Graph Theory

Graphs we've seen

data

data

data

data

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Directed Acyclic Graph

Linear Data Structure

Graph Theory

Graphs we've seen

Directed Acyclic Graph

Non-Linear Data Structure

Graph Vocab

This is a Node

Nodes represent entities in graphs

Node(s) || Vertex (Vertices)

A Node class should have:

• content

• relationships

Graph Vocab

This is an Edge

Edges represent relationships in graphs

Edge

Nodes keep track of Edges

Graph Vocab

Directed / Undirected Graphs

This is an Edge with no direction

This is an Edge with a direction

Graph Vocab

a

c

b

d

g

f

h

e

i

j

This is an Undirected graph

It's edges have no direction

A relationship is two-way by default

Graph Vocab

a

c

b

d

g

f

h

e

i

j

This is a Directed Graph

A relationship is explicitly one way

To have a two way relationship, a new relationship must be formed

Example: Undirected Graph

a

c

b

d

g

f

h

e

i

j

Friendship!

An edge here represents a mutual friendship, such as a Facebook friendship

Example: Undirected Graph

var User = function(name) {
    this.name = name;
    this.friends = [];
}

User.prototype.addFriend(user) {
    this.friends.push(user);
    user.friends.push(this);
}

Example: Directed Graph

a

c

b

d

g

f

h

e

Followers on Twitter

q

x

z

b

a is pretty popular, as is e

f, c, and d follow a and e, but are not followed back

b does not follow anyone, but is followed by g, who also follows e

Example: Directed Graph

Followers on Twitter

var User = function(name) {
    this.name = name;
    this.followers = [];
    this.following = [];
}

User.prototype.follow = function (user) {
    this.following.push(user);
    user.followers.push(this);
}

Weights

a

c

b

d

g

f

h

e

i

j

15

37

-12

5

3

23

1

-3

-15

-20

-44

15

15

Weights

a

b

15

3

c

d

2

10

2 possible paths from a to d

a

b

d

a

c

d

Total Weight: 5

Total Weight: 25

∴Shortest Path:

a

b

d

sum(

)

sum(

)

Objectives

1. Use Graph Theory vocabulary
2. Use Graph Theory Notation
3. Model Real World Relationships with Graphs

Discuss:

What does it mean for a Graph to be Undirected?

What is a real-world example of an Undirected Graph?

Turn and Talk: 2 mins

Whiteboards

Draw an undirected graph

Everyone Draws: 2 mins

Draw one of the examples of undirected graphs from the earlier discussion

Discuss:

What does it mean for a Graph to be Directed?

What is a real-world example of a Directed Graph?

Turn and Talk: 2 mins

Whiteboards

Draw a directed graph

Draw one of the real-world examples of directed graphs from the earlier discussion

Everyone Draws: 2 mins

Discuss:

What does it mean for a graph to be Weighted?

When might we use Weights on our edges?

Turn and Talk: 2 mins

Whiteboards

Draw an undirected, weighted graph

Use an undirected weighted graph to represent major cities in the US, then use their distance from each other as weights

Everyone Draws: 4 mins

Graph Notation

Here's how to read a textbook

This is a graph in mathematical notation

Represents the whole graph

Refers to the vertices in the graph

Refers to the edges in the graph

Graph Notation

Here's how to read a textbook

Is an ordered pair

Is an unordered set

Graph Notation

Here's how to read a textbook

a

b

c

d

an undirected cyclic graph

Unordered pairs mean that the edge is not explicitly one-way

Graph Notation

Here's how to read a textbook

a

b

c

d

a directed acyclic graph

{

}

Ordered pairs mean we are making explicit connections in one direction

Objectives

1. Use Graph Theory vocabulary
2. Use Graph Theory Notation
3. Define and describe the relationships depicted in a Graph

Whiteboards

Everybody Draws: 5 mins

Given:

Draw a graph such that:

Then use Graph Vocab to describe this graph

Whiteboards

Everybody Draws: 5 mins

Given:

Draw a graph such that:

Then use Graph Vocab to describe this graph