Advertisement

Maximum energy trees with two maximum degree vertices

  • Xueliang Li
  • Xiangmei Yao
  • Jianbin Zhang
  • Ivan Gutman
Article

Abstract

The energy E of a graph G is equal to the sum of the absolute values of the eigenvalues of G. In 2005 Lin et al. determined the trees with a given maximum vertex degree Δ and maximum E, that happen to be trees with a single vertex of degree Δ. We now offer a simple proof of this result and, in addition, characterize the maximum energy trees having two vertices of maximum degree Δ.

Keywords

Energy of graph Tree Maximum degree 

References

  1. 1.
    Cvetković D., Doob M., Sachs H.: Spectra of Graphs—Theory and Application. Academic Press, New York (1980)Google Scholar
  2. 2.
    Gutman I.: Ber. Math.–Statist. Sekt. Forschungsz. Graz 103, 1 (1978)Google Scholar
  3. 3.
    Gutman I.: J. Math. Chem. 1, 123 (1987)Google Scholar
  4. 4.
    Coulson C.A.: Proc. Cambridge Phil. Soc. 36, 201 (1940)CrossRefGoogle Scholar
  5. 5.
    C.A. Coulson, J. Jacobs, J. Chem. Soc. 2805 (1949)Google Scholar
  6. 6.
    Hall G.G.: Proc. R. Soc. A 229, 251 (1955)CrossRefGoogle Scholar
  7. 7.
    Hall G.G.: Trans. Faraday Soc. 53, 573 (1957)CrossRefGoogle Scholar
  8. 8.
    Ruedenberg K.: J. Chem. Phys. 29, 1232 (1958)CrossRefGoogle Scholar
  9. 9.
    Ruedenberg K.: J. Chem. Phys. 34, 1884 (1961)CrossRefGoogle Scholar
  10. 10.
    McClelland B.J.: J. Chem. Phys. 54, 640 (1971)CrossRefGoogle Scholar
  11. 11.
    Gutman I.: Chem. Phys. Lett. 24, 283 (1974)CrossRefGoogle Scholar
  12. 12.
    Gutman I.: Theor. Chim. Acta 35, 355 (1974)CrossRefGoogle Scholar
  13. 13.
    Gutman I.: Theor. Chim. Acta 45, 79 (1977)CrossRefGoogle Scholar
  14. 14.
    Koolen J., Moulton V.: Adv. Appl. Math. 26, 47 (2001)CrossRefGoogle Scholar
  15. 15.
    Koolen J., Moulton V.: Graph Combin. 19, 131 (2003)CrossRefGoogle Scholar
  16. 16.
    W.H. Haemers, Lin. Algebra Appl. (in press)Google Scholar
  17. 17.
    Wang M., Hua H., Wang D.: J. Math. Chem. 43, 1389 (2008)CrossRefGoogle Scholar
  18. 18.
    Yan W., Ye L.: Appl. Math. Lett. 18, 1046 (2005)CrossRefGoogle Scholar
  19. 19.
    Yan W., Ye L.: MATCH Commun. Math. Comput. Chem. 53, 449 (2005)Google Scholar
  20. 20.
    Lin W., Guo X., Li H.: MATCH Commun. Math. Comput. Chem. 54, 363 (2005)Google Scholar
  21. 21.
    Zhou B., Li F.: J. Math. Chem. 39, 465 (2006)CrossRefGoogle Scholar
  22. 22.
    Yu A., Lv X.: Lin. Algebra Appl. 418, 625 (2006)CrossRefGoogle Scholar
  23. 23.
    Ye L., Yuan X.: MATCH Commun. Math. Comput. Chem. 57, 193 (2007)Google Scholar
  24. 24.
    Li N., Li S.: MATCH Commun. Math. Comput. Chem. 59, 291 (2008)Google Scholar
  25. 25.
    Gutman I., Radenković S., Li N., Li S.: MATCH Commun. Math. Comput. Chem. 59, 315 (2008)Google Scholar
  26. 26.
    Ou J.: J. Math. Chem. 43, 328 (2008)CrossRefGoogle Scholar
  27. 27.
