6.1 Introduction to Graphs 1 Introduction to Graphs Section 6.1 Animations.
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Transcript of 6.1 Introduction to Graphs 1 Introduction to Graphs Section 6.1 Animations.
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 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 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 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 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 26
Non-traversable
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 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 =