I've just found the internet

1. I've just found the internet

2. How does information travel across the internet? • TCP/IP • TCP wiki • IP wiki • Request generated by user (“click”) • Response sent as set of packets with time stamps • Receipt acknowledged • Response regenerated if ack not received.

3. Bandwidth • Packets seek shortest/fastest path • Determined by number of hops • Queues form at hubs; bottlenecks can occur • Repeat requests can add to traffic

4. Main problem • Determining the shortest path • Presumes: lookup table of possible routes • Presumes: knowledge of structure of internet • Mathematical structure: directed, weighted graph. • Other related problems: railroad networks, interstate network, google search problem, etc.

5. Graph theory • A graph consists of: • set of vertices • A set of edges connecting vertex pair • Incidence matrix: which edges are connected

6. The incidence matrix of a graph gives the (0,1)-matrix which has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge e

7. These are all equivalent

8. Al qaeda graph

9. Euler and the Konigsberg bridges

10. Types of graphs • Eulerian: circuit that traverses each edge exactly once • Which graphs possess Euler circuits?

11. Problem: does this graph have an Euler cycle?

12. Theorem: If every vertex has even degree then there is an Eulerian path

13. Clicker question:What is a theorem? A) A statement that no one can understand B) A statement that only a mathematician can understand C) A statement that can be verified from “first principles” D) A statement that is “always true”

14. Heuristic argument • An argument that appeals to intuition, but may not be compelling by itself. • In the case of the Eulerian graph theorem, think of the vertex as a room and the edges as hallways connecting rooms. • If you leave using one hallway then you have to return using a different one. • “Induction argument”

15. Hamiltonian graph

17. Hamilton’s puzzle: find a path in the dodecahedron graph that traverses each vertex exactly once

18. Hamilton’s puzzle: find a path in the dodecahedron graph that traverses each of the twenty vertices exactly once

19. Hamiltonian graph • A graph is said to be Hamiltonian if, starting from a vertex v, it is possible to visit each vertex of the graph exactly once, and end up back at v • Such a path is called a Hamiltonian cycle

20. Clicker Question:Is the following graph Hamiltonian? • Yes • No

21. Rhetorical question:Is the following graph Hamiltonian?

22. Fullerenes

23. Petersen graph: symmetry

24. Other types of graphs

25. Other properties • Diameter • Girth • Chromatic number • etc

26. Graph colorings

27. Graph coloring and map coloring • The four color problem

28. Which continent is this?

29. [Clicker Question: What continent does this graph represent? ] • [Asia] • [Africa] • [Europe] • [North America] • [South America] • [Australia] • [Antarctica]

35. Boss’s dilemna • Six employees, A,B,C,D,E,F • Some do not get along with others • Find smallest number of compatible work groups

36. Other examples of problems whose solutions are simplified using graph theory

37. What does this graph have to do with the Boss’s dilemma?

38. Complementary graph

39. Complete subgraph • Subgraph: vertices subset of vertex set, edges subset of edge set • Complete: every vertex is connected to every other vertex.

40. Complementary graph

41. Clicker question: How many men are in the room • There are several men and 15 women in a room. Each man shakes hands with exactly 6 women, and each woman shakes hands with exactly 8 men. • How many men are in the room?

42. Clicker question • There are several men and 15 women in a room. Each man shakes hands with exactly 6 women, and each woman shakes hands with exactly 8 men. How many men are in the room? • A) 15 • B) 8 • C) 20 • D) 6

43. Visualize whirled peas • Samantha the sculptress wishes to make “world peace” sculpture based on the following idea: she will sculpt 7 pillars, one for each continent, placing them in circle. Then she will string gold thread between the pillars so that each pillar is connected to exactly 3 others. • Can Samantha do this?

44. Clicker Question: Can Samantha do this? • A) Yes • B) No

45. Solution: • Think of the “continents” as vertices of a graph • Think of the strings as edges • Is it possible to have a graph with seven vertices each with degree 3? • No: Each edge joins two vertices, so contribute one to each vertex degree. The sum of the vertex degree over all vertices equals twice the number of edges, so has to be even.

46. 7persons Each limb that is connected to another represents an edge. Some have four connections, some have three.

47. Some additional exercises in graph theory • There are 7 guests at a formal dinner party. The host wishes each person to shake hands with each other person, for a total of 21 handshakes, according to: • Each handshake should involve someone from the previous handshake • No person should be involved in 3 consecutive handshakes • Is this possible?

48. Clicker question: Is this possible • A) Yes • B) No

49. Camelot • King Arthur and his knights wish to sit at the round table every evening in such a way that each person has different neighbors on each occasion. If KA has 10 knights, for how long can he do this? • Suppose he wants to do this for 7 nights. How many knights does he need, at a minimum?