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

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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...
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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: • Walksare 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

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7

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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

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

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The Bridges of Koenigsberg

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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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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

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6 5

A

C

D

B

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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

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A

C

D

B

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The degree of a node is the number of incident arcs

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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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