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Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?
- a)1/8
- b)1
- c)7
- d)8
Correct answer is option 'C'. Can you explain this answer?
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Consider an undirected random graph of eight vertices. The probability...
A cycle of length 3 can be formed with 3 vertices. There can be total 8C3 ways to pick 3 vertices from 8. The probability that there is an edge between two vertices is 1/2. So expected number of unordered cycles of length 3 = (8C3)*(1/2)^3 = 7
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Consider an undirected random graph of eight vertices. The probability...
Expected number of unordered cycles of length three
To find the expected number of unordered cycles of length three in the given random graph, we can consider each possible cycle of length three and calculate the probability of its existence.
Number of possible cycles of length three
To determine the number of possible cycles of length three in the graph, we need to select three vertices out of the eight available. This can be done using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of vertices and r is the number of vertices to be selected. In this case, n = 8 and r = 3.
C(8, 3) = 8! / (3!(8-3)!) = 8! / (3!5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
Therefore, there are 56 possible cycles of length three in the graph.
Probability of existence of each cycle
Since the probability of having an edge between any pair of vertices is 1/2, the probability of a cycle of length three existing depends on the presence of edges between the three selected vertices.
For each possible cycle of length three, there are three edges that need to exist. The probability of each edge existing is 1/2. Therefore, the probability of a cycle of length three existing is (1/2)^3 = 1/8.
Expected number of unordered cycles
To find the expected number of unordered cycles of length three, we multiply the number of possible cycles by the probability of each cycle existing.
Expected number = Number of possible cycles * Probability of existence of each cycle
Expected number = 56 * (1/8) = 7
Therefore, the expected number of unordered cycles of length three in the given random graph is 7.
Hence, the correct answer is option C) 7.
To find the expected number of unordered cycles of length three in the given random graph, we can consider each possible cycle of length three and calculate the probability of its existence.
Number of possible cycles of length three
To determine the number of possible cycles of length three in the graph, we need to select three vertices out of the eight available. This can be done using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of vertices and r is the number of vertices to be selected. In this case, n = 8 and r = 3.
C(8, 3) = 8! / (3!(8-3)!) = 8! / (3!5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
Therefore, there are 56 possible cycles of length three in the graph.
Probability of existence of each cycle
Since the probability of having an edge between any pair of vertices is 1/2, the probability of a cycle of length three existing depends on the presence of edges between the three selected vertices.
For each possible cycle of length three, there are three edges that need to exist. The probability of each edge existing is 1/2. Therefore, the probability of a cycle of length three existing is (1/2)^3 = 1/8.
Expected number of unordered cycles
To find the expected number of unordered cycles of length three, we multiply the number of possible cycles by the probability of each cycle existing.
Expected number = Number of possible cycles * Probability of existence of each cycle
Expected number = 56 * (1/8) = 7
Therefore, the expected number of unordered cycles of length three in the given random graph is 7.
Hence, the correct answer is option C) 7.
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Consider an undirected random graph of eight vertices. The probability...
Answer is 7
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Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?a)1/8b)1c)7d)8Correct answer is option 'C'. Can you explain this answer?
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