Graph Models: Understanding Graphs and Directed Graphs in Mathematics

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

a 1 2 c b d e 3 4 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).

Directed Graphs • Directed edges, such as (a!,b) a b c d

Example 1: Matching A a • A-E, people • a-e, jobs • Can each person get a job? • Answer: No! Why? • Bipartite graph B b C c D d E e

Example 2: Spelling Checker M • 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 S J P Z D

Example 3: Network Reliabiliy. a b c d • 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? e f g h k j i

Example 6: Interval Graph Modeling. b d f • 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. a c e

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

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.

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

Complete graph Kn. • 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 Kn. K20

Graph Models: Understanding Graphs and Directed Graphs in Mathematics