Webnotes on the mathematics of information networks

• Chapter 1: Introduction
  • Seven Bridges of Königsberg and the handshake theorem.
  • Examples:
    • Platonic graphs.
    • Planar graphs
    • Buckyball, more.
    • The Cartesian product of two graphs.
  • Paths in a graph.
    • Shortest paths .
    • Eulerian cycle , by Yekaterina Tsipenyuk.
    • Erd"os and Gallai's pretty result of graphic sequences for simple graphs.
    • de Bruijn sequence.
  • Archimedes' Stomachion and its associated graphs
  • A cop-robber game.
• Chapter 2: Planar graphs
  • Euler's formula: v - e + f = 2
    
    • 17 proofs of Euler's formula.
    • Consequences of Euler's formula
    • Examples:
      Euler's formula for polyhedra.
  • Drawing planar graphs
    • Kuratowski's Theorem
    • Planarity testing .
      
    • Crossing numbers and many interesting problems.
    • Conway's Thrackle Problem.
  • Duality of planar graphs.
    • Dual graphs.
    • Examples
      Self-dual graphs, more and more.
  • Disc representation of planar graphs.
    
    • Straightline embedding and disc representation of planar graphs.
    • An outline for an algorithm for disc representations of planar graphs.
    • An interesting research paper by David Eppstein.
    • A neat JAVA Applet for disc representation by Shameendra Rajapaksa.
    • Stereographic projection, more.
  • Steinitz representation of planar graphs.
    
    • Some ppt files based on drawings of Lovasz.
    • An outline of an algorithm for Steinitz representation.
    • An implementation by Brian Goldsmith and Alexander Smith.
  • Coloring planar graphs
    
    • The chromatic number of a graph
    • The four color theorem.
    • Network resources for graph coloring by Michael Trick.
• Chapter 3: Discrete probability and random graphs
  • A text book on discrete probability by Charles M. Grinstead and J. Laurie Snell, freely available in pdf.
  • Simulation applets of the above book.
  • What is the meaning of the word "random" ?
  • Bernoulli trials
  • What is a random graph? Here are some definitions of random graphs that can be found on the web. For example, this one is by Mathworld and another by Wekipedia . Are these statements correct? Why?
  • Books on random graphs for G(n,p).
  • The Chebyshev inequality.
  • Central limit theorem
• Chapter 4: Network models and power law graphs
  • Some parts of the old notes of Math 262:
    • A random graph model and some useful concentration inequalities
      The first page.
    • Expected degrees in the random graph model G(w)
      The first page and a third of the second page.