View by Category

The presentation will start after a short

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

(15 second) video ad from one of our sponsors.

Hot tip: Video ads won’t appear to registered users who are logged in. And it’s free to register and free to log in!

Loading...

PPT – Discrete Mathematics Lecture 9 PowerPoint presentation | free to download - id: 23822-NjNlZ

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Discrete MathematicsLecture 9

Alexander Bukharovich New York University

Graphs

- Graph consists of two sets set V of vertices and

set E of edges. - Terminology endpoints of the edge, loop edges,

parallel edges, adjacent vertices, isolated

vertex, subgraph, bridge edge - Directed graph (digraph) has each edge as an

ordered pair of vertices

Special Graphs

- Simple graph is a graph without loop or parallel

edges - A complete graph of n vertices Kn is a simple

graph which has an edge between each pair of

vertices - A complete bipartite graph of (n, m) vertices

Kn,m is a simple graph consisting of vertices,

v1, v2, , vm and w1, w2, , wn with the

following properties - There is an edge from each vertex vi to each

vertex wj - There is no edge from any vertex vi to any vertex

vj - There is no edge from any vertex wi to any vertex

wj

The Concept of Degree

- The degree of a vertex deg(v) is a number of

edges that have vertex v as an endpoint. Loop

edge gives vertex a degree of 2 - In any graph the sum of degrees of all vertices

equals twice the number of edges - The total degree of a graph is even
- In any graph there are even number of vertices of

odd degree

Exercises

- Two jugs have capacities of of 3 and 5 gallons.

Can you use these jugs to measure out exactly one

gallon? - Bipartite graphs
- Complement of a graph
- What is the relationship between the number of

edges between a graph and its complement - Can there be a simple graph that has vertices

each of different degree? - In a group of two or more people, must there be

at least two people who are acquainted with the

same number of people?

Paths and Circuits

- A walk in a graph is an alternating sequence of

adjacent vertices and edges - A path is a walk that does not contain a repeated

edge - Simple path is a path that does not contain a

repeated vertex - A closed walk is a walk that starts and ends at

the same vertex - A circuit is a closed walk that does not contain

a repeated edge - A simple circuit is a circuit which does not have

a repeated vertex except for the first and last

Connectedness

- Two vertices of a graph are connected when there

is a walk between two of them. - The graph is called connected when any pair of

its vertices is connected - If graph is connected, then any two vertices can

be connected by a simple path - If two vertices are part of a circuit and one

edge is removed from the circuit then there still

exists a path between these two vertices - Graph H is called a connected component of graph

G when H is a subgraph of G, H is connected and H

is not a subgraph of any bigger connected graph - Any graph is a union of connected components

Euler Circuit

- Euler circuit is a circuit that contains every

vertex and every edge of a graph. Every edge is

traversed exactly once. - If a graph has Euler circuit then every vertex

has even degree. If some vertex of a graph has

odd degree then the graph does not have an Euler

circuit - If every vertex of a graph has even degree and

the graph is connected then the graph has an

Euler circuit - A Euler path is a path between two vertices that

contains all vertices and traverces all edge

exactly ones - There is an Euler path between two vertices v and

w iff vertices v and w have odd degrees and all

other vertices have even degrees

Hamiltonian Circuit

- Hamiltonian circuit is a simple circuit that

contains all vertices of the graph (and each

exactly once) - Traveling salesperson problem

Exercises

- For what values of m and n, does the complete

bipartite graph of (m, n) vertices have an Euler

circuit, a Hamiltonian circuit? - What is the maximum number of edges a simple

disconnected graph with n vertices can have? - Show that a graph is bipartite iff it does not

have a circuit with an odd number of edges

Matrix Representation of a Graph

- Adjacency matrix
- Undirected graphs and symmetric matrices
- Number of walks of a particular length between

two vertices

Isomorphism of Graphs

- Two graphs G (V, E) and G (V, E) are

called isomorphic when there exist two bijective

functions g V ? V and h E ? E so that if v

is an endpoint of e iff g(v) is an endpoint of

h(e) - Property P is called an isomorphic invariant when

given any two isomorphic graphs G and G, G has

property P, then G has property P as well - The following properties are isomorphic

invariants - Number of vertices, number of edges
- Number of vertices of a particular degree
- Connectedness
- Possession of a circuit of a particular length
- Possession of Euler circuit, Hamiltonian

circuit22222

Trees

- Connected graph without circuits is called a tree
- Graph is called a forest when it does not have

circuits - A vertex of degree 1 is called a terminal vertex

or a leaf, the other vertices are called internal

nodes - Decision tree
- Syntactic derivation tree
- Any tree with more than one vertex has at least

one vertex of degree 1 - Any tree with n vertices has n 1 edges
- If a connected graph with n vertices has n 1

edges, then it is a tree

Rooted Trees

- Rooted tree is a tree in which one vertex is

distinguished and called a root - Level of a vertex is the number of edges between

the vertex and the root - The height of a rooted tree is the maximum level

of any vertex - Children, siblings and parent vertices in a

rooted tree - Ancestor, descendant relationship between vertices

Binary Trees

- Binary tree is a rooted tree where each internal

vertex has at most two children left and right.

Left and right subtrees - Full binary tree
- Representation of algebraic expressions
- If T is a full binary tree with k internal

vertices then T has a total of 2k 1 vertices

and k 1 of them are leaves - Any binary tree with t leaves and height h

satisfies the following inequality t ? 2h

Spanning Trees

- A subgraph T of a graph G is called a spanning

tree when T is a tree and contains all vertices

of G - Every connected graph has a spanning tree
- Any two spanning trees have the same number of

edges - A weighted graph is a graph in which each edge

has an associated real number weight - A minimal spanning tree (MST) is a spanning tree

with the least total weight of its edges

Finding Minimal Spanning Tree

- Kruskals algorithm
- Prims algorithm

Exercises

- If all edges in a graph have different weights,

does this graph have a unique MST?

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 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Discrete Mathematics Lecture 9" 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 Nyu University and other schools with their online training by sharing educational presentations for free