Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is ½. What is the expected number of unordered cycles of length three?
(A) 1/8
(B) 1
(C) 7
(D) 8
Answer: C
Explanation :
To make a cycle of 3, we need 3 vertices.
The number of such sets of 3 vertices from set of 8 is = 8C3
[Not 8P3, because the order of nodes is not important]
For such each set of such 3 vertices, a loop/cycle will be made only of 3 edges exist between those nodes. Probability of having a cycle between those 3 nodes is = (1/2)(1/2)(1/2) = 1/8
Expected number of cycles will be = 8C3 * (1/8) = 7
Topics to revise :
Probability
Undirected Graphs and Cycles
References :
(A) 1/8
(B) 1
(C) 7
(D) 8
Answer: C
Explanation :
To make a cycle of 3, we need 3 vertices.
The number of such sets of 3 vertices from set of 8 is = 8C3
[Not 8P3, because the order of nodes is not important]
For such each set of such 3 vertices, a loop/cycle will be made only of 3 edges exist between those nodes. Probability of having a cycle between those 3 nodes is = (1/2)(1/2)(1/2) = 1/8
Expected number of cycles will be = 8C3 * (1/8) = 7
Topics to revise :
Probability
Undirected Graphs and Cycles
References :
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