Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphs

  • Jing Zhao
  • Jia-Bao LiuEmail author
  • Sakander Hayat
Original Research


Resistance distance is a novel distance function, also a new intrinsic graph metric, which makes some extensions of ordinary distance. Let \(O_n\) be a linear crossed octagonal graph. Recently, Pan and Li (Int J Quantum Chem 118(24):e25787, 2018) derived the closed formulas for the Kirchhoff index, multiplicative degree-Kirchhoff index and the number of spanning trees of \(H_n\). They pointed that it is interesting to give the explicit formulas for the Kirchhoff and multiplicative degree-Kirchhoff indices of \(O_n\). Inspired by these, in this paper, two resistance distance-based graph invariants, namely, Kirchhoff and multiplicative degree-Kirchhoff indices are studied. We firstly determine formulas for the Laplacian (normalized Laplacian, resp.) spectrum of \(O_n\). Further, the formulas for those two resistance distance-based graph invariants and spanning trees are given. More surprising, we find that the Kirchhoff (multiplicative degree-Kirchhoff, resp.) index is almost one quarter to Wiener (Gutman, resp.) index of a linear crossed octagonal graph.


Laplacian Normalized Laplacian Kirchhoff index Multiplicative degree-Kirchhoff index Spanning tree 

Mathematics Subject Classification

05C50 05C90 



This work is supported by National Natural Science Foundation of China (Nos. 11601006, 11801007), China Postdoctoral Science Foundation under Grant 2017M621579; the Postdoctoral Science Foundation of Jiangsu Province under Grant 1701081B; Project of Anhui Jianzhu University under Grant no. 2016QD116 and 2017dc03, Natural Science Foundation of Anhui Province, 1808085MA17.


