Extremal linear triangular chains with respect to the Kirchhoff index
-
Wensheng Sun
,
Yujun Yang
Abstract
Let G be a connected graph. The resistance distance among two vertices is defined as the effective resistance between the corresponding nodes in the electrical network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index is an important distance-based topological index corresponding to graphs, which is the sum of all the resistance distances pairs of G. Let B n be a linear polyomino chain with n squares. The linear triangular chain is a graph with 2n triangles, characterized by randomly adding an edge to each square of B n so as to make it into two triangular faces. In this paper, by standard techniques of electrical networks and the recursion formula for resistance distances, we characterize the linear triangular chains with extreme Kirchhoff index.
Funding source: Taishan Scholars Special Project of Shandong Province
Funding source: National Natural Science Foundation of China
Acknowledgments
The authors are grateful to the referees for their valuable comments, corrections, and suggestions, which lead to a great improvement of this paper.
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Use of Large Language Models, AI and Machine Learning Tools: None declared.
-
Conflict of interest: The authors declared that they have no conflicts of interest to this work.
-
Research funding: The second author is supported by the Taishan Scholars Special Project of Shandong Province. The third author is support by the National Natural Science Foundation of China with the Grant no. 12071194.
-
Data availability: Not applicable.
References
[1] F. Buckley and F. Harary, Distance in Graphs, Reading, MA, Addison-Wesley, 1989.Suche in Google Scholar
[2] D. B. West, Introduction to Graph Theory, Upper Saddle River, Prentice-Hall, 2001.Suche in Google Scholar
[3] D. J. Klein and M. Randić, “Resistance distance,” J. Math. Chem., vol. 12, nos. 1–4, pp. 81–95, 1993. https://doi.org/10.1007/bf01164627.Suche in Google Scholar
[4] L. Lovász and K. Vesztergombi, “Geometric representations of graphs,” in Paul Erdős and his Mathematics, Proc. Conf, Budapest, 1999.Suche in Google Scholar
[5] H. Wiener, “Structural determination of paraffin boiling points,” J. Am. Chem. Soc., vol. 69, no. 1, pp. 17–20, 1947. https://doi.org/10.1021/ja01193a005.Suche in Google Scholar
[6] I. Lukovits, S. Nikolić, and N. Trinajstić, “Resistance distance in regular graphs,” Int. J. Quantum Chem., vol. 71, no. 3, pp. 217–225, 1999. https://doi.org/10.1002/(sici)1097-461x(1999)71:3<217::aid-qua1>3.3.co;2-3.10.1002/(SICI)1097-461X(1999)71:3<217::AID-QUA1>3.3.CO;2-3Suche in Google Scholar
[7] W. Xiao and I. Gutman, “Resistance distance and Laplacian spectrum,” Theor. Chem. Acc., vol. 110, no. 4, pp. 284–289, 2003. https://doi.org/10.1007/s00214-003-0460-4.Suche in Google Scholar
[8] M. Karelson, V. S. Lobanov, and A. R. Katritzky, “Quantum-chemical descriptors in QSAR/QSPR studies,” Chem. Rev., vol. 96, no. 3, pp. 1027–1044, 1996. https://doi.org/10.1021/cr950202r.Suche in Google Scholar
[9] I. Gutman and B. Mohar, “The Quasi-Wiener and the Kirchhoff indices coincide,” J. Chem. Inf. Comput. Sci., vol. 36, no. 5, pp. 982–985, 1996. https://doi.org/10.1021/ci960007t.Suche in Google Scholar
[10] R. B. Bapat, I. Gutmana, and W. Xiao, “A simple method for computing resistance distance,” Z. Naturforsch. A, vol. 58, nos. 9–10, pp. 494–498, 2003. https://doi.org/10.1515/zna-2003-9-1003.Suche in Google Scholar
[11] J. Ge, Y. Liao, and B. Zhang, “Resistance distances and the Moon-type formula of a vertex-weighted complete split graph,” Discrete Appl. Math., vol. 359, pp. 10–15, 2024. https://doi.org/10.1016/j.dam.2024.07.040.Suche in Google Scholar
[12] S. Xu, H. Zhou, and X. Pan, “A method for constructing graphs with the same resistance spectrum,” Discrete Math., vol. 348, no. 2, 2025, Art. no. 114284. https://doi.org/10.1016/j.disc.2024.114284.Suche in Google Scholar
