1. I've just found the internet

2. History of the internet
Digital computers: 1950s
ARPA, ARPANET: developed in 1960s
1980s: NSF funds CSNET, TCP/IP developed
Late 1980s: commercial ISPs;
1990: Arpanet decommissioned
1990s: , primitive VoIP
2000s: social networks
2010s: cloud computing
2020s: embedded devices?

3. 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.

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

5. 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.

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

7. 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

8. These are all equivalent

9. Al qaeda graph

10. Euler and the Konigsberg bridges

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

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

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

14. Clicker question: The following graph is Eulerian
True
False

15. 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”

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

17. 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

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

19. Hamiltonian graph

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

22. Rhetorical question: Is the following graph Hamiltonian?

23. Fullerenes

25. Petersen graph: symmetry

26. Other types of graphs

27. Other properties
Diameter
Girth
Chromatic number
etc

28. Graph colorings

29. Graph coloring and map coloring
The four color problem

30. Which continent is this?

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

37. 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
Worker A B C D E F
Doesn’t like
B,C
A,C
A,B,D,E
C,F
D,E

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

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

40. Complementary graph

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

42. Complementary graph

43. 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?

44. 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

45. 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?

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

47. 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.

48. 7persons
Each limb that is connected to another represents an edge. Some have four connections, some have three.

49. 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?

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

51. 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?