Graphs Eulerian and Hamiltonian Applications Graph layout software. Graphs. SET07106 Mathematics for Software Engineering
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1 Graphs SET76 Mathematics for Software Engineering School of Computing Edinburgh Napier University Module Leader: Uta Priss 2 Copyright Edinburgh Napier University Graphs Slide /3
2 Outline Graphs Eulerian and Hamiltonian Applications Graph layout software Copyright Edinburgh Napier University Graphs Slide 2/3
3 Binary relations a c d z e a c d e z { (a,5), (e,5),(e,4),(z,7) } Copyright Edinburgh Napier University Graphs Slide 3/3
4 A binary relation can be represented as a graph Students Subjects Tim Mary John Sue Pete maths biology computing science John Pete computing Tim Sue maths science biology Mary Copyright Edinburgh Napier University Graphs Slide 4/3
5 A graph consists of nodes (vertices) and edges John Pete computing Tim Sue maths science biology Mary Copyright Edinburgh Napier University Graphs Slide 5/3
6 Undirected and directed graphs John Pete computing Tim maths biology Sue science Mary John Pete computing Tim maths biology Sue science Mary Copyright Edinburgh Napier University Graphs Slide 6/3
7 The complete graphs for n 6 Copyright Edinburgh Napier University Graphs Slide 7/3
8 Exercises Draw the complete graph with 7 nodes. How many edges does every node have in a complete graph? How many edges does a complete graph have? Copyright Edinburgh Napier University Graphs Slide 8/3
9 A graph can be connected or disconnected John Pete computing Tim Sue maths science biology Mary Copyright Edinburgh Napier University Graphs Slide 9/3
10 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 science Mary Copyright Edinburgh Napier University Graphs Slide /3
11 Exercise Are there any complete graphs which are bipartite? Copyright Edinburgh Napier University Graphs Slide /3
12 Can you draw this figure... in one go without starting and stopping in between? Copyright Edinburgh Napier University Graphs Slide 2/3
13 Eulerian path Each edge is visited once. Copyright Edinburgh Napier University Graphs Slide 3/3
14 Seven bridges of Königsberg (735) Walk through the city and cross each bridge exactly once. Copyright Edinburgh Napier University Graphs Slide 4/3
15 Seven bridges of Königsberg 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. Copyright Edinburgh Napier University Graphs Slide 5/3
16 Hamiltonian path Each node is visited once. Copyright Edinburgh Napier University Graphs Slide 6/3
17 Travelling salesman problem What is the shortest path for a salesman to visit a given set of cities? A problem of optimisation 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. Copyright Edinburgh Napier University Graphs Slide 7/3
18 Sudoku 2 2 A bipartite graph. Copyright Edinburgh Napier University Graphs Slide 8/3
19 Sudoku A tripartite graph. Copyright Edinburgh Napier University Graphs Slide 9/3
20 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. Copyright Edinburgh Napier University Graphs Slide 2/3
21 Small-world effect Every node can be reached by every other node by a short path. Examples: Social networks Internet Road maps Electric power grids Copyright Edinburgh Napier University Graphs Slide 2/3
22 Which of these have a small-world effect? clique hub resource. Copyright Edinburgh Napier University Graphs Slide 22/3
23 Small-world networks Small world effect: small average node-to-node distance (shortest path length) 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. Copyright Edinburgh Napier University Graphs Slide 23/3
24 Other graph applications Links between webpages Sitemaps Flow charts, UML diagrams Database schemata, ER diagrams Class hierarchies Web site paths traversed by users XML tree structures and DTDs Copyright Edinburgh Napier University Graphs Slide 24/3
25 Picture of a Null Graph: Copyright Edinburgh Napier University Graphs Slide 25/3
26 Harary and Read (973): 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. Copyright Edinburgh Napier University Graphs Slide 26/3
27 Harary and Read (973): 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 Copyright Edinburgh Napier University Graphs Slide 26/3
28 Graphs are special kinds of vector graphics Moving or removing a node affects its edges. Graph editors provide graph layout algorithms. John Susan knows knows knows Paul friend Robert friend Mary friend knows Paul knows John friend Robert knows Mary Susan Copyright Edinburgh Napier University Graphs Slide 27/3
29 Graph layout software/editors TouchGraph, spring embedder algorithms Java toolkits: Prefuse,... Graphviz: open source graph visualisation software Copyright Edinburgh Napier University Graphs Slide 28/3
30 Graphviz Directed and undirected graphs. Graph layouts: hierarchies, spring, radial, circular. Simple text-based format (called dot format ). APIs for different programming languages exist. Many output formats: gif, jpg, svg, pdf,... Copyright Edinburgh Napier University Graphs Slide 29/3
31 The dot format digraph names { node [label= John ] node [label= Mary ] node2 [label= Paul ] node -> node node -> node2 node2 -> node } John Paul Mary Copyright Edinburgh Napier University Graphs Slide 3/3
32 Mary John Paul Sue Bob Dora Bart Lisa Jim Rita Hierarchical, radial, circular layouts: Rita Sue Lisa Mary Jim Bart Bob John Paul John Sue Bob Dora Paul Dora Bart Lisa Mary Jim Rita Spring layouts: Bob Sue Bob Sue Rita Paul Mary Lisa Mary Paul John Dora Bart John Dora Bart Lisa Rita Jim Jim Copyright Edinburgh Napier University Graphs Slide 3/3
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