[13] M. S. Sardar, S. Xu, and X. Pan, “Computation of the resistance distance and the Kirchhoff index for the two types of claw-free cubic graphs,” Appl. Math. Comput., vol. 473, 2024, Art. no. 128670. https://doi.org/10.1016/j.amc.2024.128670.Suche in Google Scholar
[14] W. Kook and K. J. Lee, “Simplicial Kirchhoff index of random complexes,” Adv. Appl. Math., vol. 159, 2024, Art. no. 102733. https://doi.org/10.1016/j.aam.2024.102733.Suche in Google Scholar
[15] H. Chen, “Resistance distances and Kirchhoff index in generalised join graphs,” Z. Naturforsch. A, vol. 72, no. 3, pp. 207–215, 2017. https://doi.org/10.1515/zna-2016-0295.Suche in Google Scholar
[16] W. Sun, Y. Yang, and S. Xu, “Resistance distance and Kirchhoff index of unbalanced blowups of graphs,” Discrete Math., vol. 348, no. 3, 2025, Art. no. 114327. https://doi.org/10.1016/j.disc.2024.114327.Suche in Google Scholar
[17] S. Li, D. Li, and W. Yan, “Combinatorial explanation of the weighted Wiener (Kirchhoff) index of trees and unicyclic graphs,” Discrete Math., vol. 345, no. 12, 2022, Art. no. 113109. https://doi.org/10.1016/j.disc.2022.113109.Suche in Google Scholar
[18] S. Xu and K. Xu, “Resistance distances in generalized join graphs,” Discrete Appl. Math., vol. 362, pp. 18–33, 2025. https://doi.org/10.1016/j.dam.2024.11.013.Suche in Google Scholar
[19] Y. Yang and D. J. Klein, “Comparison theorems on resistance distances and Kirchhoff indices of S, T-isomers,” Discrete Appl. Math., vol. 175, pp. 87–93, 2014. https://doi.org/10.1016/j.dam.2014.05.014.Suche in Google Scholar
[20] Y. Yang and D. Wang, “Extremal phenylene chains with respect to the Kirchhoff index and degree-based topological indices,” IAENG Int. J. Appl. Math., vol. 49, no. 3, pp. 274–280, 2019.Suche in Google Scholar
[21] Y. Yang and W. Sun, “Minimal hexagonal chains with respect to the Kirchhoff index,” Discrete Math., vol. 345, no. 12, 2022, Art. no. 113099. https://doi.org/10.1016/j.disc.2022.113099.Suche in Google Scholar
[22] W. Sun and Y. Yang, “Extremal pentagonal chains with respect to the Kirchhoff index,” Appl. Math. Comput., vol. 437, 2023, Art. no. 127534. https://doi.org/10.1016/j.amc.2022.127534.Suche in Google Scholar
[23] L. Zhang, “The minimum Kirchhoff index of phenylene chains,” Discrete Appl. Math., vol. 340, pp. 69–75, 2023. https://doi.org/10.1016/j.dam.2023.06.043.Suche in Google Scholar
[24] Q. Ma, “Extremal polygonal chains with respect to the Kirchhoff index,” Discrete Appl. Math., vol. 342, pp. 218–226, 2024. https://doi.org/10.1016/j.dam.2023.09.022.Suche in Google Scholar
[25] H. Liu and L. You, “Extremal Kirchhoff index in polycyclic chains,” Discrete Appl. Math., vol. 348, pp. 292–300, 2024. https://doi.org/10.1016/j.dam.2024.01.046.Suche in Google Scholar
[26] C. Li, H. Bian, and H. Yu, “Extremal polyphenyl chains with respect to the Kirchhoff index,” Theor. Comput. Sci., vol. 1023, 2024, Art. no. 114893. https://doi.org/10.1016/j.tcs.2024.114893.Suche in Google Scholar
[27] Y. Sun, et al.., “Structural evolution of boron clusters on Ag (111) surfaces-from atomic chains to triangular sheets with hexagonal holes,” Chem. Phys. Chem., vol. 22, no. 9, pp. 894–903, 2021. https://doi.org/10.1002/cphc.202001019.Suche in Google Scholar PubMed
[28] A. Amer, M. Irfan, and H. U. Rehman, “Computational aspects of line graph of boron triangular nanotubes Math,” Methods Appl. Sci., pp. 1–7, 2021.10.1002/mma.7333Suche in Google Scholar
[29] J. Liu, et al.., “Topological aspects of boron nanotubes,” Adv. Mater. Sci. Eng., vol. 1, 2018, Art. no. 5729291.10.1155/2018/5729291Suche in Google Scholar
[30] E. Cheng, L. Lipták, and F. Sala, “Linearly many faults in 2 tree-generated networks,” Networks, vol. 55, no. 2, pp. 90–98, 2010. https://doi.org/10.1002/net.20319.Suche in Google Scholar
[31] A. Ullah, S. Zaman, and A. Hamraz, “Zagreb connection topological descriptors and structural property of the triangular chain structures,” Phys. Scr., vol. 98, no. 2, 2023, Art. no. 025009.10.1088/1402-4896/acb327Suche in Google Scholar
