1. CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory

2. Graph formalism –G = (V, E) –Vertices –Edges Directed Graph –Edges ordered pairs Undirected Graph –Edges sets of size two

3. Graph examples Communication Networks Road networks

4. Social networks Community Graph –Linked In, Face Book Transactions –Ebay Authorship –Erdos Number

5. The web graph

6. Page Rank Determine the value of a page based on link analysis Model of randomly traversing a graph –Weighting factors on nodes –Damping (random transitions)

7. Graph terminology NeighborhoodDegree

8. Degree sequence Find a graph with degree sequence –3, 3, 2, 1, 1 Find a graph with degree sequence –3, 3, 3, 3, 3

9. Handshake Theorem

10. Directed Degree Theorem

11. Special Graphs Complete Graphs K n Cycle C n Hypercube Q n Mesh M n,m

12. Bipartite Graphs

13. 2-coloring A graph is two colorable iff all cycles have even length

14. Graph Representations Adjacency Lists Adjacency Matrices Incidence Matrices

15. Graph Connectivity

16. Strong connectivity vs. Weak Connectivity

17. Strongly Connected Components

18. Counting Paths Let A be the Adjacency Matrix. What is A 2 ? d c b e a

19. Graph Isomorphism I Are these two graphs the same? ad cb w x y z

20. Graph Isomorphism II Are these graphs the same?

21. Graph Isomorphism III Are these graphs the same?