New Results for 2D Tilings: Wiener Index & Statistical Mechanics of Graphs

ArizMATYC & MAA Southwestern SectionScottsdale, AZ

April 9-10, 2010

Forrest H Kaatz 

Maricopa Community Colleges, AZUniversity of Advancing Technology, Tempe, AZ

 Adhemar Bultheel

 Department of Computer Science

K.U.Leuven, Celestijnenlaan 200A, 3001 Heverlee, Belgium 

Andrej Vodopivec 

Department of MathematicsIMFM, 1000 Ljubljana, Slovenia

 Ernesto Estrada

 Institute of Complexity Science

Department of Physics and Department of MathematicsUniversity of Strathclyde, Glasgow G1 1XH, United Kingdom

Periodic and Non-periodic Tilings

Archimedean Lattices -Tiling in the Plane

Regular Tilings

Squares Triangles

Hexagons Random

Methods

Image SXM determines array coordinates

Excel macros for distances (adjacency matrix)

Maxima used for distance matrices, WienerIndex, and eigenvalues

MATLAB used for graphing and statistical mechanics

Adjacency Matrix

€

Aij =0 if no link between i and j1 if a link between i and j ⎧ ⎨ ⎩

Euclidean adjacency matrix:

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A ij =0 if no link between i and jaij if a link between i and j ⎧ ⎨ ⎩

aij = Euclidean distance between i and j

Adjacency Matrices

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AS =

0 1 0 11 0 1 00 1 0 11 0 1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

AH =

0 1 0 0 0 11 0 1 0 0 00 1 0 1 0 00 0 1 0 1 00 0 0 1 0 11 0 0 0 1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Statistical Mechanics of Graphs

The partition function is defined as:

where the Hamiltonian H = -A and A is the adjacency

matrix, G is a graph, and ß = 1 for an unweighted network

Statistical Mechanics of Graphs, cont’d

The probability that the system will occupythe jth microstate is given by:

where is an eigenvalue of the adjacency matrix

We can then define the thermodynamic functionsas follows:

Statistical Mechanics of Graphs, cont’d

The (Shannon) entropy is :

which can be rearranged as:

The energy relationships are then:

Statistical Mechanics of Graphs, cont’d

The total energy can be written as:

and the Helmholtz free energy is:

Hexagonal Arrays

10 Coordinates11 Links

1000 Coordinates1446 Links

Statistical Mechanics ResultsPartition Function and Enthalpy

Results, cont’dEntropy and Free Energy

Power Laws

If a relationship has a power fit

A log-log plot produces a straight line, slope k

Summary

Wiener Index used to model real 2D tilings, i.e. porous arrays

Porous arrays are increasing (expanding) in size

Distributions of normalized link length determined

Statistical mechanics functions are fit with power regression

Work to be Done

Still need a model for the thermodynamic behavior of graphs