Day 25, MAT115

Try it yourself. Here is the problem abstracted:

• 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.

    Otherwise there would be an odd total degree (sum of all the degrees), call it T; but that can't be, because each edge contributes two to the total degree T, so T=2*E, where E is the number of edges.

• An alternative solution -- thanks to spiked math

• 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.

• A cycle is a route from a node back to itself that doesn't retrace steps. One of these graphs has a cycle.

• One of these graphs is a tree: a graph that has no cycles. (Where have we seen trees used before?)

• Some graphs are sort of negatives of others -- for example, we can use this principle to form a graph from another: wherever two vertices are connected in one graph, we disconnect them in the other; and if two vertices are unconnected, we connect them in the other. This creates something which we could call the dual graph.

  What is the dual of each of the simple graphs with three vertices?

• Use duality to cut your work in half!
• Have you ever seen any of these before? Could we give any of them names?

• 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:

• Chapter 3: The Enemy of My Enemy (complete graphs)
  • The focus of this reading is negative numbers. The Indians also 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! p. 15).

  • One of the confusing aspects of negative numbers from Strogatz's perspective is that the product of two negative numbers is a positive number. The story about the two linguists is priceless (p. 17).

  • 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:

    
    Why is the all negative graph also unbalanced? Because "the enemy of my enemy is my friend!"

    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."

• Chapter 24: Untangling the Web (directed graphs)

  This section describes how Google uses the "Pagerank" algorithm to determine the importance of a webpage (and hence where it falls in the search results for a particular topic).

  Graphs provide a useful way of illustrating how pages interact. If there's a link between two pages, then a directed arrow indicates it. Here's the graph of the "toy web" Strogatz considers, with the final rankings:

  

  He justifies this ranking in a series of graphs, and a set of equations on page 195. Let's see how these equations work (we'll use this Excel spreadsheet).

• Chapter 26: Group Think (complete, directed graphs)

  In this chapter, we discover how to rotate and flip our mattress over the course of the year, so that the mattress gets even wear (while avoiding having to flip the mattress end-over-end the long way -- the "death-defying vertical flip" of p. 214).

  "Spin in the spring, flip in the fall" is what some manufacturers suggest (p. 217 -- so today is the day to spin!).

  

  There is a certain amount of curious algebra here, on p. 217:

  HR=V

  means that a horizontal flip followed by a rotation is equivalent to a vertical flip. Using only horizontal flips and rotations, we can achieve each of the four possible configurations of a matress. So over the course of two years, you'll have visited all four configurations.