Skip to main content
SpringerLink
Log in

Maximum number of subtrees in cacti and block graphs

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

For a graph G, we denote by N(G) the number of non-empty subtrees of G. As a topological index based on counting, N(G) has some correlations to other well studied topological indices, including the Wiener index W(G). In this paper we characterize the extremal graphs with the maximum number of subtrees among all cacti of order n with k cycles. Similarly, the extremal graphs with the maximum number of subtrees among all block graphs of order n with k blocks are also determined and shown to have the minimum Wiener index within the same collection of graphs. Analogous results are also obtained for the number of connected subgraphs C(G). Finally, a general question is posed concerning the relation between the number of subtrees and the Wiener index of graphs.

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

Similar content being viewed by others

References

  1. Andriantiana, E.O.D., Wagner, S., Wang, H.: Greedy trees, subtrees and antichains. Electron. J. Comb. 20, P28 (2013)

    Article  MathSciNet  Google Scholar 

  2. Andriantiana, E.O.D., Wagner, S., Wang, H.: Extremal problems for trees with given segment sequence. Discrete Appl. Math. 220, 20–34 (2017)

    Article  MathSciNet  Google Scholar 

  3. Andriantiana, E.O.D., Wang, H.: Subtrees and independent subsets in unicyclic graphs and unicyclic graphs with fixed segment sequence. MATCH Commun. Math. Comput. Chem. 84, 537–566 (2020)

    MATH  Google Scholar 

  4. Bessy, S., Dross, F., Knor, M., Škrekovski, R.: Graphs with the second and third maximum Wiener indices over the 2-vertex connected graphs. Discrete Appl. Math. 284, 195–200 (2020)

    Article  MathSciNet  Google Scholar 

  5. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan Press, New York (1976)

    Book  Google Scholar 

  6. Cavaleri, M., Donno, A., Scozzari, A.: Total distance, Wiener index and opportunity index in wreath products of star graphs. Electron. J. Comb. 26, P1-21 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Cambie, S., Wagner, S., Wang, H.: On the maximum mean subtree order of trees. Eur. J. Comb. 97, 103388 (2021)

    Article  MathSciNet  Google Scholar 

  8. Davenport, H., Pólya, G.: On the product of two power series. Can. J. Math. 1, 1–5 (1949)

    Article  MathSciNet  Google Scholar 

  9. Dobryinin, A.A.: On the Wiener index of the forest induced by contraction of edges in a tree. MATCH Commun. Math. Comput. Chem. 86, 321–326 (2021)

    Google Scholar 

  10. Harary, F., Palmer, E.M.: Graphical Enumeration. Academic Press, New York-London (1973)

    MATH  Google Scholar 

  11. Kirk, R., Wang, H.: Largest number of subtrees of trees with a given maximum degree. SIAM J. Discrete Math. 22, 985–995 (2008)

    Article  MathSciNet  Google Scholar 

  12. Li, S., Wang, S.: Further analysis on the total number of subtrees of trees. Electron. J. Comb. 19(4), P48 (2012)

    Article  MathSciNet  Google Scholar 

  13. Liu, H., Lu, M.: A unified approach to cacti for different indices. MATCH Commun. Math. Comput. Chem. 58, 193–204 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Liu, L., Wang, Y.: On the log-convexity of combinatorial sequences. Adv. Appl. Math. 39(4), 453–476 (2007)

    Article  MathSciNet  Google Scholar 

  15. Liu, M., Yao, Y., Das, K.C.: Extremal results for cacti. Bull. Malays. Math. Sci. Soc. 43, 2783–2798 (2020)

    Article  MathSciNet  Google Scholar 

  16. Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York-London (1973)

    MATH  Google Scholar 

  17. Schmuck, N., Wagner, S., Wang, H.: Greedy trees, caterpillars, and Wiener-type graph invariants. MATCH Commun. Math. Comput. Chem. 68, 273–292 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Székely, L.A., Wang, H.: On subtrees of trees. Adv. Appl. Math. 34, 138–155 (2005)

    Article  MathSciNet  Google Scholar 

  19. Székely, L.A., Wang, H.: Binary trees with the largest number of subtrees. Discrete Appl. Math. 155, 374–385 (2007)

    Article  MathSciNet  Google Scholar 

  20. Tian, J., Xu, K.: The general position number of Cartesian products involving a factor with small diameter. Appl. Math. Comput. 403, 126206 (2021)

    MathSciNet  MATH  Google Scholar 

  21. Wiener, H.: Structrual determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)

    Article  Google Scholar 

  22. Xu, K., Das, K.C., Klavžar, S., Li, H.: Comparison of Wiener index and Zagreb eccentricity indices. MATCH Commun. Math. Comput. Chem. 84, 595–610 (2020)

    MATH  Google Scholar 

  23. Xu, K., Liu, M., Das, K.C., Gutman, I., Furtula, B.: A survey on graphs extremal with respect to distance-based topological indices. MATCH Commun. Math. Comput. Chem. 71, 461–508 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Xu, K., Li, J., Wang, H.: The number of subtrees in graphs with given number of cut edges. Discrete Appl. Math. 304, 283–296 (2021)

    Article  MathSciNet  Google Scholar 

  25. Xu, K., Wang, M., Tian, J.: Relations between Merrifield-Simmons and Wiener indices. MATCH Commun. Math. Comput. Chem. 85, 147–160 (2021)

    MATH  Google Scholar 

  26. Yan, W., Yeh, Y.N.: Enumeration of subtrees of trees. Theor. Comput. Sci. 369, 256–268 (2006)

    Article  MathSciNet  Google Scholar 

  27. Zhang, X.-M., Zhang, X.-D., Gray, D., Wang, H.: The number of subtrees of trees with given degree sequence. J. Graph Theory 73, 280–295 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

S. Wagner was supported by the Knut and Alice Wallenberg Foundation. The authors are very grateful to PhD student Zuwen Luo for the helpful discussion with him.

Author information

Authors and Affiliations

  1. College of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Jie Li, Kexiang Xu & Tianli Zhang

  2. MIIT Key Laboratory of Mathematical Modelling and High Performance, Computing of Air Vehicles, Nanjing, 210016, China

    Jie Li, Kexiang Xu & Tianli Zhang

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460, USA

    Hua Wang

  4. Department of Mathematics, Uppsala University, 75106, Uppsala, Sweden

    Stephan Wagner

  5. Department of Mathematical Sciences, Stellenbosch University, Stellenbosch, 7602, South Africa

    Stephan Wagner

Authors
  1. Jie Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kexiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tianli Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Stephan Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kexiang Xu.

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, J., Xu, K., Zhang, T. et al. Maximum number of subtrees in cacti and block graphs. Aequat. Math. 96, 1027–1040 (2022). https://doi.org/10.1007/s00010-022-00879-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-022-00879-1

Keywords

Mathematics Subject Classification

Search

Navigation