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I've just found the internetPowerPoint Presentation

I've just found the internet

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**kaipo** - Follow User

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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.

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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 has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge e

Euler and the Konigsberg bridges has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge e

Types of graphs 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?

Problem: does this graph have an Euler cycle? has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge e

Theorem: If every vertex has even degree then there is an has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge eEulerian path

What is a theorem? has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge e

- 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 has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge e

- 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 has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge e

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

Is the following graph Hamiltonian? that traverses each vertex exactly once

Is the following graph Hamiltonian? that traverses each vertex exactly once

Petersen graph: symmetry that traverses each vertex exactly once

Graph colorings that traverses each vertex exactly once

Other types of graphs that traverses each vertex exactly once

Other properties that traverses each vertex exactly once

- Diameter
- Girth
- Chromatic number
- etc

Graph coloring and map coloring that traverses each vertex exactly once

- The four color problem

Which continent is this? that traverses each vertex exactly once

Boss’s dilemna that traverses each vertex exactly once

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

Other examples of problems whose solutions are simplified using graph theory

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

Complementary graph using graph theory

Complete subgraph using graph theory

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

Complementary graph using graph theory

Handshakes, part 2 using graph theory

- 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 using graph theory

- 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 using 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 using graph theory

- 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?

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