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I've just found the internet

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- 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.

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

- 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.

- 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

- Eulerian: circuit that traverses each edge exactly once
- Which graphs possess Euler circuits?

- 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”

- 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”

- Diameter
- Girth
- Chromatic number
- etc

- The four color problem

- Six employees, A,B,C,D,E,F
- Some do not get along with others
- Find smallest number of compatible work groups

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

- 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?

- 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?

- 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?

- 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?