[32] P. Creed, “Sampling Eulerian orientations of triangular lattice graphs,” J. Discrete Algorithms, vol. 7, no. 2, pp. 168–180, 2009. https://doi.org/10.1016/j.jda.2008.09.006.Suche in Google Scholar
[33] A. Ali and A. A. Bhatti, “Extremal triangular chain graphs for bond incident degree (BID) indices,” Ars Combin., vol. 141, pp. 213–227, 2016.Suche in Google Scholar
[34] X. Yao, S. Dong, H. Yu, and J. Liu, “Monte Carlo simulation of magnetic behavior of a spin-chain system on a triangular lattice,” Phys. Rev. B, vol. 74, no. 13, 2006, Art. no. 134421. https://doi.org/10.1103/physrevb.74.134421.Suche in Google Scholar
[35] K. Boya, et al.., “Magnetic properties of the S=52$S=\frac{5}{2}$ anisotropic triangular chain compound Bi3FeMo2O12,” Phys. Rev. B, vol. 104, no. 18, 2021, Art. no. 184402. https://doi.org/10.1103/physrevb.104.184402.Suche in Google Scholar
[36] W. Barrett, E. J. Evans, and A. E. Francis, “Resistance distance in straight linear 2-trees,” Discrete Appl. Math., vol. 258, pp. 13–34, 2019. https://doi.org/10.1016/j.dam.2018.10.043.Suche in Google Scholar
[37] E. Teufl and S. Wagner, “Determinant identities for Laplace matrices,” Linear Algebra Appl., vol. 432, no. 1, pp. 441–457, 2010. https://doi.org/10.1016/j.laa.2009.08.028.Suche in Google Scholar
[38] A. E. Kennelly, “The equivalence of triangles and three-pointed stars in conducting networks,” Electr. World Eng., vol. 34, no. 1, pp. 413–414, 1899.Suche in Google Scholar
[39] A. Nikseresht and Z. Sepasdar, “On the Kirchhoff and the Wiener indices of graphs and block decomposition,” Electron. J. Comb., vol. 25, no. 1, p. P1, 2014. https://doi.org/10.37236/3508.Suche in Google Scholar
[40] Y. Yang and D. J. Klein, “A recursion formula for resistance distances and its applications,” Discrete Appl. Math., vol. 161, nos. 16–17, pp. 2702–2715, 2013. https://doi.org/10.1016/j.dam.2012.07.015.Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Atomic, Molecular & Optical Physics
- Artificial intelligence assisted photonic bio sensing for rapid bacterial diseases
- A compact analytical solution of the Dicke superradiance master equation via residue calculus
- Chemical Physics
- Extremal linear triangular chains with respect to the Kirchhoff index
- Hydrodynamics & Plasma Physics
- Heat and mass transfer enhancement for MHD nanofluid coupled to elastic interface
- Effect of Combined Kappa–Cairns distributed electrons on ion-acoustic solitary structures in electron–ion dusty plasma
- The solitary periodic soliton and formation of shock with phase shift in multi-component unmagnetized dusty plasmas
- Study of propagation dynamics of intense Laguerre–Gaussian laser beam in preformed plasma channel created by ignitor–heater method
- Solid State Physics & Materials Science
- Effects of substrate bias on the properties of TiCrN films deposited on 316L by RF magnetron sputtering
- Generalized Debye scattering formula for strained amorphous materials
Artikel in diesem Heft
- Frontmatter
- Atomic, Molecular & Optical Physics
- Artificial intelligence assisted photonic bio sensing for rapid bacterial diseases
- A compact analytical solution of the Dicke superradiance master equation via residue calculus
- Chemical Physics
- Extremal linear triangular chains with respect to the Kirchhoff index
- Hydrodynamics & Plasma Physics
- Heat and mass transfer enhancement for MHD nanofluid coupled to elastic interface
- Effect of Combined Kappa–Cairns distributed electrons on ion-acoustic solitary structures in electron–ion dusty plasma
- The solitary periodic soliton and formation of shock with phase shift in multi-component unmagnetized dusty plasmas
- Study of propagation dynamics of intense Laguerre–Gaussian laser beam in preformed plasma channel created by ignitor–heater method
- Solid State Physics & Materials Science
- Effects of substrate bias on the properties of TiCrN films deposited on 316L by RF magnetron sputtering
- Generalized Debye scattering formula for strained amorphous materials
