6.1 Introduction to Graphs 1 Introduction to Graphs Section 6.1 Animations.

6.1 Introduction to Graphs 1

Introduction to Graphs

Section 6.1

Animations

6.1 Introduction to Graphs 2

Konigsberg Bridge Problem

Rules

1.

2.

A

DB

C

6.1 Introduction to Graphs 3

Your solution

1. Did it every time

2. Did it at least once

3. Can’t seem to do it

6.1 Introduction to Graphs 4

Konigsberg Bridge (8th bridge)

6.1 Introduction to Graphs 5

Your solution

1. Did it every time

2. Did it at least once

3. Can’t seem to do it

6.1 Introduction to Graphs 6

Konigsberg Bridge (9th bridge)

6.1 Introduction to Graphs 7

Your solution

1. Did it every time

2. Did it at least once

3. Can’t seem to do it

6.1 Introduction to Graphs 8

3 Cases for Konigsberg

1. 7 Bridges (Non-traversable)

2. 8 Bridges (Euler Path)

3. 9 Bridges (Euler circuit)

Picture

Picture

Picture

6.1 Introduction to Graphs 9

Euler’s View

A

BC

D

Map Graph

6.1 Introduction to Graphs 10

You try one

A

D

CBIsland Island

River

Graph

6.1 Introduction to Graphs 11

Definition - Graph

A graph is any collection of

• Dots (Vertices)

• Arcs/Lines (Edges) that join the points

Examples

6.1 Introduction to Graphs 12

Two Special Cases

1. Loops

2. Isolated Points

A

B CD

We will avoid isolated points

6.1 Introduction to Graphs 13

The Degree of a Vertex

The degree of a vertex is the number of times the vertex is touched by an edge

A

B

C

D

A

B

CD

E

E

Degree Evenness/

oddness

Counting vertices, edges, degrees applet

6.1 Introduction to Graphs 14

This graph has

1. 6 edges, 4 vertices (exactly 2 of which are odd)

2. 4 edges, 6 vertices (all of which are odd)

3. 6 edges, 4 vertices (all of which are odd)

4. 4 edges, 4 vertices (exactly 2 of which are odd)

6.1 Introduction to Graphs 15

This graph has

B

D

C

AE

1. 8 edges, 5 vertices (none of which are odd)

2. 8 edges, 5 vertices (exactly 2 of which are odd)

3. 8 edges, 5 vertices (exactly 4 of which are odd)

4. 8 edges, 5 vertices (all of which are odd)

6.1 Introduction to Graphs 16

Draw a graph with

A. 4 vertices (all odd) and 5 edges

B. 4 vertices (all odd) and 3 edges (no loops)

6.1 Introduction to Graphs 17

Draw a graph with

C. 3 vertices (exactly 1 even) and 4 edges

D. 3 vertices (exactly 1 odd) and 4 edges

6.1 Introduction to Graphs 18

All vertices of a graph could be odd

1. True

2. False

6.1 Introduction to Graphs 19

All vertices of a graph could be even

1. True

2. False

6.1 Introduction to Graphs 20

End of 6.1

6.1 Introduction to Graphs 21

1. GRAPHS Lots of explorations. Discovery. Hit theory a bit harder. Discover sum og degrees in agrpah is even., etc

6.1 Introduction to Graphs 22

Arrrgh!Expleti

ve deleted

!

7 bridges

2 islands

Ouch!

SPLAT!

6.1 Introduction to Graphs 23

7 bridges

2 islands

I did

it!

But…

6.1 Introduction to Graphs 24

7 bridges

2 islands

I did it! I did

it!

And …

6.1 Introduction to Graphs 25

Leonard Euler (“Oiler”)1706 - 1783

6.1 Introduction to Graphs 26

Non-traversable

6.1 Introduction to Graphs 27

Euler Path

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Euler Circuit

6.1 Introduction to Graphs 29

Genealogy

Vertices =

Edges =

Abraham Lincoln

Bathsheba Herring

James Hanks

LucyShipley

Thomas Lincoln

Nancy Hanks

16th President Abraham Lincoln

6.1 Introduction to Graphs 30

Constellations

Vertices =

Edges =

6.1 Introduction to Graphs 31

The nine members of the Supreme Court in 1973 were Justices Marshall, Burger, White, Blackman, Powell, Rhenquist, Brennan, Douglas, and Stewart. The conservative block of Burger, Rhenquist, Powell and Blackman voted together on 70+ percent of the votes. Justice White joined with Justice Blackman 70+ percent of the time. The liberal block of Brennan, Douglas, and Marshall voted together 70+ percent of the time. Justice Stewart was the maverick who voted with no one 70+ percent of the time

6.1 Introduction to Graphs 32

Political Science

Burger

Marshall

DouglasBlackman

Brennan

Rhenquist

Stewart

Powell

White

Vertices =

Edges =

6.1 Introduction to Graphs 33

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