me test

Question

A simple graph with n vertices is constructed by randomly and independently placing an edge between every two vertices with probability p. What is the expected no. of nodes with degree 2?

Answer 1

A vertex will have n-1 edges going out of it. For node to be of degree two, 2 of these n-1 edges need to be selected.

$P=\binom{n-1}{2}\left ( p \right )^{2}\left ( 1-p \right )^{n-3}$

We have n nodes, so expected no of nodes of degree 2

E= n*P

$E= n\binom{n-1}{2}\left ( p \right )^{2}\left ( 1-p \right )^{n-3}$