Download PDF

Graphs and Combinatorics

, Volume 30, Issue 1, pp 109–118 | Cite as

Extremal Zagreb Indices of Graphs with a Given Number of Cut Edges

Open Access
Original Paper
First Online:
Received:
Revised:

Abstract

For a graph, the first Zagreb index M 1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M 2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. Denote by \({\mathcal{G}_{n,k}}\) the set of graphs with n vertices and k cut edges. In this paper, we showed the types of graphs with the largest and the second largest M 1 and M 2 among \({\mathcal{G}_{n,k}}\).

Keywords

Zagreb index Cut edges Grafting transformation 

Mathematics Subject Classification (2000)

05C05 92E10 05C35 
Download to read the full article text

References

  1. 1.
    Bondy J.A., Murty U.S.R.: Graph Theory with Applications. Macmillan, New York (1976)MATHGoogle Scholar
  2. 2.
    Gutman I., Trinajstić N.: Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)CrossRefGoogle Scholar
  3. 3.
    Gutman, I., Ruščić, B., Trinajstić, N., Wilcox, C.F.: Graph theory and molecular orbitals. XII. Acyclic Polyenes J. Chem. Phys. 62(195), 3399–3405Google Scholar
  4. 4.
    Nikolić S., Kovačević G., Miličević A., Trinajstić N.: The Zagreb indices 30 years after. Croat. Chem. Acta 76, 113–124 (2003)Google Scholar
  5. 5.
    Gutman I., Das K.C.: The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem 50, 83–92 (2004)MATHMathSciNetGoogle Scholar
  6. 6.
    Balaban A.T., Motoc I., Bonchev D., Mekenyan O.: Topological indices for structure-activity corrections. Topics Curr. Chem 114, 21–55 (1983)CrossRefGoogle Scholar
  7. 7.
    Gutman I., Das K.C.: The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem 50, 83–92 (2004)MATHMathSciNetGoogle Scholar
  8. 8.
    Xia F., Chen S.: Ordering unicyclic graphs with respect to Zagreb indices. MATCH Commun. Math. Comput. Chem 58, 663–673 (2007)MATHMathSciNetGoogle Scholar
  9. 9.
    Zhang H., Zhang S.: Uncyclic graphs with the first three smallest and largest first general Zagreb index. MATCH Commun. Math. Comput. Chem. 55, 427–438 (2006)MATHMathSciNetGoogle Scholar
  10. 10.
    Chen S., Deng H.: Extremal (n, n + 1)-graphs with respected to zeroth-order Randic index. J. Math. Chem. 42, 555–564 (2007)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Deng H.: A unified approach to the extremal Zagreb indices for trees, unicyclic graphs and bicyclic graphs. MATCH Commun. Math. Comput. Chem 57, 597–616 (2007)MATHMathSciNetGoogle Scholar
  12. 12.
    Zhou B.: Zagreb indices. MATCH Commun. Math. Comput. Chem 52, 113–118 (2004)MATHGoogle Scholar
  13. 13.
    Liu B.: Upper bounds for Zagreb indices of connected graphs. MATCH Commun. Math. Comput. Chem. 55, 439–446 (2006)MATHMathSciNetGoogle Scholar
  14. 14.
    Zhou B.: Further properties of Zagreb indices. MATCH Commun. Math. Comput. Chem 54, 233–239 (2005)MATHMathSciNetGoogle Scholar
  15. 15.
    Vukičević D., Trinajstić N.: On the discriminatory power of the Zagreb indices for molecular graphs. MATCH Commun. Math. Comput. Chem. 53, 111–138 (2005)MATHMathSciNetGoogle Scholar
  16. 16.
    Gutman I., Furtula B., Toropov A.A., Toropov A.P.: The grpah of atomic orbitals and its basic properties. 2. Zagreb indices. MATCH Commun. Math. Comput. Chem 53, 111–138 (2005)MathSciNetGoogle Scholar
  17. 17.
    Nikolić S., Tolić I.M., Trinajstić N., Baučić I.: On the Zagreb indices as complexity indices. Croat. Chem. Acta 73, 909–921 (2000)Google Scholar
  18. 18.
    Das K., Gutman I., Zhou B.: New upper bounds on Zagreb indices. J. Math. Chem 46, 514–521 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Xu K.: The Zagreb indices of graphs with a given clique number. Appl. Math. Lett 24, 1026–1030 (2011)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Behtoei A., Jannesari M., Taeri B.: Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity. Appl. Math. Lett. 22, 1571–1576 (2009)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Das K.: On comparing Zagreb indices of graphs. MATCH Commun. Math. Comput. Chem. 63, 433–440 (2010)MATHMathSciNetGoogle Scholar
  22. 22.
    Hfath-Tabar G.H.: Old and new Zagreb indices of graphs. MATCH Commun. Math. Comput. Chem. 65, 79–84 (2011)MathSciNetGoogle Scholar
  23. 23.
    Ashrafi A.R., DoŠlić T., Hamzeh A.: Extremal graphs with respect to the Zagreb coindices. MATCH Commun. Math. Comput. Chem 65, 85–92 (2011)MATHMathSciNetGoogle Scholar
  24. 24.
    Zhao Q., Li S.C.: On the maximum Zagreb indices of graphs with k cut vertices. Acta Appl. Math. 111, 93–106 (2010)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Li S.C., Zhao Q.: Sharp upper bounds on Zagreb indices of bicyclic graphs with a given matching number. Math. Comput. Model. 54, 2869–2879 (2011)CrossRefMATHGoogle Scholar
  26. 26.
    Liu H., Lu M., Tian F.: On the spectral radius of graphs with cut edges. Linear Algebra Appl. 389, 139–145 (2004)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Wu Y.R., He S., Shu J.L.: Largest spectral radius among graphs with cut edges. J. East China Norm. Univ. Nat. Sci. Ed. 3, 67–74 (2007)MathSciNetGoogle Scholar
  28. 28.
    Pepper R., Henry G., Sexton D.: Cut edges and independence number. MATCH Commun. Math. Comput. Chem 56, 403–408 (2006)MATHMathSciNetGoogle Scholar
  29. 29.
    Deng H.: On the Minimum Kirchhoff index of graphs with a given cut edges. MATCH Commun. Math. Comput. Chem. 63, 171–180 (2110)Google Scholar
  30. 30.
    Balakrishnan R., Sridharan N., Viswanathan Iyer K.: Wiener index of graphs with more than one cut vertex. Appl. Math. Lett. 21, 922–927 (2008)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Berman A., Zhang X.D.: On the spectral radius of graphs with cut vertices. J. Comb. Theory, Ser. B 83, 233–240 (2001)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2012

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.College of MathematicsHunan City UniversityYiyangPeople’s Republic of China
  2. 2.College of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China

Personalised recommendations