Skip to main content
SpringerLink
Log in

Analytic properties of sextet polynomials of hexagonal systems

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

In this paper we investigate analytic properties of sextet polynomials of hexagonal systems. For the pyrene chains, we show that zeros of the sextet polynomials \(P_n(x)\) are real, located in the open interval \((-3-2\sqrt{2},-3+2\sqrt{2})\) and dense in the corresponding closed interval. We also show that coefficients of \(P_n(x)\) are symmetric, unimodal, log-concave, and asymptotically normal. For general hexagonal systems, we show that real zeros of all sextet polynomials are dense in the interval \((-\infty ,0]\), and conjecture that every sextet polynomial has log-concave coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. J. Aihara, Aromatic sextets and aromaticity in benzenoid hydrocarbons. Bull. Chem. Soc. Jpn. 50, 2010–2012 (1977)

    Article  CAS  Google Scholar 

  2. S. Barnard, J.F. Child, Higher Algebra (Macmillan, London, 1955)

    Google Scholar 

  3. E.A. Bender, Central and local limit theorems applied to asymptotic enumeration. J. Combin. Theory Ser. A 15, 91–111 (1973)

    Article  Google Scholar 

  4. S. Beraha, J. Kahane, N. Weiss, Limits of zeros of recursively defined families of polynomials, in Studies in Foundations and Combinatorics, Advances in Mathematics, vol. 1, Supplementary Studies, ed. by G. Rota (Academic Press, New York, 1978), pp. 213–232

    Google Scholar 

  5. B. Brešar, S. Klavžar, R. Škrekovski, Roots of cube polynomials of median graphs. J. Graph Theory 52, 37–50 (2006)

    Article  Google Scholar 

  6. J.I. Brown, C.A. Hickman, R.J. Nowakowski, On the location of roots of independence polynomials. J. Algebr. Combin. 19, 273–282 (2004)

    Article  Google Scholar 

  7. M. Chudnovsky, P. Seymour, The roots of the independence polynomials of a clawfree graph. J. Combin. Theory Ser. B 97, 350–357 (2007)

    Article  Google Scholar 

  8. E. Clar, The Aromatic Sextet (Wiley, London, 1972)

    Google Scholar 

  9. C.D. Godsil, Matching behavior is asymptotically normal. Combinatorica 1, 369–376 (1981)

    Article  Google Scholar 

  10. C.D. Godsil, I. Gutman, On the theory of the matching polynomial. J. Graph Theory 5, 137–144 (1981)

    Article  Google Scholar 

  11. I. Gutman, Topological properties of benzenoid systems. An identity for the sextet polynomial. Theoret. Chem. Acta (Berl.) 45, 309–315 (1977)

    Article  CAS  Google Scholar 

  12. I. Gutman, Topological properties of benzenoid systems IX. On the sextet polynomial. Z. Naturforsch. A 37, 69–73 (1982)

    Article  Google Scholar 

  13. I. Gutman, Topological properties of benzenoid systems. XXI. Theorems, conjectures, unsolved problems. Croat. Chem. Acta 56, 365–374 (1983)

    CAS  Google Scholar 

  14. I. Gutman, S.J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons (Springer, Berlin, 1989)

    Book  Google Scholar 

  15. I. Gutman, H. Hosoya, T. Yamaguchi, A. Motoyama, N. Kuboi, Topological properties of benzenoid systems III. Bull. Soc. Chim. Beograd. 42, 503–510 (1977)

    CAS  Google Scholar 

  16. S.G. Hoggar, Chromatic polynomials and logarithmic concavity. J. Combin. Theory Ser. B 16, 248–254 (1974)

    Article  Google Scholar 

  17. C. Hoede, X.L. Li, Clique polynomials and independent set polynomials of graphs. Discrete Math. 125, 219–228 (1994)

    Article  Google Scholar 

  18. H. Hosoya, T. Yamaguchi, Sextet polynomial. A new enumeration and proof technique for the resonance theory applied to the aromatic hydrocarbons. Tetrahedron Lett. 52, 4659–4662 (1975)

    Article  Google Scholar 

  19. G.R. Li, Y.N. Pei, Y. Wang, Clar covering polynomials with only real zeros. MATCH Commun. Math. Comput. Chem. 84, 217–228 (2020)

    Google Scholar 

  20. L.L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros. Adv. Appl. Math. 38, 542–560 (2007)

    Article  Google Scholar 

  21. N. Ohkami, H. Hosoya, Topological dependency of the aromatic sextets in polycyclic benzenoid hydrocarbons Recursive relations of the sextet polynomial. Theoret. Chem. Acta (Berl.) 64, 153–170 (1983)

    Article  CAS  Google Scholar 

  22. N. Ohkami, H. Hosoya, Tables of the topological characteristics of polyhexes (condensed aromatic hydrocarbons). II. Sextet polynomial. Nat. Sci. Rep. Ochanomizu Univ. 35, 71–83 (1984)

    Google Scholar 

  23. Y. Wang, S.-N. Zheng, X. Chen, Analytic aspects of Delannoy numbers. Discrete Math. 342, 2270–2277 (2019)

    Article  Google Scholar 

  24. Y. Wang, B.-X. Zhu, On the unimodality of independence polynomials of some graphs. Eur. J. Combin. 32, 10–20 (2011)

    Article  Google Scholar 

  25. H.S. Wilf, Generating Functionology, 3rd edn. (A. K. Peters, Wellesley, 2006)

    Google Scholar 

  26. H.P. Zhang, W.C. Shiu, P.K. Sun, A relation between Clar covering polynomial and cube polynomial. MATCH Commun. Math. Comput. Chem. 70, 477–492 (2013)

    CAS  Google Scholar 

  27. H.P. Zhang, F.J. Zhang, The Clar covering polynomial of hexagonal systems I. Discrete Appl. Math. 69, 147–167 (1996)

    Article  Google Scholar 

  28. H.P. Zhang, F.J. Zhang, The Clar covering polynomial of hexagonal systems III. Discrete Math. 212, 261–269 (2000)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported partially by the National Natural Science Foundation of China (Nos. 11771065, 11871304), the Young Talents Invitation Program of Shandong Province.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Guanru Li & Yi Wang

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, People’s Republic of China

    Lily Li Liu

Authors
  1. Guanru Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lily Li Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, G., Liu, L.L. & Wang, Y. Analytic properties of sextet polynomials of hexagonal systems. J Math Chem 59, 719–734 (2021). https://doi.org/10.1007/s10910-021-01213-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-021-01213-x

Keywords

Mathematics Subject Classification

Search

Navigation