Discrete Mathematics

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Discrete Mathematics

• Graphs and Networks

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Graphs and networks

• One of the most well known graphs is the map of
  the London Underground.

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Definitions

• Nodes - the vertices of the graph
• Arcs or edges the lines joining the arcs

Loop an arc connecting a node to itself
Multiple arcs two or more arcs joining the same
nodes
Simple graph a graph without loops or multiple
arcs
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Definitions

• A sub-graph is part of a graph.

(a) and (b) are sub-graphs of graph (c)
Graph (c) is an example of a complete graph which
is a graph in which each node is connected to
every other node by precisely one arc. The
notation used is Kn where n is the number of
nodes Graph (c) is K3 Graphs (d) and (e) are
further examples.
(e)
K5
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Definitions

• Bipartite graphs

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The Königsberg bridge problem.

• The city of Königsberg has seven bridges
  crossing the River Pregel. Can you devise a
  circular route which crosses each bridge once
  only.

The problem can be represented by a graph with
four nodes (land masses) and seven arcs (bridges).
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Leonhard Euler

• Euler, an 18th Century mathematician, is known
  as the father of graph theory. He was supposed
  to have devised one of his theories based on the
  Königsberg bridge problem.
• He realised that for a circular route
• a land mass is entered the same number of times
  it is left.
• each land mass would need to be linked to the
  other land masses by an even number of bridges.
• Therefore each node has to have an even
• number of arcs.
• The nodes in this problem have orders
• 3, 3, 3 and 5. (the order is the number
• of arcs meeting at the node)
• Therefore no circular tour is possible.

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More definitions

• A trail or route is a continuous sequence of
  arcs leading from one node to another.
• A path is a trail in which no node is passed
  more than once.
• A closed trail is one in which the starting node
  and finishing node are the same.
• A cycle is a closed trail where only the first
  and last nodes are the same.
• A connected graph is one in which, for any two
  nodes, a path can be found connecting them.

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Eulerian Graphs

• A Eulerian Graph is a graph which has a closed
  trail containing every arc precisely once.

A closed trail exists A B C D B E
A It is therefore Eulerian
A closed trail does not exist (it would have to
contain BC twice) It is not Eulerian
A connected graph is Eulerian if, and only if,
every node has an even order.
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Semi-Eulerian graphs

• Can you draw this diagram
• without taking the pencil
• off the paper?

To do this you need to draw a trail which is not
necessarily closed. It is a semi-Eulerian graph.
i.e. there is a trail which is not closed that
contains every arc precisely once.
A connected graph is semi-Eulerian if, and only
if, precisely two nodes have an odd order.
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Practice questions

• Exercise 2A page 21

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Planar graphs

• A planar graph is one which can be drawn in
  one plane in such a way that arcs only meet at
  nodes and do not cross.
• e.g.

The first graph can be redrawn so that no arcs
cross, so it is planar.
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Eulers relationship

• Euler noticed a relationship between the
  regions (R), the nodes (N) and the arcs (A) of a
  planar graph.
• R N A 2
• For a graph to be planar it must obey Eulers
  relationship.

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Trees

• A connected graph with no cycles is a tree.

Features of trees Every connected graph contains
a tree which contains every node of the graph. A
family of saturated hydro-carbons can be drawn as
trees.
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Practice questions

• Exercise 2B page 25

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Network problems
Directed graphs The arcs may have arrows
indicating a direction representing one-way
streets, flow of a manufacturing process etc.
These graphs are directed graphs or digraphs.

• Weighted graphs
• A numerical value may be attached to an arc
  representing distance, cost etc. This value is
  called the weight of the arc.

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Matrices
A B C D
A - 2 4 -
B 2 - 3 -
C 4 3 - 1
D - - 1 -
An undirected graph will have a symmetrical matrix
To
A directed graph is not symmetrical. It is
important to indicate the direction in the matrix.
A B C D
A - 2 4 -
B - - 3 -
C 5 - - 1
D - - 1 -
From
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Practice questions

• Cambridge Advanced Level Maths
• Discrete Mathematics 1
• Chapter 2
• Exercise 2 A page 21
• Exercise 2 B page 25
• Exercise 2 C page 28
• Miscellaneous Exercise 2 page 29