Journal of Mathematical Sciences

, Volume 209, Issue 6, pp 809–825 | Cite as

Chip Removal. Urban Renewal Revisited

  • V. Aksenov
  • K. Kokhas

We describe a new combinatorial-algebraic transformation on graphs which we call the “chip removal.” It generalizes the well-known urban renewal trick of Propp and Kuperberg. The chip removal is useful in calculations of the determinants of adjacency matrices and matching numbers of graphs. A beautiful example of applying this technique is a theorem on removing 4-contact chips, which generalizes Kuo’s graphical condensation method. Numerous examples are given. Bibliography: 6 titles.


Adjacency Matrix Chebyshev Polynomial Urban Renewal Outgoing Edge Adjacency Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. Aksenov and K. Kokhas, “Domino tilings and determinants,” J. Math. Sci., 200, No. 6, 647–653 (2014).zbMATHCrossRefGoogle Scholar
  2. 2.
    K. Bibak and R. Tauraso, “Determinants of grids, tori, cylinders and M¨obius ladders,” arXiv:1212.4816v1.Google Scholar
  3. 3.
    M. Ciucu, “Enumeration of perfect matchings in graphs with reflective symmetry,” J. Combin. Theory Ser. A, 77, No. 1, 67–97 (1997).zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    E. Kuo, “Application of graphical condensation for enumerating matchings,” Theoret. Comput. Sci., 319, 29–57 (2004); arXiv:math.CO/0304090.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Propp, “Generalized domino-shuffling,” Theoret. Comput. Sci., 303, Nos. 2–3, 267–301 (2003), arXiv:math/0111034.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    H.M. Rara, “Reduction procedures for calculating the determinant of the adjacency matrix of some graphs and the singularity of square planar grids,” Discrete Math., 151, 213–219 (1996).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.ITMO UniversitySt.PetersburgRussia

Personalised recommendations