Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs

  • Jia-Bao Liu
  • Chunxiang Wang
  • Shaohui WangEmail author
  • Bing Wei


For a graph \(G = (V(G), E(G))\), let d(u), d(v) be the degrees of the vertices uv in G. The first and second Zagreb indices of G are defined as \( M_1(G) = \sum _{u \in V(G)} d(u)^2\) and \( M_2(G) = \sum _{uv \in E(G)} d(u)d(v)\), respectively. The first (generalized) and second Multiplicative Zagreb indices of G are defined as \(\Pi _{1,c}(G) = \prod _{v \in V(G)}d(v)^c\) and \(\Pi _2(G) = \Pi _{uv \in E(G)} d(u)d(v)\), respectively. The (Multiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Narumi and Katayama in 1980s. Denote by \({\mathcal {G}}_{n}\) the set of all Eulerian graphs of order n. In this paper, we characterize Eulerian graphs with first three smallest and largest Zagreb indices and Multiplicative Zagreb indices in \({\mathcal {G}}_{n}\).


Extremal bounds Zagreb index Multiplicative Zagreb index Eulerian graphs 

Mathematics Subject Classification

05C12 05C05 



This work is partially supported by National Natural Science Foundation of China (Nos. 11601006, 11471016, 11401004, 11571134, 11371162), Anhui Provincial Natural Science Foundation (Nos. KJ2015A331, KJ2013B105, 1408085QA03), the Self-determined Research Funds of Central China Normal University from the colleges basic research and operation of MOE. The authors would like to express their sincere gratitude to the anonymous referees and the editor for many friendly and helpful suggestions, which led to great deal of improvement of the original manuscript.


