Graphs Eulerian and Hamiltonian Applications Graph layout software
Graphs
SET07106 Mathematics for Software Engineering
School of Computing Edinburgh Napier University Module Leader: Uta Priss
2010
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Outline
Graphs
Eulerian and Hamiltonian
Applications
Graph layout software
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Binary relations
5 1 3 45 7 9 a e a 4 c 7 d d z 9 3 e c 1 z
{ (a,5), (e,5),(e,4),(z,7) }
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A binary relation can be represented as a graph
Students Subjects
Tim maths Mary biology John computing Sue Pete science
John
Tim biology Petecomputing maths
Sue Mary science
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A graph consists of nodes (vertices) and edges
John
Tim biology Petecomputing maths
Sue Mary science
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Undirected and directed graphs
John
Tim biology Petecomputing maths
Sue Mary science
John
Tim biology Petecomputing maths
Sue Mary science
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The complete graphs for n ≤ 6
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Exercises
I Draw the complete graph with 7 nodes.
I How many edges does every node have in a complete graph?
I How many edges does a complete graph have?
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A graph can be connected or disconnected
John
Tim biology Petecomputing maths
Sue Mary science
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A bipartite graph has two sets of nodes
Edges are from one set of nodes to the other. There are no edges within the same set of nodes.
Tim
Pete computing maths
Sue Mary science
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Exercise
Are there any complete graphs which are bipartite?
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Can you draw this figure ...
in one go without starting and stopping in between?
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Eulerian path
Each edge is visited once.
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Seven bridges of K¨onigsberg (1735)
Walk through the city and cross each bridge exactly once.
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Seven bridges of K¨onigsberg
This is graph problem.
Euler asserted that a graph has a Eulerian path if the graph is connected and has either no or two nodes with an odd number of edges.
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Hamiltonian path
Each node is visited once.
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Travelling salesman problem
What is the shortest path for a salesman to visit a given set of cities? A problem of
I optimisation
I planning A graph where the edges are labelled with the distances between the cities. Among all the Hamiltonian paths, find the one which minimises distances.
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Sudoku
1 2
2 1
A bipartite graph.
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Sudoku
1 2 3
2 3 1 3 1 2
1 2 3
4 5 6
7 8 9
A tripartite graph.
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6 degrees of separation
Do you know the Prime Minister? Do you know someone who knows the Prime Minister? Do you know someone who knows someone who knows the PM? ...
The claim: everybody is connected to everybody else by at most 6 degrees of separation. =⇒ It is a small world.
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Small-world effect
Every node can be reached by every other node by a short path.
Examples:
I Social networks
I Internet
I Road maps
I Electric power grids
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Which of these have a small-world effect?
clique
hub resource .
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Small-world networks
I Small world effect: small average node-to-node distance (shortest path length)
I Clustering coefficient that is larger than the clustering coefficient of a random graph with the same number of nodes and edges. Having a large clustering coefficient means that “the people you know also know each other”.
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Other graph applications
I Links between webpages
I Sitemaps
I Flow charts, UML diagrams
I Database schemata, ER diagrams
I Class hierarchies
I Web site paths traversed by users
I XML tree structures and DTDs
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Picture of a Null Graph:
Copyright Edinburgh Napier University Graphs Slide 25/31 From Wolfram MathWorld: “the only good null graph is a dead null graph”
Graphs Eulerian and Hamiltonian Applications Graph layout software
Harary and Read (1973): Is the Null Graph a Pointless Concept?
“The graph with no points and no lines is discussed critically. ... No conclusion is reached.”
The question is not whether it exists, but whether there is a point in it.
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Harary and Read (1973): Is the Null Graph a Pointless Concept?
“The graph with no points and no lines is discussed critically. ... No conclusion is reached.”
The question is not whether it exists, but whether there is a point in it.
From Wolfram MathWorld: “the only good null graph is a dead null graph”
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Graphs are special kinds of vector graphics
I Moving or removing a node affects its edges.
I Graph editors provide graph layout algorithms.
John Susan Robert Mary friend friend Robert knows knows friend knows Paul knows Paul knows knows John friend
Mary Susan
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Graph layout software/editors
I TouchGraph, spring embedder algorithms
I Java toolkits: Prefuse, ...
I Graphviz: open source graph visualisation software
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Graphviz
I www.graphviz.org
I Directed and undirected graphs.
I Graph layouts: hierarchies, spring, radial, circular.
I Simple text-based format (called “dot format”).
I APIs for different programming languages exist.
I Many output formats: gif, jpg, svg, pdf, ...
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The “dot format” digraph names { node0 [label=’’John’’] node1 [label=’’Mary’’] node2 [label=’’Paul’’] node0 -> node1 node0 -> node2 node2 -> node1 }
John
Paul
Mary
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Hierarchical, radial, circular layouts:
Rita
Rita Sue Lisa
Bart Lisa
Mary Dora
Jim Jim Bart Bob John Mary
Paul Paul John John Sue Bob Dora
Bart Paul Dora Lisa
Bob Jim Mary Rita
Sue
Spring layouts:
Bob Sue Bob
Sue Rita
Paul Mary Paul Mary Lisa John John Dora Bart Lisa Rita Dora Bart
Jim Jim
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