  1. 1.
    Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  2. 2.
    Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)CrossRefGoogle Scholar
  3. 3.
    Gutman, I.: Selected properties of the Schultz molecular topological index. J. Chem. Inf. Comput. Sci. 34, 1087–1089 (1994)CrossRefGoogle Scholar
  4. 4.
    Pan, Y., Li, J.: Graphs that minimizing symmetric division degree index. MATCH Commun. Math. Comput. Chem. 82, 43–55 (2019)Google Scholar
  5. 5.
    Liu, J.-B., Wang, C., Wang, S., Wei, B.: Zagreb indices and multiplicative Zagreb indices of Eulerian graphs. Bull. Malays. Math. Sci. Soc. 42, 67–78 (2019)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Liu, J.-B., Zhao, J., Min, J., Cao, J.D.: On the Hosoya index of graphs formed by a fractal graph. Fractals (2019). CrossRefGoogle Scholar
  7. 7.
    Klein, D.J., Randić, M.: Resistance distances. J. Math. Chem. 12, 81–95 (1993)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Klein, D.J., Ivanciuc, O.: Graph cyclicity, excess conductance, and resistance deficit. J. Math. Chem. 30, 271–287 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Klein, D.J.: Resistance–distance sum rules. Croat. Chem. Acta 75, 633–649 (2002)Google Scholar
  10. 10.
    Chen, H.Y., Zhang, F.J.: Resistance distance and the normalized Laplacian spectrum. Discrete Appl. Math. 155, 654–661 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zhang, H.H., Li, S.C.: On the Laplacian spectral radius of bipartite graphs with fixed order and size. Discrete Appl. Math. 229, 139–147 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liu, J.-B., Pan, X.F., Hu, F.T., Hu, F.F.: Asymptotic Laplacian-energy-like invariant of lattices. Appl. Math. Comput. 253, 205–214 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Huang, J., Li, S.C.: On the normalised laplacian spectrum, degree-Kirchhoff index and spanning trees of graphs. Bull. Aust. Math. Soc. 91, 353–367 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Huang, J., Li, S.C.: The normalized Laplacians on both \(k\)-triangle graph and \(k\)-quadrilateral graph with their applications. Appl. Math. Comput. 320, 213–225 (2018)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mohar, B.: The Laplacian spectrum of graphs. In: Alavi, Y., Chartrand, G., Oellermann, O.R., Schwenk, A.J. (eds.) Graph Theory, Combinatorics, and Applications, vol. 2, pp. 871–898. Wiley, Hoboken (1991)Google Scholar
  16. 16.
    Chen, Y., Dai, M., Wang, X., Sun, Y., Su, W.: Spectral analysis for weighted iterated triangulations of graphs. Fractals 26(01), 1850017 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dai, M., Shen, J., Dai, L., Ju, T., Hou, Y., Su, W.: Generalized adjacency and Laplacian spectra of the weighted corona graphs. Physica A 528, 121285 (2019)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pan, Y., Li, J., Li, S.C., Luo, W.: On the normalized Laplacians with some classical parameters involving graph transformations. Linear Multilinear A (2018). CrossRefGoogle Scholar
  19. 19.
    Gutman, I., Mohar, B.: The quasi-Wiener and the Kirchhoff indices coincide. J. Chem. Inf. Comput. Sci. 36, 982–985 (1996)CrossRefGoogle Scholar
  20. 20.
    Huang, J., Li, S.C., Li, X.C.: The normalized laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains. Appl. Math. Comput. 289, 324–334 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Peng, Y.J., Li, S.C.: On the Kirchhoff index and the number of spanning trees of linear phenylenes. MATCH Commun. Math. Comput. Chem. 77(3), 765–780 (2017)MathSciNetGoogle Scholar
  22. 22.
    He, C., Li, S.C., Luo, W., Sun, L.: Calculating the normalized Laplacian spectrum and the number of spanning trees of linear pentagonal chains. J. Comput. Appl. Math. 344, 381–393 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhu, Z.X., Liu, J.-B.: The normalized Laplacian, degree-Kirchhoff index and the spanning tree numbers of generalized phenylenes. Discrete Appl. Math. 254, 256–267 (2019)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li, S.C., 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)CrossRefGoogle Scholar
  25. 25.
    Ma, X., Bian, H.: The normalized Laplacians, degree-Kirchhoff index and the spanning trees of hexagonal M\(\ddot{o}\)bius graphs. Appl. Math. Comput. 355, 33–46 (2019)MathSciNetGoogle Scholar
  26. 26.
    Liu, J.-B., Zhao, J., Zhu, Z.X.: On the number of spanning trees and normalized Laplacian of linear octagonal-quadrilateral networks. Int. J. Quantum Chem. 119(17), e25971 (2019)CrossRefGoogle Scholar
  27. 27.
    Yang, Y.J., Zhang, H.P.: Kirchhoff Index of Linear Hexagonal Chains. Int. J. Quantum Chem. 108(3), 503–512 (2008)CrossRefGoogle Scholar
  28. 28.
    Huang, J., Li, S.C., Sun, L.: The normalized Laplacians degree-Kirchhoff index and the spanning trees of linear hexagonal chains. Discrete Appl. Math. 207, 67–79 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pan, Y., Li, J.: Kirchhoff index, multiplicative degree-Kirchhoff index and spanning trees of the linear crossed hexagonal chains. Int. J. Quantum Chem. 118(24), e25787 (2018)CrossRefGoogle Scholar
  30. 30.
    Yang, Y.L., Yu, T.Y.: Graph theory of viscoelasticities for polymers with starshaped, multiple-ring and cyclic multiple-ring molecules. Makromol. Chem. 186, 609 (1985) CrossRefGoogle Scholar

Copyright information

© Korean Society for Informatics and Computational Applied Mathematics 2019

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsAnhui Jianzhu UniversityHefeiPeople’s Republic of China
  2. 2.School of MathematicsSoutheast UniversityNanjingPeople’s Republic of China
  3. 3.Faculty of Engineering SciencesGIK Institute of Engineering Sciences and TechnologyTopiPakistan

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