1. Group 10 Project Part 3 Derrick Jasso Rodolfo Garcia Ivan Lopez M.

2. Section 10.1 #36 36. Recall that K m,n denotes a complete bipartite graph on (m, n) vertices. a. Draw K 4,2 b. Draw K 1,3 c. Draw K 3,4 d. How many vertices of K m,n have degree m? degree n? e. What is the total degree of K m,n ? f. Find a formula in terms of m and n for the number of edges of K m,n. Explain.

3. Solution a) K 4,2 b)K 1,3

4. Solution c)K 3,4 d) If n≠m, the vertices of K m,n are divided into two groups: one of size m and the other of size n. Every vertex in the group of size m has degree n because each is connected to every vertex in the group of size n. So K m,n has n vertices of degree m. Similarly, every vertex in the group of size n has degree m since each is connected to every vertex in the group of size m. So K m,n has n vertices of degree m.

5. Solution e) The total degree of K m,n is 2mn because K m,n has m vertices of degree n and n vertices of degree m. f)The number of edges of K m,n = mn. This is because the total degree of K m,n is 2mn, and so, by Theorem 11.1.1, K m,n has 2mn = mn edges. 2

6. Section 10.1 #37 Find which of the following graphs are bipartite. Redraw the bipartite graphs so that their bipartite nature is evident.

7. Solution a.) b.)Not bipartite. No possible way to create two subsets without vertices relating to one another within the same subset. c.) d.) Not bipartite. No possible way to create two subsets without vertices relating to one another within the same subset. v1v1 v4v4 v2v2 v3v3 v1v1 v4v4 v3v3 v2v2 v5v5 v6v6

8. Solution Continued d.) Not bipartite. No possible way to create two subsets without vertices relating to one another within the same subset. e.) f.) Not bipartite. No possible way to create two subsets without vertices relating to one another within the same subset. v5v5 v2v2 v3v3 v1v1 v4v4

9. Summary of Main Results Definition: Let n be a positive integer. A complete graph on n vertices, denoted K n, is a simple graph with n vertices and exactly one edge connecting each pair of distinct vertices.

10. Definition: Let m and n be positive integers. A complete bipartite graph on (m,n) vertices, denoted K m,n, is a simple graph with distinct vertices v 1, v 2, …,v m and w 1, w 2, …,w n that satisfies the following properties: For all i,k = 1, 2, …,m and for all j, l = 1, 2, …, n, 1. There is an edge from each vertex v i to each vertex w j. 2. There is no edge from any vertex v i to any other vertex v k. 3. There is no edge from any vertex w j to any other vertex w l.

11. Leonhard Euler Euler lived from April 15, 1707 to September 18, 1783. Euler is considered one of the greatest mathematician of all time. With contributions to calculus and wrote the first paper on graph theory. Not only was he a great mathematician but, he was also renowned for his work in mechanics, fluid dynamics, optics, and astronomy.