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

Discrete Mathematics

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 Lines, Nodes and Vertices, Arcs and Edges Or Links, Set of Two Nodes, Bridges of Koenigsberg, Graph Problem, Existence of Eulerian Cycle
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Chapter 10 Graphs and Trees

This handout:

• Terminology of Graphs

• Eulerian Cycles

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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:

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Graphs in our daily lives

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

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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

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Terminology of Graphs: Paths

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

Example: • Walks are paths that can repeat nodes and arcs.

a b

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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

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The Bridges of Koenigsberg

A

D

C B

1 2

4

3

7

6 5

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

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The Bridges of Koenigsberg

A

D

C B

1 2

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3

7

6 5

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

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The Bridges of Koenigsberg

1 2

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3

7

6 5

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

A

C

D

B

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The Bridges of Koenigsberg

1 2

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3

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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

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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.

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Existence of Eulerian Cycle

1 2

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A

C

D

B

8

9

The degree of a node is the number of incident arcs

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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).

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