Day 13, Mat115H

Announcements:
- Your papers are going on-line. Thanks to those who've gotten us started.
  I'd like these submitted by Friday, please both your hurting/healthy paper, and your group presentation or paper).
- A few articles from the New York Times to bring to your attention:
  - On cereal
  - On access: How Segregation Destroys Black Wealth
- Your homework for today is due:
  Find the "five (or so) dimensions of mental well-being", and then five (or so) short yes/no (or easily answered) questions that people can answer through a survey that we might administer to folks in Newport.
  Type up and justify your work!
  Your questions will also be posted, once revised. But those can wait.
Today we have two presentations before us: we'll hear from the Kara/Haley/Kensey group, and Marisa and Alli will also present.
- Psychology and the hope metric -- Kara (and Kensey and Haley)
- Income inequality -- Marisa and Alli
- Done: Photography -- Betsy and Cassidy
- Done: History -- Emily and Sandra
- Done: Public art and Little Free Libraries -- Melody, Grace, and Robin
Last time I talked about the kind of mathematics that we'll be studying the rest of the semester. Today we begin a unit on what I call "trees and graphs". And we'll start at the very beginning: with a drinking game.
Question of the Day:

How did a drinking game lead to new mathematics?
- The "Bridges of Konigsberg" problem started graph theory: "is it possible to set off and walk around Konigsberg crossing each bridge exactly once?" (p. 116 of this reading on graphs):
  Try it yourself. Here is the problem abstracted:
  
  "It being customary among the townsmen, after some hours in the Gasthaus [on the island], to attempt to walk the bridges, many have returned for more refreshment claiming success. However, none have been able to repeat the feat by the light of day."
  - Solved by Euler (1735), the beginning of graph theory.
    - Graphs are made up of vertices (points) and edges that connect the vertices. More formally, a graph is
      1. A collection of points, called vertices.
      2. A collection of edges, each of which connects two vertices.
    - The degree of a vertex is the number of edges coming into it.
  - Euler's solution:
    - Konigsberg: To be traversable, there must be at most two vertices of odd degree.
    - Euler noticed the so-called "hand-shaking theorem": in any graph, the number of vertices with odd degree must be even.
  - An alternative solution -- thanks to spiked math
- You've probably been making graphs for a long time.... (the pentagram)
- Definition of a graph: a graph is
  - a collection of vertices,
  - a collection of edges, and
  - a set of rules, one for each edge, telling how each edge is connected to its pair of vertices.
- Graphs don't change by bending edges, but breaking them or detaching edges from their vertices (and hence creating new vertices) gives new graphs.
- A graph is simple if it doesn't have any loops (edges connected from a vertix to itself), or multiple edges with the same two vertices.
- Exercise: draw all the simple graphs with three vertices.
  - A cycle is a route from a node back to itself that doesn't retrace steps.
  - One of these graphs is a tree: a graph that has no cycles.
    - Where have we seen trees used before?
- Exercise: you draw all the simple graphs with four vertices.
  - Have you ever seen any of these before? Could we give any of them names?
- Planar graphs are graphs that can be drawn such that no two edges intersect.
- Complete graphs are simply graphs with connections between every pair of vertices (but no loops).
- Exercise: Let's see if we can show that the complete graph with five vertices is not planar:
  - We can show that the complete graph with four vertices is planar.
  - Why isn't the five graph planar? What happens?
  - The three utilities and three houses graph is not planar, either:
  - Any non-planar graph has a copy of one of these two graphs in it somewhere, as a subgraph:

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