M-polynomial revisited: Bethe cacti and an extension of Gutman’s approach

  • Emeric Deutsch
  • Sandi KlavžarEmail author
Original Research


The M-polynomial of a graph G is defined as \(\sum _{i\le j} m_{i,j}(G)x^iy^j\), where \(m_{i,j}(G)\), \(i,j\ge 1\), is the number of edges uv of G such that \(\{d_v(G), d_u(G)\} = \{i,j\}\). Knowing the M-polynomial, formulas for bond incident degree indices (an important subclass of degree-based topological indices) can be obtained by means of specific operators defined on differentiable functions in two variables. This is illustrated on three infinite families of Bethe cacti. Gutman’s approach for the computation of the coefficients of the M-polynomial is also recalled and an extension of it is given. This extension is used to determine the M-polynomial of a two-parameter infinite family of lattice graphs.


M-polynomial Bethe cacti Degree-based topological index Bond incident degree index Graph polynomial 

Mathematics Subject Classification

05C07 05C31 92E10 



Sandi Klavžar acknowledges the financial support from the Slovenian Research Agency (Research Core Funding No. P1-0297).


  1. 1.
    Ali, A., Nazeer, W., Munir, M., Kang, S.M.: M-polynomials and topological indices of zigzag and rhombic benzenoid systems. Open Chem. 16, 73–78 (2018)CrossRefGoogle Scholar
  2. 2.
    An, M., Das, K.C.: First Zagreb index, \(k\)-connectivity, beta-deficiency and \(k\)-hamiltonicity of graphs. MATCH Commun. Math. Comput. Chem. 80, 141–151 (2018)MathSciNetGoogle Scholar
  3. 3.
    Balasubramanian, K.: Recent developments in tree-pruning methods and polynomials for cactus graphs and trees. J. Math. Chem. 4, 89–102 (1990)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bollobás, B., Erdös, P.: Graphs with extremal weights. Ars Comb. 50, 225–233 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, Z., Su, G., Volkmann, L.: Sufficient conditions on the zeroth-order general Randić index for maximally edge-connected graphs. Discrete Appl. Math. 218, 64–70 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Das, K.C., Balachandran, S., Gutman, I.: Inverse degree, Randić index and harmonic index of graphs. Appl. Anal. Discrete Math. 11, 304–313 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deutsch, E., Klavžar, S.: Computing the Hosoya polynomial of graphs from primary subgraphs. MATCH Commun. Math. Comput. Chem. 70, 627–644 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Deutsch, E., Klavžar, S.: \(M\)-polynomial and degree-based topological indices. Iran. J. Math. Chem. 6, 93–102 (2015)zbMATHGoogle Scholar
  9. 9.
    Doslić, T., Sedghi, S., Shobe, N.: Stirling numbers and generalized Zagreb indices. Iran. J. Math. Chem. 8, 1–5 (2017)zbMATHGoogle Scholar
  10. 10.
    Eliasi, M., Iranmanesh, A.: Hosoya polynomial of hierarchical product of graphs. MATCH Commun. Math. Comput. Chem. 69, 111–119 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gutman, I.: Molecular graphs with minimal and maximal Randić indices. Croat. Chem. Acta 75, 357–369 (2002)Google Scholar
  12. 12.
    Gutman, I.: Degree-based topological indices. Croat. Chem. Acta 86, 351–361 (2013)CrossRefGoogle Scholar
  13. 13.
    Gutman, I., Tošović, J.: Testing the quality of molecular structure descriptors. Vertex-degree-based topological indices. J. Serb. Chem. Soc. 78, 805–810 (2013)CrossRefGoogle Scholar
  14. 14.
    Hollas, B.: The covariance of topological indices that depend on the degree of a vertex. MATCH Commun. Math. Comput. Chem. 54, 177–187 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hosoya, H.: On some counting polynomials in chemistry. Discrete Appl. Math. 19, 239–257 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kang, S.M., Nazeer, W., Gao, W., Afzal, D., Gillani, S.N.: M-polynomials and topological indices of dominating David derived networks. Open Chem. 16, 201–213 (2018)CrossRefGoogle Scholar
  17. 17.
    Kwun, Y.C., Munir, M., Nazeer, W., Rafique, S., Kang, S.M.: M-polynomials and topological indices of V-phenylenic nanotubes and nanotori. Sci. Rep. 7, 8756 (2017)CrossRefGoogle Scholar
  18. 18.
    Lin, X., Xu, S.J., Yeh, Y.N.: Hosoya polynomials of circumcoronene series. MATCH Commun. Math. Comput. Chem. 69, 755–763 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Liu, J.-B., Wang, S., Wang, C., Hayat, S.: Further results on computation of topological indices of certain networks. IET Control Theory Appl. 11, 2065–2071 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ma, Y., Cao, S., Shi, Y., Gutman, I., Dehmer, M., Furtula, B.: From the connectivity index to various Randić-type descriptors. MATCH Commun. Math. Comput. Chem. 80, 85–106 (2018)MathSciNetGoogle Scholar
  21. 21.
    Milivojević, M., Pavlović, L.: The variation of the Randić index with regard to minimum and maximum degree. Discrete Appl. Math. 217, 286–293 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Munir, M., Nazeer, W., Rafique, S., Kang, S.M.: \(M\)-polynomial and related topological indices of nanostar dendrimers. Symmetry 8, 97 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Munir, M., Nazeer, W., Rafique, S., Kang, S.M.: \(M\)-polynomial and degree-based topological indices of polyhex nanotubes. Symmetry 8, 149 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Munir, M., Nazeer, W., Rafique, S., Nizami, A.R., Kang, S.M.: Some computational aspects of boron triangular nanotubes. Symmetry 9, 6 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rezaei, M., Gao, W., Siddiqui, M.K., Farahani, M.R.: Computing hyper Zagreb index and \(M\)-polynomials of titania nanotubes \({\rm TiO}_2[m, n]\). Sigma J. Eng. Nat. Sci. 35, 707–714 (2017)Google Scholar
  26. 26.
    Tratnik, N., Žigert, P.: Pleteršek, Relationship between the Hosoya polynomial and the edge-Hosoya polynomial of trees. MATCH Commun. Math. Comput. Chem. 78, 181–187 (2017)MathSciNetGoogle Scholar
  27. 27.
    Vetrík, T.: Degree-based topological indices of hexagonal nanotubes. J. Appl. Math. Comput. (2017). MathSciNetzbMATHGoogle Scholar
  28. 28.
    Vukičević, D., Đurđević, J.: Bond additive modeling 10. Upper and lower bounds of bond incident degree indices of catacondensed fluoranthenes. Chem. Phys. Lett. 515, 186–189 (2011)CrossRefGoogle Scholar
  29. 29.
    Vukičević, D., Sedlar, J., Stevanović, D.: Comparing Zagreb indices for almost all graphs. MATCH Commun. Math. Comput. Chem. 78, 323–336 (2017)MathSciNetGoogle Scholar
  30. 30.
    Wang, S., Wang, C., Liu, J.-B.: On extremal multiplicative Zagreb indices of trees with given domination number. Appl. Math. Comput. 332, 338–350 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice-Hall, New York (2001)Google Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Polytechnic Institute of New York UniversityBrooklynUSA
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  4. 4.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

Personalised recommendations