Skip to main content
SpringerLink
Log in

On Resistance Distance and Kirchhoff Index of Cacti Networks

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Resistance distance in electrical circuits measures how much a component or an entire circuit resists the flow of electric current. When dealing with intricate circuits, this term explicitly denotes the total resistance observed between any two points, which varies based on the configuration and resistance values of the components within the circuit. The Kirchhoff index is a metric used to quantify the mean resistance distance across all pairs of nodes in an electrical network. In graph theory, these networks are depicted as graphs with nodes representing electrical components and edges symbolizing the connecting wires. The resistance distance between any two nodes is calculated as if the graph were an electrical circuit, with each edge functioning as a resistor. We focus on a particular type of graph known as a cacti graph, denoted by \(\mathcal {C}(n,s)\), which features interconnected cycles that share a single common vertex, with n representing the total number of nodes and s the number of cycles. This paper explores cacti networks to establish the maximum possible values of the Kirchhoff index for these structures.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Data Availibility

All the data used to finding the results is included in the manuscript.

References

  1. Chen, S.: Cacti with the smallest, second smallest, and third smallest Gutman index. J. Comb. Optim. 31(1), 327–332 (2016)

    Article  MathSciNet  Google Scholar 

  2. Chen, T., Yuan, Z., Peng, J.: The normalized Laplacian spectrum of n-polygon graphs and applications. Linear Multilinear Algebra (2022). https://doi.org/10.1080/03081087.2022.2158293

    Article  Google Scholar 

  3. Choi, M.C.: On resistance distance of Markov chain and its sum rules. Linear Algebra Appl. 571, 14–25 (2019)

    Article  MathSciNet  Google Scholar 

  4. Estrada, E., Hatano, N.: Resistance distance, information centrality, node vulnerability and vibrations in complex networks. Netw. Sci. 13–29 (2020)

  5. Fan, J., Zhu, J., Tian, L., Wang, Q.: Resistance distance in potting networks. Physica A 540, 123053 (2020)

    Article  MathSciNet  Google Scholar 

  6. Gutman, I., Mohar, B.: The quasi-Wiener and the Kirchhoff indices coincide. J. Chem. Inf. Comput. Sci. 36(5), 982–985 (1996)

    Article  Google Scholar 

  7. Hayat, F., Zhou, B.: On cacti with large Mostar index. Filomat 33(15), 4865–4873 (2019)

    Article  MathSciNet  Google Scholar 

  8. Huang, S., Li, S.: On the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain. Physica A 558, 124999 (2020)

    Article  MathSciNet  Google Scholar 

  9. Kagan, M., Mata, B.: A physics perspective on the resistance distance for graphs. Math. Comput. Sci. 13, 105–115 (2019)

    Article  MathSciNet  Google Scholar 

  10. Klein, D.J., Randić, M.: Resistance distance. J. Math. Chem. 12(1), 81–95 (1993)

    Article  MathSciNet  Google Scholar 

  11. Li, S., Wei, W., Yu, S.: On normalized Laplacians, multiplicative degree Kirchhoff indices, and spanning trees of the linear [n] phenylenes and their dicyclobutadieno derivatives. Int. J. Quantum Chem. 119(8), e25863 (2019)

    Article  Google Scholar 

  12. Li, S., Yang, H., Zhao, Q.: Sharp bounds on Zagreb indices of cacti with K pendant vertices. Filomat 26(6), 1189–1200 (2012)

    Article  MathSciNet  Google Scholar 

  13. Li, W., Zhu, H., Li, S., Wang, H., Dai, H., Wang, C., Jin, Q.: Evolutionary community discovery in dynamic social networks via resistance distance. Expert Syst. Appl. 171, 114536 (2021)

    Article  Google Scholar 

  14. Liu, J.B., Pan, X.F.: Minimizing Kirchhoff index among graphs with a given vertex bipartiteness. Appl. Math. Comput. 291, 84–88 (2016)

    MathSciNet  Google Scholar 

  15. Liu, J.B., Pan, X.F.: Asymptotic incidence energy of lattices. Physica A 422, 193–202 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  16. Liu, J.B., Pan, X.F., Cao, J., Hu, F.F.: A note on some physical and chemical indices of clique-inserted lattices. J. Stat. Mech. 2014(6), P06006 (2014)

