1. 1.1. Graph Models Two basic notions: Graphs Directed Graphs

2. Graphs (Simple) graph G=(V,E) V = {1,2,3,4} – vertices E = {a,b,c,d,e} – edges Vertices 1 and 2 are adjacent: 1 ~ 2; a = (1,2). 1 34 2 a e cb d

3. Directed Graphs Directed edges, such as (a !,b) a cd b

4. Example 1: Matching A-E, people a-e, jobs Can each person get a job? Answer: No! Why? Bipartite graph B Dd b Aa Cc Ee

5. Example 2: Spelling Checker Testing if letter x is in the tree. Say x = Q x · M or x > M. x · S or x > S. x · P or x > P. G PZ S M D J

6. Example 3: Network Reliabiliy. Two questions: What is the minimal number of edges whose removal will disconnect the graph? What is the minimum number of edges needed to link together the eleven vertices? i g ad efh k cb j

7. Example 6: Interval Graph Modeling. A competition graph used in ecology, has a vertex for each pair of species that feed on a comon prey. One may sometimes consider competition graphs as interval graphs. b ac d e f

8. 1.2. Isomorphism Homework: Read 1.3. Do Exercises1.2: 3,4,8,10,16 Volunteers: ____________ Problem: 16. News: News: We have a class website at: We have a class website at: http://math.colgate.edu/math310 http://math.colgate.edu/math310/ Warning: In 1.1.#5 use Figure 1.5(a) [and NOT figure 1.3] Warning: In 1.1.#5 use Figure 1.5(a) [and NOT figure 1.3]

9. Isomorphism Two graphs G and G’ are isomorphic if there exists a one-to-one correspondence between the vertices in G and the vertices in G’ such that a pair of vertices are adjacent in G if and only if the corresponding pair of vertices is adjacent in G’. Such a correspondence is called an isomorphism.

10. Subgraph A subgraph G’ of a graph G is a graph formed by a subset of vertices and edges of G.

11. Complete graph K n. A graph on n vertices in which each vertex is adjacent to all other vertices is called a complete graph on n vertices, denoted by K n. K 20