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.railroad networks interstate networkgoogle search problem

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?

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 WorkerABCDEF Doesn’t like B,CA,CA,B,D,EC,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?