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Introduction to Graphs

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Introduction to Graphs

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    1. 6.1 Introduction to Graphs 1 Introduction to Graphs Section 6.1

    2. 6.1 Introduction to Graphs 2 Konigsberg Bridge Problem Write the rules more formally Start on any land mass Cross each bridge once and only once Can you do it?Write the rules more formally Start on any land mass Cross each bridge once and only once Can you do it?

    3. 6.1 Introduction to Graphs 3 Your solution Did it every time Did it at least once Can’t seem to do it 33

    4. 6.1 Introduction to Graphs 4 Konigsberg Bridge (8th bridge) Have student present solution. Then make them start at an even vertex.Have student present solution. Then make them start at an even vertex.

    5. 6.1 Introduction to Graphs 5 Your solution Did it every time Did it at least once Can’t seem to do it 22

    6. 6.1 Introduction to Graphs 6 Konigsberg Bridge (9th bridge)

    7. 6.1 Introduction to Graphs 7 Your solution Did it every time Did it at least once Can’t seem to do it 11

    8. 6.1 Introduction to Graphs 8 3 Cases for Konigsberg 7 Bridges (Non-traversable) 2. 8 Bridges (Euler Path) 3. 9 Bridges (Euler circuit) Define Euler Circuit (EC) etc. hereDefine Euler Circuit (EC) etc. here

    9. 6.1 Introduction to Graphs 9 Euler’s View Hand draw Euler's graph What mathematicians do Bring in clay: “donut = coffee cup” Hand draw Euler's graph What mathematicians do Bring in clay: “donut = coffee cup”

    10. 6.1 Introduction to Graphs 10 You try one Practice moving from map to graph.Practice moving from map to graph.

    11. 6.1 Introduction to Graphs 11 Definition - Graph A graph is any collection of Dots (Vertices) Arcs/Lines (Edges) that join the points Notes: The singular of “vertices” is “vertex” Notes: The singular of “vertices” is “vertex”

    12. 6.1 Introduction to Graphs 12 Two Special Cases

    13. 6.1 Introduction to Graphs 13 The Degree of a Vertex The degree of a vertex is the number of times the vertex is touched by an edge Switching gears Emphasize “leaving-entering” here. Ask class for answersSwitching gears Emphasize “leaving-entering” here. Ask class for answers

    14. 6.1 Introduction to Graphs 14 This graph has 6 edges, 4 vertices (exactly 2 of which are odd) 4 edges, 6 vertices (all of which are odd) 6 edges, 4 vertices (all of which are odd) 4 edges, 4 vertices (exactly 2 of which are odd) 44

    15. 6.1 Introduction to Graphs 15 This graph has 8 edges, 5 vertices (none of which are odd) 8 edges, 5 vertices (exactly 2 of which are odd) 8 edges, 5 vertices (exactly 4 of which are odd) 8 edges, 5 vertices (all of which are odd) 44

    16. 6.1 Introduction to Graphs 16 Draw a graph with 4 vertices (all odd) and 5 edges 4 vertices (all odd) and 3 edges (no loops) The “reverse” problem – frequently asked in mathematics Box with both diagonals Paw print The “reverse” problem – frequently asked in mathematics Box with both diagonals Paw print

    17. 6.1 Introduction to Graphs 17 Draw a graph with 3 vertices (exactly 1 even) and 4 edges 3 vertices (exactly 1 odd) and 4 edges Ice cream cone Can’t Be Done – presages future results – explain why – maybe make a homework problem for bonusIce cream cone Can’t Be Done – presages future results – explain why – maybe make a homework problem for bonus

    18. 6.1 Introduction to Graphs 18 All vertices of a graph could be odd True False

    19. 6.1 Introduction to Graphs 19 All vertices of a graph could be even True False

    20. 6.1 Introduction to Graphs 20 End of 6.1

    21. 6.1 Introduction to Graphs 21 GRAPHS Lots of explorations. Discovery. Hit theory a bit harder. Discover sum og degrees in agrpah is even., etc

    22. 6.1 Introduction to Graphs 22 Entirely animated Entirely animated

    23. 6.1 Introduction to Graphs 23

    24. 6.1 Introduction to Graphs 24

    25. 6.1 Introduction to Graphs 25 Leonard Euler (“Oiler”) 1706 - 1783 Swiss—Dominant mathematical figure of the 18th c. Authored 500+ books and articles in his lifetime “Euler calculates just as men breath and as eagles sustain themselves in air” He and his wife Katharina had 13 children although only five survived their infancy Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet Blind in right eye at age 29. Completely blind at 60 Swiss—Dominant mathematical figure of the 18th c. Authored 500+ books and articles in his lifetime “Euler calculates just as men breath and as eagles sustain themselves in air” He and his wife Katharina had 13 children although only five survived their infancy Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet Blind in right eye at age 29. Completely blind at 60

    26. 6.1 Introduction to Graphs 26 Non-traversable

    27. 6.1 Introduction to Graphs 27 Euler Path

    28. 6.1 Introduction to Graphs 28 Euler Circuit

    29. 6.1 Introduction to Graphs 29 Genealogy Examples of graphs you may seenExamples of graphs you may seen

    30. 6.1 Introduction to Graphs 30 Constellations

    31. 6.1 Introduction to Graphs 31 The nine members of the Supreme Court in 1973 were Justices Marshall, Burger, White, Blackman, Powell, Rhenquist, Brennan, Douglas, and Stewart. The conservative block of Burger, Rhenquist, Powell and Blackman voted together on 70+ percent of the votes. Justice White joined with Justice Blackman 70+ percent of the time. The liberal block of Brennan, Douglas, and Marshall voted together 70+ percent of the time. Justice Stewart was the maverick who voted with no one 70+ percent of the time Ugh. Lots of information. hard to dissect 79 words Versus ...Ugh. Lots of information. hard to dissect 79 words Versus ...

    32. 6.1 Introduction to Graphs 32 Political Science

    33. 6.1 Introduction to Graphs 33 Meta - Material

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