Journal of Mathematical Chemistry

, Volume 51, Issue 6, pp 1599–1607 | Cite as

Planar polycyclic graphs and their Tutte polynomials

  • Tomislav DošlićEmail author
Original Paper


We consider several classes of planar polycyclic graphs and derive recurrences satisfied by their Tutte polynomials. The recurrences are then solved by computing the corresponding generating functions. As a consequence, we obtain values of several chemically and combinatorially interesting enumerative invariants of considered graphs. Some of them can be expressed in terms of values of Chebyshev polynomials of the second kind.


Span Tree Planar Graph Chebyshev Polynomial Span Forest Combinatorial Interpretation 
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.



Partial support of the Ministry of Science, Education and Sport of the Republic of Croatia (Grants No. 177-0000000-0884 and 037-0000000-2779) is gratefully acknowledged.


  1. 1.
    B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics 184. (Springer, Berlin, 1998)Google Scholar
  2. 2.
    S.J. Cyvin, I. Gutman Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry 46, (Springer, New York, 1988)Google Scholar
  3. 3.
    T. Došlić, F. Maløy, Chain hexagonal cacti: matchings and independent sets. Discrete Math. 310, 1176–1190 (2010)Google Scholar
  4. 4.
    T. Došlić, M.-S. Litz, Matchings and independent sets in polyphenylene chains. MATCH Commun. Math. Comput. Chem. 67, 313–330 (2012)Google Scholar
  5. 5.
    G.H. Fath-Tabar, Z. Gholam-Rezaei, A.R. Ashrafi, On the Tutte polynomial of benzenoid chains. Iran. J. Math. Chem. 3, 113–119 (2012)Google Scholar
  6. 6.
    I. Gutman, S.J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons (Springer, Berlin, 1989)CrossRefGoogle Scholar
  7. 7.
    The Online Encyclopedia of Integer Sequences,
  8. 8.
    W.T. Tutte, A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)CrossRefGoogle Scholar
  9. 9.
    W.T. Tutte, On dichromatic polynomials. J. Comb. Theory 2, 301–320 (1967)CrossRefGoogle Scholar
  10. 10.
    H. Whitney, The coloring of graphs. Ann. Math. 33, 688–718 (1932)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Civil EngineeringUniversity of ZagrebZagrebCROATIA

Personalised recommendations