    Li S., Li X.: MATCH Commun. Math. Comput. Chem. 61, 383 (2009)Google Scholar
  28. 28.
    Hou Y.: J. Math. Chem. 29, 163 (2001)CrossRefGoogle Scholar
  29. 29.
    Gutman I., Hou Y.: MATCH Commun. Math. Comput. Chem. 43, 17 (2001)Google Scholar
  30. 30.
    Hou Y., Gutman I., Woo C.W.: Lin. Algebra Appl. 356, 27 (2002)CrossRefGoogle Scholar
  31. 31.
    Li F., Zhou B.: MATCH Commun. Math. Comput. Chem. 54, 379 (2005)Google Scholar
  32. 32.
    Wang W.H., Chang A., Zhang L.Z., Lu D.Q.: J. Math. Chem. 39, 231 (2006)CrossRefGoogle Scholar
  33. 33.
    Hua H.: MATCH Commun. Math. Comput. Chem. 57, 351 (2007)Google Scholar
  34. 34.
    Hua H.: MATCH Commun. Math. Comput. Chem. 58, 57 (2007)Google Scholar
  35. 35.
    Gutman I., Furtula B., Hua H.: MATCH Commun. Math. Comput. Chem. 58, 85 (2007)Google Scholar
  36. 36.
    Hua H., Wang M.: Lin. Algebra Appl. 426, 478 (2007)CrossRefGoogle Scholar
  37. 37.
    Wang W.H., Chang A., Lu D.Q.: J. Math. Chem. 42, 311 (2007)CrossRefGoogle Scholar
  38. 38.
    Li X., Zhang J., Zhou B.: J. Math. Chem. 42, 729 (2007)CrossRefGoogle Scholar
  39. 39.
    Li F., Zhou B.: J. Math. Chem. 43, 476 (2008)CrossRefGoogle Scholar
  40. 40.
    Li S., Li X., Zhu Z.: MATCH Commun. Math. Comput. Chem. 61, 325 (2009)Google Scholar
  41. 41.
    Hou Y.: Lin. Multilin. Algebra 49, 347 (2002)CrossRefGoogle Scholar
  42. 42.
    Zhang J., Zhou B.: J. Math. Chem. 37, 423 (2005)CrossRefGoogle Scholar
  43. 43.
    Li X., Zhang J.: Lin. Algebra Appl. 427, 87 (2007)CrossRefGoogle Scholar
  44. 44.
    Yang Y., Zhou B.: MATCH Commun. Math. Comput. Chem. 59, 321 (2008)Google Scholar
  45. 45.
    Liu Z., Zhou B.: MATCH Commun. Math. Comput. Chem. 59, 381 (2008)Google Scholar
  46. 46.
    Li S., Li X., Zhu Z.: MATCH Commun. Math. Comput. Chem. 59, 397 (2008)Google Scholar
  47. 47.
    Zhang F., Li Z., Wang L.: Chem. Phys. Lett. 337, 125 (2001)CrossRefGoogle Scholar
  48. 48.
    Zhang F., Li Z., Wang L.: Chem. Phys. Lett. 337, 131 (2001)CrossRefGoogle Scholar
  49. 49.
    Rada J., Tineo A.: Lin. Algebra Appl. 372, 333 (2003)CrossRefGoogle Scholar
  50. 50.
    Ren H., Zhang F.: J. Math. Chem. 42, 1041 (2007)CrossRefGoogle Scholar
  51. 51.
    I. Gutman, in Algebraic Combinatorics and Applications, ed. by A. Betten, A. Kohnert, R. Laue, A. Wassermann (Springer-Verlag, Berlin, 2001), pp. 196–211Google Scholar
  52. 52.
    Gutman I., Zhang F.: Discr. Appl. Math. 15, 25 (1986)CrossRefGoogle Scholar
  53. 53.
    Gutman I., Polansky O.E.: Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin (1986)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Xueliang Li
    • 1
  • Xiangmei Yao
    • 1
  • Jianbin Zhang
    • 1
  • Ivan Gutman
    • 2
  1. 1.Center for Combinatorics and LPMC-TJKLCNankai UniversityTianjinPeople’s Republic of China
  2. 2.Faculty of ScienceUniversity of KragujevacKragujevacSerbia

Personalised recommendations