I've just found the internet

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.

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

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.

Graph theory

• • •  A graph consists of: set of vertices A set of edges connecting vertex pair Incidence matrix: which edges are connected

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

These are all equivalent

Euler and the Konigsberg bridges

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

Problem: does this graph have an Euler cycle?

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

What is a theorem?

• • • • A statement that no one can understand A statement that only a mathematician can understand A statement that can be verified from “first principles” A statement that is “always true”

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”

Hamiltonian graph

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

Is the following graph Hamiltonian?

Is the following graph Hamiltonian?

Petersen graph: symmetry

Graph colorings

Other types of graphs

Other properties • • • • Diameter Girth Chromatic number etc

Graph coloring and map coloring • The four color problem

Which continent is this?

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

Doesn’t like

A

B,C

B

A,C

C

A,B,D,E

D

C,F

E

C,F

F

D,E

Other examples of problems whose solutions are simplified using graph theory

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

Complementary graph

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

Complementary graph

Handshakes, part 2 • • 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?

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?

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?

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?