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Graph and Network Theory
Math 137A
HOMEWORK 1
8 PROBLEMS
- Let A = {3, 4, 5}, B = {3, 4}, C = {4}. Find D = A4 B4 C.
- Suppose 70% of Californians like cheese, 80% like apples and 10% like neither. What percentage of Californians like both cheese and apples?
- Use the Principle of Mathematical Induction to prove that for n ∈ N, n 3 − n is always divisible by 3.
- Find a surjective function from N to Z. Find an injective function from Z to N.
- Write an explicit description of the edgemap for the complete bipartite (3, 5)-graph.
- Is there a simple graph on 6 vertices such that the vertices all have distinct degree? If not, why not? If so, draw one.
- Let G be a k-regular graph, where k is an odd number. Prove that the number of edges in G is a multiple of k.
- Prove that it is impossible to have a group of nine people at a party such that each one knows exactly five of the others in the group.