A random graph G(n, p) with 0 p 1 is a graph with n labeled vertices such that the probability that there is an edge between any pair of vertices is p. (a) (2 points) If p = 0.5, then it is equally likely that G(n, p) is any graph. What is the probability that G(n, 0.5) is the complete graph on n vertices? (b) ( 2 points) What is the probability that G(3, 0.5) is a triangle? (c) (2 points) What is the probability that G(3, 0.5) is connected? (d) ( 2 points) What is the probability that G(4, 0.5) is a square, i.e., a cycle with 4 vertices and 4 edges? (e) (2 points) What is the expected number of edges in G(n, 0.5)? (f) (2 points) What is the expected number of triangles in G(n, 0.5)?

3. A random graph G(n,p) with 0 Sp 1 is a graph with n labeled vertices such that the probability that there is an edge between any pair of vertices is p. (a) (2 points) If p 0.5, then it is equally likely that G(n, p) is any graph. What is the probability that G(n, 0.5) is the complete graph on n vertices? b) (2 points) What is the probability that G(3,0.5) is a triangle? (c) (2 points) What is the probability that G(3,0.5) is connected? (d) ( 2 points) what is the probability that G(4.05) is a square, i.e., a cycle with 4 vertices and 4 edges? e) (2 points) What is the expected number of edges in G(n, 0.5)? (f) (2 points) What is the expected number of triangles in G(n, 0.5)