Journal of Mathematical Chemistry

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

Topographical distance matrices for porous arrays

  • Andrej Vodopivec
  • Forrest H. Kaatz
  • Bohan Mohar
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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wiener H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)CrossRefGoogle Scholar
  2. 2.
    Hosoya H.: Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. Jpn. 44, 2332–2339 (1971)CrossRefGoogle Scholar
  3. 3.
    B. Bogdanov, S. Nikolic, N. Trinajstic, On the three-dimensional Wiener number. J. Math. Chem. 3, 299–309 (1989)CrossRefGoogle Scholar
  4. 4.
    Mihalic Z., Trinajstic N.: The algebraic modeling of chemical structures: on the development of three-dimensional molecular descriptors. J. Mol. Struct. (Theochem) 232, 65–78 (1991)CrossRefGoogle Scholar
  5. 5.
    Nikolic S., Trinajstic N., Mihalic Z., Carter S.: On the geometric-distance matrix and the corresponding structural invariants of molecular systems. Chem. Phys. Lett. 179, 21–28 (1991)CrossRefGoogle Scholar
  6. 6.
    Mihalic Z., Nikolic S., Trinajstic N.: Comparative study of molecular descriptors derived from the distance matrix. J. Chem. Inf. Comput. Sci. 32, 28–37 (1992)Google Scholar
  7. 7.
    Masuda H., Asoh H., Watanabe M., Nishio K., Nakao M., Tamamura T.: Square and triangular nanohole array architectures in anodic alumina. Adv. Mater. 13(3), 189–192 (2001)CrossRefGoogle Scholar
  8. 8.
    Toth L.F.: Regular Figures, pp. 21–38. MacMillan, New York (1964)Google Scholar
  9. 9.
    The website of Image SXM:
  10. 10.
    Mohar B., Pisanski T.: How to compute the Wiener index of a graph. J. Math. Chem. 2, 267–277 (1988)CrossRefGoogle Scholar
  11. 11.
    Juvan M., Mohar B.: Bond contributions to the Wiener index. J. Chem. Inf. Comput. Sci. 35, 217–219 (1995)Google Scholar
  12. 12.
    Choi J., Luo Y., Wehrspohn R.B., Hillebrand R., Schilling J., Gösele U.: Perfect two-dimensional porous alumina photonic crystals with duplex oxide layers. J. Appl. Phys. 94(8), 4757–4762 (2003)CrossRefGoogle Scholar
  13. 13.
    Krishnan R., Nguyen H.Q., Thompson C.V., Choi W.K., Foo Y.L.: Wafer-level ordered arrays of aligned carbon nanotubes with controlled size and spacing on silicon. Nanotechnology 16, 841–845 (2005)CrossRefGoogle Scholar
  14. 14.
    Hulteen J.C., Treichel D.A., Smith M.T., Duval M.L., Jensen T.R., Van Duyne R.P.: Nanosphere lithography: size-tunable silver nanoparticle and surface cluster arrays. J. Phys. Chem. B 103, 3854–3863 (1999)CrossRefGoogle Scholar
  15. 15.
    Randic M., Vracko M.: On the similarity of DNA primary sequences. J. Chem. Inf. Comput. Sci. 40, 599–606 (2000)Google Scholar
  16. 16.
    Randic M., Kleiner A.F., DeAlba L.M.: Distance/distance matrices. J. Chem. Inf. Comput. Sci. 34, 277–286 (1994)Google Scholar
  17. 17.
    The website for Maxima:
  18. 18.
    Contact one of the authors at for more information on the Lisp programGoogle Scholar
  19. 19.
    Kaatz F.H., Bultheel A., Egami T.: Order parameters from image analysis: a honeycomb example. Naturwissenschaften 95(11), 1033–1040 (2008)CrossRefGoogle Scholar
  20. 20.
    Kaatz F.H., Bultheel A., Egami T.: Real and reciprocal space order parameters for porous arrays from image analysis. J. Mater. Sci. 44, 40–46 (2009)CrossRefGoogle Scholar
  21. 21.
    Shiu W.C., Lam P.C.B., Poon K.K.: On Wiener numbers of polygonal nets. Discret. Appl. Math. 122, 251–261 (2002)CrossRefGoogle Scholar
  22. 22.
    Gutman I., Klavzar S., Rajapakse A.: Average distances in square-cell configurations. Int. J. Quan. Chem. 76, 611–617 (1999)CrossRefGoogle Scholar
  23. 23.
    Yang B.Y., Yeh Y.N.: A crowning moment for Wiener indices. Stud. Appl. Math. 112, 333–340 (2004)CrossRefGoogle Scholar
  24. 24.
    Shiu W.C., Lam P.C.B.: The Wiener number of the hexagonal net. Discret. Appl. Math. 73, 101–111 (1997)CrossRefGoogle Scholar
  25. 25.
    Hales T.C.: The honeycomb conjecture. Discret. Comput. Geom. 25, 1–22 (2001)Google Scholar

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

Personalised recommendations