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On the Maximal Independence Polynomial of the Covering Graph of the Hypercube up to n=6

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Formal Concept Analysis (ICFCA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13934))

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Abstract

There are well-known problems in extremal set theory that can be formulated as enumeration of the maximal independent sets or counting their total number in certain graphs. Here we provide an FCA-based solution on the number of maximal independent sets of the covering graph of a hypercube. In addition, we consider the related maximal independence polynomials for n up to 6, and prove several properties of the polynomials’ coefficients and the corresponding concept lattices.

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Notes

  1. 1.

    We use MIS both for the plural and singular forms depending on context.

  2. 2.

    https://oeis.org/A284707.

  3. 3.

    https://ipython.org/notebook.html.

  4. 4.

    https://github.com/dimachine/CubeIndSets.

  5. 5.

    https://cython.readthedocs.io/.

  6. 6.

    https://pypi.org/project/multiprocess/.

  7. 7.

    \({[n] \atopwithdelims ()k}\) are subsets of [n] of size k.

  8. 8.

    https://oeis.org/A001608.

  9. 9.

    https://oeis.org/A000931.

References

  1. Boginski, V., Butenko, S., Pardalos, P.M.: Statistical analysis of financial networks. Comput. Stat. Data Anal. 48(2), 431–443 (2005). https://doi.org/10.1016/j.csda.2004.02.004. https://www.sciencedirect.com/science/article/pii/S0167947304000258

  2. Butenko, S., Pardalos, P., Sergienko, I., Shylo, V., Stetsyuk, P.: Estimating the size of correcting codes using extremal graph problems. In: Pearce, C., Hunt, E. (eds.) Optimization. Springer Optimization and Its Applications, vol. 32, pp. 227–243. Springer, New York (2009). https://doi.org/10.1007/978-0-387-98096-6_12

    Chapter  Google Scholar 

  3. Duffus, D., Frankl, P., Rödl, V.: Maximal independent sets in bipartite graphs obtained from Boolean lattices. Eur. J. Comb. 32(1), 1–9 (2011). https://doi.org/10.1016/j.ejc.2010.08.004

    Article  MathSciNet  MATH  Google Scholar 

  4. Duffus, D., Frankl, P., Rödl, V.: Maximal independent sets in the covering graph of the cube. Discret. Appl. Math. 161(9), 1203–1208 (2013). https://doi.org/10.1016/j.dam.2010.09.003

    Article  MathSciNet  MATH  Google Scholar 

  5. Erdös, P.: On cliques in graphs. Israel J. Math. 4(4), 233–234 (1966). https://doi.org/10.1007/BF02771637

    Article  MathSciNet  MATH  Google Scholar 

  6. Ganter, B., Obiedkov, S.A.: Conceptual Exploration. Springer, Cham (2016). https://doi.org/10.1007/978-3-662-49291-8

  7. Ganter, B., Wille, R.: Formal Concept Analysis - Mathematical Foundations. Springer, Cham (1999). https://doi.org/10.1007/978-3-642-59830-2

  8. Hu, H., Mansour, T., Song, C.: On the maximal independence polynomial of certain graph configurations. Rocky Mt. J. Math. 47(7), 2219–2253 (2017). https://doi.org/10.1216/RMJ-2017-47-7-2219

    Article  MathSciNet  MATH  Google Scholar 

  9. Ignatov, D.I.: Introduction to formal concept analysis and its applications in information retrieval and related fields. In: Braslavski, P., Karpov, N., Worring, M., Volkovich, Y., Ignatov, D.I. (eds.) RuSSIR 2014. CCIS, vol. 505, pp. 42–141. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25485-2_3

    Chapter  Google Scholar 

  10. Ilinca, L., Kahn, J.: Counting maximal antichains and independent sets. Order 30(2), 427–435 (2013). https://doi.org/10.1007/s11083-012-9253-5

    Article  MathSciNet  MATH  Google Scholar 

  11. Kahn, J., Park, J.: The number of maximal independent sets in the hamming cube. Comb. 42(6), 853–880 (2022). https://doi.org/10.1007/s00493-021-4729-9

    Article  MathSciNet  MATH  Google Scholar 

  12. Kuznetsov, S.O.: A fast algorithm for computing all intersections of objects from an arbitrary semilattice. Nauchno-Tekhnicheskaya Informatsiya Seriya 2-Informatsionnye Protsessy i Sistemy (1), 17–20 (1993)

    Google Scholar 

  13. Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, New Delhi (1998)

    Book  MATH  Google Scholar 

  14. von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007). https://doi.org/10.1007/s11222-007-9033-z

    Article  MathSciNet  Google Scholar 

  15. Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6(3), 505–517 (1977). https://doi.org/10.1137/0206036

    Article  MathSciNet  MATH  Google Scholar 

  16. Weisstein, E.W.: Independence Number. From MathWorld-A Wolfram Web Resource (2023). https://mathworld.wolfram.com/IndependenceNumber.htm

  17. Weisstein, E.W.: Independence Polynomial. From MathWorld-A Wolfram Web Resource (2023). https://mathworld.wolfram.com/IndependencePolynomial.html

  18. Weisstein, E.W.: Linear Recurrence Equation. From MathWorld-A Wolfram Web Resource (2023). https://mathworld.wolfram.com/LinearRecurrenceEquation.html

  19. Xiao, M., Nagamochi, H.: Exact algorithms for maximum independent set. Inf. Comput. 255, 126–146 (2017). https://doi.org/10.1016/j.ic.2017.06.001

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

This paper is an output of a research project implemented as part of the Basic Research Program at the National Research University Higher School of Economics (HSE University). This research was also supported in part through computational resources of HPC facilities at HSE University

We would like to thank N.J.A. Sloane and OEIS editors for their assistance and the anonymous reviewers for their useful suggestions.

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Authors and Affiliations

  1. HSE University, Moscow, Russia

    Dmitry I. Ignatov

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  1. Dmitry I. Ignatov
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Correspondence to Dmitry I. Ignatov .

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Editors and Affiliations

  1. Universität Kassel, Kassel, Germany

    Dominik Dürrschnabel

  2. Universidad de Málaga, Málaga, Spain

    Domingo López Rodríguez

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Ignatov, D.I. (2023). On the Maximal Independence Polynomial of the Covering Graph of the Hypercube up to n=6. In: Dürrschnabel, D., López Rodríguez, D. (eds) Formal Concept Analysis. ICFCA 2023. Lecture Notes in Computer Science(), vol 13934. Springer, Cham. https://doi.org/10.1007/978-3-031-35949-1_11

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  • DOI: https://doi.org/10.1007/978-3-031-35949-1_11

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  • Print ISBN: 978-3-031-35948-4

  • Online ISBN: 978-3-031-35949-1

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