Graphs and Trees - Discrete Mathematics - Lecture Slides, Slides for Discrete Mathematics. Alagappa University
aslesha
aslesha27 April 2013

Graphs and Trees - Discrete Mathematics - Lecture Slides, Slides for Discrete Mathematics. Alagappa University

PDF (184.5 KB)
12 pages
1000+Number of visits
Description
During the study of discrete mathematics, I found this course very informative and applicable.The main points in these lecture slides are:Graphs and Trees, Terminology of Graphs, Eulerian Cycles, Set of Points, Set of Li...
20points
Download points needed to download
this document
Download the document
Preview3 pages / 12
PowerPoint Presentation

Chapter 10 Graphs and Trees

This handout:

• Terminology of Graphs

• Eulerian Cycles

Docsity.com

Terminology of Graphs • A graph (or network) consists of

– a set of points – a set of lines connecting certain pairs of the points.

The points are called nodes (or vertices). The lines are called arcs (or edges or links). • Example:

Docsity.com

Graphs in our daily lives

• Transportation • Telephone • Computer • Electrical (power) • Pipelines • Molecular structures in biochemistry

Docsity.com

Terminology of Graphs • Each edge is associated with a set of two nodes, called its endpoints. Ex:a and b are the two endpoints of edge e • An edge is said to connect its endpoints. Ex: Edge e connects nodes a and b. • Two nodes that are connected by an edge are called adjacent. Ex: Nodes a and b are adjacent.

a

b

c e f

Docsity.com

Terminology of Graphs: Paths

• A path between two nodes is a sequence of distinct nodes and edges connecting these nodes.

Example: • Walksare paths that can repeat nodes and arcs.

a b

Docsity.com

A little history: the Bridges of Koenigsberg

• “Graph Theory” began in 1736 • Leonhard Eüler

– Visited Koenigsberg – People wondered whether it is possible to take a

walk, end up where you started from, and cross each bridge in Koenigsberg exactly once

Docsity.com

The Bridges of Koenigsberg

A

D

C B

12

4

3

7

65

Is it possible to start in A, cross over each bridge exactly once, and end up back in A?

Docsity.com

The Bridges of Koenigsberg

A

D

C B

12

4

3

7

65

Translation into a graph problem: Land masses are “nodes”.

Docsity.com

The Bridges of Koenigsberg

12

4

3

7

65

Translation into a graph problem: Bridges are “arcs.”

A

C

D

B

Docsity.com

The Bridges of Koenigsberg

1 2

4

3

7

6 5

Is there a “walk” starting at A and ending at A and passing through each arc exactly once? Such a walk is called an eulerian cycle.

A

C

D

B

Docsity.com

Adding two bridges creates such a walk

A, 1, B, 5, D, 6, B, 4, C, 8, A, 3, C, 7, D, 9, B, 2, A

1 2

4

3

7

6 5

A

C

D

B

8

9

Here is the walk.

Note: the number of arcs incident to B is twice the number of times that B appears on the walk.

Docsity.com

Existence of Eulerian Cycle

1 2

4

3

7

6 5

A

C

D

B

8

9

The degree of a node is the number of incident arcs

6

4

4

4

Theorem. An undirected graph has an eulerian cycle if and only if (1) every node degree is even and (2) the graph is connected (that is, there is a path from each node to each other node).

Docsity.com

comments (0)
no comments were posted
be the one to write the first!
This is only a preview
See and download the full document
Docsity is not optimized for the browser you're using. In order to have a better experience please switch to Google Chrome, Firefox, Internet Explorer 9+ or Safari! Download Google Chrome