I. Consider the complete graph K, on n vertices. (a) How many balanced assignments are there? (b) How many weakly balanced assignments are there? 2. Consider the complete graph Kn on n vertices. Assign or - on edges at random each with probability 1/2 (a) What is the expected number of balanced triangles? (b) What is the expected number of weakly balanced triangles? 3 (c) What is the probability that a random assignment is balanced? 3. Let E be the set of edges of Kn Fix a cycle Cn of Kn of length n Consider the complete graph Kn[C] in which all edges in C are assigned the sign +. Answer all the questions above for the graph K[C I. Consider the complete graph K, on n vertices. (a) How many balanced assignments are there? (b) How many weakly balanced assignments are there? 2. Consider the complete graph Kn on n vertices. Assign or - on edges at random each with probability 1/2 (a) What is the expected number of balanced triangles? (b) What is the expected number of weakly balanced triangles? 3 (c) What is the probability that a random assignment is balanced? 3. Let E be the set of edges of Kn Fix a cycle Cn of Kn of length n Consider the complete graph Kn[C] in which all edges in C are assigned the sign +. Answer all the questions above for the graph K[C