices. If the vertices of G are labelled, then the number of distinct cycles of length 4 in G isCorrect answer is '45'. Can you explain this answer?.
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