  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Liu, J.B., Pan, X.F.: Asymptotic incidence energy of lattices. Phys. A 422, 193–202 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    Khadikar, P.V.: On a novel structural descriptor PI. Natl. Acad. Sci. Lett. 23, 113–118 (2000)MathSciNetGoogle Scholar
  5. 5.
    Chen, H., Wu, R., Deng, H.: The extremal values of some topological indices in bipartite graphs with a given matching number. Appl. Math. Comput. 280, 103–109 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Deng, H., Huang, G., Jiang, X.: A unified linear programming modeling of some topological indices. J. Comb. Optim. 30(3), 826–837 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Liu, J.B., Pan, X.F., Yu, L., Li, D.: Complete characterization of bicyclic graphs with minimal Kirchhoff index. Discrete Appl. Math. 200, 95–107 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Liu, J.B., Wang, W.R., Pan, X.F., Zhang, Y.M.: On degree resistance distance of cacti. Discrete Appl. Math. 203, 217–225 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Das, K.C.: Maximizing the sum of the squares of the degrees of a graph. Discrete Math. 285, 57–66 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Wang, S., Farahani, M., Kanna, M., Kumar, R.: Schultz polynomials and their topological indices of Jahangir graphs \(\text{ J }_{2, m}\). Appl. Math. 7, 1632–1637 (2016)CrossRefGoogle Scholar
  11. 11.
    Wang, S., Farahani, M., Kanna, M., Jamil, M., Kumar, R.: The Wiener Index and the Hosoya Polynomial of the Jahangir Graphs. Appl. Comput. Math. 5, 138–141 (2016)CrossRefGoogle Scholar
  12. 12.
    Wang, S., Farahani, M., Baig, A., Sajja, W.: The sadhana polynomial and the sadhana index of polycyclic aromatic hydrocarbons PAHk. J. Chem. Pharm. Res. 8, 526–531 (2016)Google Scholar
  13. 13.
    Lang, R., Deng, X., Lu, H.: Bipartite graphs with the maximal value of the second Zagreb index. Bull. Malays. Math. Sci. Soc. 36, 1–6 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Duan, S., Zhu, Z.: Extremal bicyclic graph with perfect matching for different indices. Bull. Malays. Math. Sci. Soc. 36, 733–745 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gutman, I., Rus̆čić, B., Trinajstić, N., Wilcox, C.F.: Graph theory and molecular orbitals. XII. Acyclic polyenes. J. Chem. Phys. 62, 3399–3405 (1975)CrossRefGoogle Scholar
  16. 16.
    Gutman, I., Trinajstić, N.: Graph theory and molecular orbitals. Total \(\uppi \)-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)CrossRefGoogle Scholar
  17. 17.
    Estes, J., Wei, B.: Sharp bounds of the Zagreb indices of \(k\)-trees. J. Comb. Optim. 27, 271–291 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gutman, I.: Multiplicative Zagreb indices of trees. Bull. Soc. Math. Banja Luka 18, 17–23 (2011)zbMATHGoogle Scholar
  19. 19.
    Li, S., Yang, H., Zhao, Q.: Sharp bounds on Zagreb indices of cacti with \(k\) pendant vertices. Filomat 26, 1189–1200 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nikolić, S., Kovac̆ević, G., Milicc̆ević, A., Trinajstić, N.: The Zagreb indices 30 years after. Croat. Chem. Acta 76, 113–124 (2003)Google Scholar
  21. 21.
    Wang, S., Wei, B.: Multiplicative Zagreb indices of cacti. Discrete Math. Algorithm. Appl. 8, 1650040 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Narumi, H., Katayama, M.: Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons. Mem. Fac. Eng. Hokkaido Univ. 16, 209–214 (1984)Google Scholar
  23. 23.
    Wang, S., Wei, B.: Multiplicative Zagreb indices of \(k\)-trees. Discrete Appl. Math. 180, 168–175 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhao, Q., Li, S.: On the maximum Zagreb index of graphs with \(k\) cut vertices. Acta Appl. Math. 111, 93–106 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gutman, I., Cruz, R., Rada, J.: Wiener index of Eulerian graphs. Discrete Appl. Math. 162, 247–250 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Xu, K., Hua, H.: A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs. MATCH Commun. Math. Comput. Chem. 68, 241–256 (2012)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Gutman, I., Furtula, B., Vukćvić, Ž.Kovijanić, Popivoda, G.: Zagreb indices and coindices. MATCH Commun. Math. Comput. Chem. 74, 5–16 (2015)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kazemi, R.: Note on the multiplicative Zagreb indices. Discrete Appl. Math. 198, 147–154 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Siddiqui, M.K., Imran, M., Ahmad, A.: On Zagreb indices, Zagreb polynomials of some nanostar dendrimers. Appl. Math. Comput. 280, 132–139 (2016)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Wang, C., Wang, S., Wei, B.: Cacti with extremal PI index. Trans. Comb. 5, 1–8 (2016)MathSciNetGoogle Scholar
  31. 31.
    Farahani, M.R.: First and second Zagreb polynomials of \(VC_5C_7[p, q]\) and \(HC_5C_7[p, q]\) nanotubes. Int. Lett. Chem. Phys. Astron. 12, 56–62 (2014)CrossRefGoogle Scholar
  32. 32.
    Farahani, M.R.: Zagreb indices and Zagreb polynomials of pent-heptagon nanotube \(VAC_5C_7(S)\). Chem. Phys. Res. J. 6(1), 35–40 (2013)MathSciNetGoogle Scholar
  33. 33.
    Farahani, M.R.: Zagreb indices and Zagreb polynomials of polycyclic aromatic hydrocarbons PAHs. J. Chem. Acta. 2, 70–72 (2013)Google Scholar
  34. 34.
    Tache, R.M.: On degree-based topological indices for bicyclic graphs. MATCH Commun. Math. Comput. Chem. 76, 99–116 (2016)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Berrocal, L., Olivieri, A., Rada, J.: Extremal values of vertex-degree-based topological indices over hexagonal systems with fixed number of vertices. Appl. Math. Comput. 243, 176–183 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  • Jia-Bao Liu
    • 1
  • Chunxiang Wang
    • 2
  • Shaohui Wang
    • 3
    Email author
  • Bing Wei
    • 4
  1. 1.School of Mathematics and PhysicsAnhui Jianzhu UniversityHefeiPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsCentral China Normal UniversityWuhanPeople’s Republic of China
  3. 3.Department of Mathematics and Computer ScienceAdelphi UniversityGarden CityUSA
  4. 4.Department of MathematicsThe University of MississippiUniversityUSA

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