If G1 and G2 are isomorphic, must they have the same number of edges? If G1 and G2 have the same number of edges must they be isomorphic? If not, give an example.

6. Determine whether the following pair of graphs is isomorphic. Exhibit an isomorphism ( show the mapping between vertices) or provide an argument that none exists. V1 V3 V5 V1 V2 V5 VA V3 V4 V2 G1 G2

2. Say that for any pair of people, they either both know each other, or they are strangers (neither knows each other). Prove that in any party of six people, there is a group of three that either all know each other, or all are strangers. (Hint: how can you model this problem using a graph?)

3 and 4

Discrete math. Graph theory

2. In a round-robin tournament the Tigers beath the Blue Jays, the Tigers beat the Cardinals, the Tigers beat the Orioles, the Blue Jays beat the Cardinals, the Blue Jays beat the Orioles, and the Cardinals beat the Orioles. Model this outcome with a directed graph.

Consider a bipartite graph on the set A = {1,2,3,4,5,6} and B = {7,8,9,10,11,12}, where there is an edge between xe A and y e B if and only if x+y is a multiple of 4. What is the size of N({2})? What is the size of N({2,5})? What is the size of N({2,6})? Is there a matching in this graph? %3D

Consider the following sets V = {v,V1, V2, V3, V4} and a set of vertices A simple graph can be such that d(vo) = 0, d(v1) = 1, d(v2) = 2, d(v3) = 3 and d(v4) = 4, explain why

For the directed multigraph below, determine the totl number of vertices and edges.

Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. How many possible ways are there to match or pair vertices between A and B?