- Your symmetry homework is graded.
- You have a new assignment to hand in, due Thursday.
- The Bridges
of Konigsberg problem: "is it possible to set off and walk around Konigsberg crossing each bridge exactly one?"
- 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 defined as
- A collection of points, called vertices.
- A collection of edges, each of which connects two vertices.
- a set of rules, one for each edge, telling how each edge is connected to its pair of vertices.
- The degree of a vertex is the number of edges coming into it.
- Graphs are made up of vertices (points) and edges
that connect the vertices. More formally, a graph is defined as
- 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)
- Solved by Euler
(1735), the beginning of graph theory.
- You've
probably been making graphs for a long time.... (the pentagram)
- 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?
- Duality -- again!
- 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?
- We noticed last time that the tetrahedron is one of these: the complete simple graph on four vertices.
- How does this relate to Facebook?
- Planar graphs are graphs that can be drawn such that no two edges
intersect. Every Platonic
graph can be drawn this way, so they are planar graphs.
- 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 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:
- For Facebook, there is a difference between vertices: vertices are
labelled (with your name, perhaps. Pascal's triangle gives all
the ways of having a "Facebook" on any number of vertices. That's one
way that you can think about the rows of Pascal's triangle.
- One focus of this reading is negative numbers. It
was the Indians (of Asia) who put negative numbers on a
firm foundation (as well as zero), in about the 6th
century AD.
"...you can't see negative 4 cookies and you certainly can't eat them -- but you can think about them...." ("and you have to", says our author).
- One of the confusing aspects of negative numbers
from Strogatz's perspective is that the product of two
negative numbers is a positive number.
- But then Strogatz goes on to use the context of
social interactions (negative and positive) to discuss
how we maintain stable relationships.
And the key to understanding stability in three-way social relationships is that the product of two interactions (signified by either +1 or -1) must be equal to the other: so that if two legs are positive, the third in the triangle must be positive; if one leg positive, and the other negative, then the third leg must be negative as well.
Strogatz sums up the second case above in the familiar saying that "The enemy of my enemy is my friend".
The following (two) graphs are unbalanced:
Finally Strogatz shows how historical relationships settled down into this pattern of stability: in "...the run-up to World War I. The diagram that follows shows the shifting alliances among Great Britain, France, Russia, Italy, Germany, and Austria-Hungary between 1872 and 1907."
The bottom right graph (complete!) is the only stable configuration, "...balanced, but on the brink of war."