    Article  Google Scholar 

  17. Liu, J.B., Pan, X.F., Yu, L., Li, D.: Complete characterization of bicyclic graphs with minimal Kirchhoff index. Discret. Appl. Math. 200, 95–107 (2016)

    Article  MathSciNet  Google Scholar 

  18. Liu, J.B., Wang, W.-R., Zhang, Y.-M., Pan, X.-F.: On degree resistance distance of cacti. Discret. Appl. Math. 203, 217–225 (2016)

    Article  MathSciNet  Google Scholar 

  19. Liu, J.B., Zhang, T., Wang, Y., Lin, W.: The Kirchhoff index and spanning trees of Möbius/cylinder octagonal chain. Discret. Appl. Math. 307, 22–31 (2022)

    Article  Google Scholar 

  20. Sardar, M.S., Pan, X.F., Xu, S.A.: Computation of resistance distance and Kirchhoff index of the two classes of silicate networks. Appl. Math. Comput. 381, 125283 (2020)

    MathSciNet  Google Scholar 

  21. Stankov, S., Milovanovic, I., Milovanovic, E., Matejic, M.: Some lower bounds on the Kirchhoff index. Commun. Comb. Optim. (2022). https://doi.org/10.22049/cco.2022.27898.1389

    Article  Google Scholar 

  22. Thiele, J., Buchholz, S., Schirmel, J.: Using resistance distance from circuit theory to model dispersal through habitat corridors. J. Plant Ecol. 11(3), 385–393 (2018)

    Article  Google Scholar 

  23. Wang, C., Wang, S., Wei, B.: cacti with extremal PI index. Trans. Comb. 5(4), 1–8 (2016)

    MathSciNet  Google Scholar 

  24. Wang, D.F., Tan, S.W.: The Maximum hyper-Wiener index of cacti. J. Appl. Math. Comput. 47(1–2), 91–102 (2015)

    Article  MathSciNet  Google Scholar 

  25. Wang, H., Kang, L.: On the Harary index of cacti. J. Appl. Math. Comput. 43(1–2), 369–386 (2013)

    Article  MathSciNet  Google Scholar 

  26. Wang, S.: On extremal cacti with respect to the Szeged index. Appl. Math. Comput. 309, 85–92 (2017)

    ADS  MathSciNet  Google Scholar 

  27. Wang, D., Zeng, C., Zhao, Z., Wu, Z., Xue, Y.: Kirchhoff index of a class of polygon networks. Chaos Solitons Fract. 168, 113149 (2023)

    Article  MathSciNet  Google Scholar 

  28. Wang, W., Ma, T., Liu, J.B.: Resistance distance and Kirchhoff index of Q-double join graphs. IEEE Access 7, 102313–102320 (2019)

    Article  Google Scholar 

  29. Yan, T., Kosar, Z., Aslam, A., Zaman, S., Ullah, A.: Spectral techniques and mathematical aspects of K4 chain graph. Phys. Scr. 98, 045222 (2023)

    Article  ADS  Google Scholar 

  30. Zhang, T., Bu, C.: Detecting community structure in complex networks via resistance distance. Physica A 526, 120782 (2019)

    Article  Google Scholar 

  31. Zhu, W., Geng, X.: Enumeration of the multiplicative degree-Kirchhoff index in the random polygonal chains. Molecules 27, 5669 (2022)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, COMSATS University Islamabad Lahore Campus, Lahore, 54000, Pakistan

    Muhammad Faisal Nadeem & Faiza Ishfaq

  2. Department of Physical Sciences, The University of Chenab, Gujarat, Pakistan

    Ayesha Shabbir

Authors
  1. Muhammad Faisal Nadeem
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Faiza Ishfaq
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ayesha Shabbir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Faisal Nadeem.

Ethics declarations

Conflict of interest

The authors have no conflict of interest to disclose.

Additional information

Communicated by Bruno Nachtergaele.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nadeem, M.F., Ishfaq, F. & Shabbir, A. On Resistance Distance and Kirchhoff Index of Cacti Networks. J Stat Phys 191, 83 (2024). https://doi.org/10.1007/s10955-024-03300-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-024-03300-0

Keywords

Search

Navigation