View by Category

Loading...

PPT – Graphs PowerPoint presentation | free to view

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Chapter 9

- Graphs

Sec 9.1

- Graphs and Graph Models

Simple Graph

- A simple graph G is an ordered pair (V,E),

consisting of a nonempty set of vertices V, and a

set of edges E, where each edge e?E corresponding

to a unique element in u,v u?V, v?V, u ? v

- For an edge e which corresponds to u,v the

vertices u and v are called the endpoints of this

edge

Multigraph

- A multigraph G is an ordered pair (V,E),

consisting of a nonempty set of vertices V, and a

set of edges E, where each edge e ? E corresponds

to some element in u,v u?V, v?V, u ? v - Note It is possible for two edges to correspond

to the same u,v , allowing for two or more

edges to have the same endpoints.

Pseudograph

- A pseudograph G is an ordered pair (V,E),

consisting of a set of vertices V, and a set of

edges E, where each edge e ? E corresponds to

some element in u,v u?V, v?V ? u,u

u?V - Note that an edge may correspond to u thus

allowing both endpoints to be the same. Such

edges are called loops.

Directed Graph

- A directed graph G is an ordered pair (V,E),

consisting of a set of vertices V, and a set of

directed edges E, where each edge e ? E

corresponds to an element in (u,v) u?V, v?V - Note Associating directed edges to ordered

pairs means that each edge has an associated

orientation or direction. If e corresponds to

(u,v), then u is called the initial vertex and v

is called the terminal vertex of e.

Simple Directed Graph

- A directed graph G is simple if it has no loops

and no multiple directed edges. So a simple

directed graph is an ordered pair (V,E),

consisting of a set of vertices V, and a set of

edges E, where each edge e ? E corresponds to at

a unique element in (u,v) u?V, v?V, u ? v

Directed Multigraph

- A directed multigraph G is an ordered pair (V,E),

consisting of a set of vertices V, and a set of

edges E, where each edge e ? E corresponds to

some element in (u,v) u?V, v?V - Note More than one edge may correspond to the

same (u,v) allowing multiple directed edges

between two vertices.

Summary Graph G (V,E)

- Simple Each edge ei ? a unique u,v, u ? v
- Multigraph Each edge ei ? some u,v, u ? v
- Pseudograph Each edge ei ? some u,v or u
- Simple Directed Graph Each edge ei ? a unique

(u,v), u ? v - Directed Multigraph Each edge ei ? some (u,v)

Homework

- Sec 9.1
- pg. 596 1, 3, 5, 7, 9, 15, 21

Sec 9.2

- Graph Terminology

Definitions

- Adjacent Vertices Two vertices, u and v in an

undirected graph G are adjacent (or neighbors) if

there is an edge e in E which corresponds to

u,v - Incident If edge e corresponds to u,v, the

edge e is called incident with the vertices u and

v. - Connected If edge e corresponds to u,v, the

vertices u and v are said to be connected. - Endpoints If edge e corresponds to u,v, the

vertices u and v are said to be endpoints of e

Degree of a Vertex

- Deg(v) The degree of a vertex in an undirected

graph G is the number of edges incident with it.

(with the convention that a loop at v contributes

twice to the degree of that vertex) - The Handshaking Theorem. Let G (V,E) be an

undirected graph. The sum of the degrees of all

the vertices of the graph is equal to 2E. - Theorem An undirected graph has an even number

of vertices of odd degree.

Definition Directed Graphs

- Let G is a directed graph with an edge e

corresponding to (u,v), - Adjacent Then vertex u is said to be adjacent to

v and vertex v is said to be adjacent from u. - Initial and Terminal Vertices Vertex u is called

the initial vertex and vertex v is called the

terminal vertex of edge e. - For a loop in a directed graph, the single vertex

is considered both the initial and the terminal

vertex.

In-degree and Out-degree

- Deg-(v) In a directed graph G, the in-degree of

vertex v is the number of edges with v as their

terminal vertex. - Deg(v) In a directed graph G, the out-degree

of vertex v is the number of edges with v as

their initial vertex. - Theorem For any directed graph G (V,E), If

the summation is taken over all v ? V, ?deg(v)

?deg-(v) E2e, e of edges

Some Special Simple Graphs

- Complete A simple graph G is complete if it

contains an edge between each pair of vertices.

A complete graph of n vertices is denoted by Kn. - Bipartite A simple graph G (V,E) is called

bipartite if the vertices in V can be partitioned

into two disjoint sets V1 and V2 so that each

edge e ? E has exactly one endpoint in V1 and the

other in V2.

Results

- Theorem A simple graph is bipartite if and only

if it is possible to assign one of two different

colors to each vertex of the graph so that no two

adjacent vertices are assigned the same color. - Complete Bipartite A bipartite graph whose

vertices can partitioned into two sets of m and n

vertices respectively with an edge between every

pair of vertices in the two sets is called a

complete bipartite graph, denoted by Km,n.

Application Job Assignments

- Suppose that there are m employees in a group and

j different jobs that need to be done where m ?

j. Each employee is trained to do one or more of

there j jobs. - Vertices
- Employees
- Jobs
- Edges connect employees to jobs

Homework

- Sec 9.2
- pg. 608 1, 3, 5, 7, 9, 17, 21, 23

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2015 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2015 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Graphs" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Astate University and other schools with their online training by sharing educational presentations for free