Day 21, MAT115

• 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 hand-shaking theorem: in any graph, the number of vertices with odd degree must be even.

• An alternative solution -- thanks to spiked math
• Which of the Platonic solid graphs can be traversed without backtracking? (here they are)

• The Platonic solid graphs
• Graphs of the dual solids from the Platonic solid graphs
• Trees
• Graph that are
  • weighted: each edge (connection between vertices) has a number attached to it:
    

  • labelled:
    • The graph above has vertices labelled.
    • Sometimes the edges are labelled too.
    • Sometimes a label indicates a state of being (e.g. LAX); sometimes a label indicates the result of an action (e.g. a coin toss).
    • Examples:
      1. Fibonacci's Bunny Tree
      2. Hexstat Probability generator

Hence, for example,

1. the graph of the tetrahedron and the Mercedes Benz symbol are isomorphic as graphs
2. the graphs of the regular pentagons, star and convex, are isomorphic.

Exercise: draw all the distinct 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.
• Duality -- again! The dual of a graph is obtained by
  • throwing out the connections that exist between vertices, and
  • making any connections that don't exist.
  Or, as I like to say, "if you have an edge, break it; if you don't have an edge, make it."

  We can use duality to save ourselves some work.

Exercise: draw all the simple graphs with four vertices.

• Have you ever seen any of these before? Could we give any of them names?

Here are all simple graphs with five vertices -- there are 34!

• We can show that the complete graph with four vertices is planar.
• Why not? What happens?
• The three utilities and three houses graph is not planar:
• Any non-planar graph has a copy of one of these two graphs in it somewhere, as a subgraph:

Standard diagram of the Borromean rings

A realization of the Borromean rings as ellipses

Coat of arms (of the municipality of Hallsberg, Sweden) showing padlocks in Borromean rings configuration

(Thanks Wikipedia!)

• Use symmetry!
• How far off golden are we?
• Do you see the icosahedron?

• I always think of them in the context of this quote:

  "We must all hang together, or assuredly we shall all hang separately." Benjamin Franklin, at the signing of the Declaration of Independence.

• Led Zeppelin, in the album sleeve of their album Led Zeppelin IV used it:

  

  "Drummer John Bonham's symbol, the three interlocking rings, was picked by the drummer from [Rudolf Koch's Book of Signs]. It represents the triad of mother, father and child, but also happens to be the logo for Ballantine beer." (from the Wikipedia article on Led Zeppelin IV).

  

  (thanks Matt!).

• If you want to draw the Borromean rings, you'd draw three circles, as in the first figure above:	but you'd want to indicate, somehow, that one ring is below another ring (e.g. the "Irish Trinity", as Cody called it):

  One more:

  

  Scene from Stora Hammar stone

  with a closeup of the rings: