Question

Suppose that we generate a random graph G = (V, E) on the vertex set V...

Suppose that we generate a random graph G = (V, E) on the vertex set V = {1, 2, . . . , n} in the following way.

For each pair of vertices i, j ∈ V with i < j, we flip a fair coin, and we include the edge i−j in E if and only if the coin comes up heads.

How many edges should we expect G to contain?

How many cycles of length 3 should we expect G to contain?

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Answer #1

The number of edges in G depend on the flip of the coin.

Since the coin is fair, the probability to include any edge is 1/2.

Since the number of vertices is n, there are n(n-1) pair of vertices in a random graph.

So each of edges get added with probability p=1/2 and any two edges in the graph are added with probability .

Thus, there are edges in the graph.

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