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