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I've just found the internet PowerPoint PPT Presentation


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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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I've just found the internet

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I ve just found the internet l.jpg

I've just found the internet


How does information travel across the internet l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

Graph theory

  • A graph consists of:

  • set of vertices

  • A set of edges connecting vertex pair

  • Incidence matrix: which edges are connected


Slide6 l.jpg

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 l.jpg

These are all equivalent


Euler and the konigsberg bridges l.jpg

Euler and the Konigsberg bridges


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Types of graphs

  • Eulerian: circuit that traverses each edge exactly once

  • Which graphs possess Euler circuits?


Problem does this graph have an euler cycle l.jpg

Problem: does this graph have an Euler cycle?


Theorem if every vertex has even degree then there is an eulerian path l.jpg

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


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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”


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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 l.jpg

Hamiltonian graph


Hamilton s puzzle find a path in the dodecahedron graph that traverses each vertex exactly once l.jpg

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


Is the following graph hamiltonian l.jpg

Is the following graph Hamiltonian?


Is the following graph hamiltonian18 l.jpg

Is the following graph Hamiltonian?


Petersen graph symmetry l.jpg

Petersen graph: symmetry


Graph colorings l.jpg

Graph colorings


Other types of graphs l.jpg

Other types of graphs


Other properties l.jpg

Other properties

  • Diameter

  • Girth

  • Chromatic number

  • etc


Graph coloring and map coloring l.jpg

Graph coloring and map coloring

  • The four color problem


Which continent is this l.jpg

Which continent is this?


Boss s dilemna l.jpg

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


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

Other examples of problems whose solutions are simplified using graph theory


What does this graph have to do with the boss s dilemma l.jpg

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


Complementary graph l.jpg

Complementary graph


Complete subgraph l.jpg

Complete subgraph

  • Subgraph: vertices subset of vertex set, edges subset of edge set

  • Complete: every vertex is connected to every other vertex.


Complementary graph34 l.jpg

Complementary graph


Handshakes part 2 l.jpg

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?


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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?


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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?


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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?


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