Journal of Mathematical Chemistry

, Volume 47, Issue 3, pp 1145–1153 | Cite as

Topographical distance matrices for porous arrays

Original Paper


The topographical Wiener index is calculated for two-dimensional graphs describing porous arrays, including bee honeycomb. For tiling in the plane, we model hexagonal, triangular, and square arrays and compare with topological formulas for the Wiener index derived from the distance matrix. The normalized Wiener indices of C4, T13, and O(4), for hexagonal, triangular, and square arrays are 0.993, 0.995, and 0.985, respectively, indicating that the arrays have smaller bond lengths near the center of the array, since these contribute more to the Wiener index. The normalized Perron root (the first eigenvalue, λ 1), calculated from distance/distance matrices describes an order parameter, \({\phi=\lambda_1/n}\) , where \({\phi= 1}\) for a linear graph and n is the order of the matrix. This parameter correlates with the convexity of the tessellations. The distributions of the normalized distances for nearest neighbor coordinates are determined from the porous arrays. The distributions range from normal to skewed to multimodal depending on the array. These results introduce some new calculations for 2D graphs of porous arrays.


Topographical Wiener index Porous arrays Distance matrix Order parameter 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Andrej Vodopivec
    • 1
  • Forrest H. Kaatz
    • 2
    • 3
    • 4
  • Bohan Mohar
    • 5
  1. 1.Department of MathematicsIMFMLjubljanaSlovenia
  2. 2.Department of Mathematics and PhysicsUniversity of Advancing TechnologyTempeUSA
  3. 3.Department of MathematicsChandler-Gilbert Community CollegeChandlerUSA
  4. 4.Department of Physical ScienceMesa Community CollegeMesaUSA
  5. 5.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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