Consider an undirected random graph of eight vertices The probability that there is an edge between a pair of vertices is What is the expected number of unordered cycles of length three gate computer science 2012

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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 ½. What is the expected number of unordered cycles of length three? -gate-computer science-2012

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

By:Amogh

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C) C
A cycle of length 3 requires 3 vertices.
Number of ways in which we can choose 3 vertices from 8 = 8C3 =56.
Probability that 3 vertices form a cycle = Probability of edge between vertices 1 and 2 * Probability of edge between vertices 2 and 3 * Probability of edge between vertices 1 and 3
=1/2 * 1/2 * 1/2 = 1/8
So, expected number of cycles of length 3 = 56 * 1/8 = 7

milan-ransingh

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