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Embedding Distributions and Chebyshev Polynomials

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Abstract

The history of genus distributions began with J. Gross et al. in 1980s. Since then, a lot of study has given to this parameter, and the explicit formulas are obtained for various kinds of graphs. In this paper, we find a new usage of Chebyshev polynomials in the study of genus distribution, using the overlap matrix, we obtain homogeneous recurrence relation for rank distribution polynomial, which can be solved in terms of Chebyshev polynomials of the second kind. The method here can find explicit formula for embedding distribution of some other graphs. As an application, the well known genus distributions of closed-end ladders and cobblestone paths (Furst et al. in J Combin Ser B 46:22–36, 1989) are derived. The explicit formula for non-orientable embedding distributions of closed-end ladders and cobblestone paths are also obtained.

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Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

    Yichao Chen & Qian Zou

  2. Department of Mathematics, University of Haifa, 31905, Haifa, Israel

    Toufik Mansour

Authors
  1. Yichao Chen
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  2. Toufik Mansour
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  3. Qian Zou
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Corresponding author

Correspondence to Toufik Mansour.

Additional information

Y. Chen’s work was partially supported by NNSFC under Grant No. 10901048.

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Chen, Y., Mansour, T. & Zou, Q. Embedding Distributions and Chebyshev Polynomials. Graphs and Combinatorics 28, 597–614 (2012). https://doi.org/10.1007/s00373-011-1075-5

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  • DOI: https://doi.org/10.1007/s00373-011-1075-5

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