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Note on the Minimal Energy Ordering of Conjugated Trees

  • Yulan Xiao
  • Bofeng Huo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7389)

Abstract

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix A(G). Gutman proposed two conjectures on the minimal energy of the class of conjugated trees (trees having a perfect matching). Zhang and Li determined the trees in the class with the minimal and second-minimal energies, which confirms the conjectures. Zhang and Li also found that the conjugated tree with the third-minimal energy is one of the two graphs which are quasi-order incomparable. Recently, Huo, Li and Shi found there exists a fixed positive integer N 0, such that for all n > N 0, the energy of the graphs with the third-minimal through the sixth-minimal are determined. In this paper, the N 0 is fixed by a recursive method, and the problem is solved completely.

Keywords

extremal graph minimal energy quasi-order incomparable conjugated tree 

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References

  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)zbMATHCrossRefGoogle Scholar
  2. 2.
    Coulson, C.A.: On the Calculation of the Energy in Unsaturated Hydrocarbon Molecules. Proc. Cambridge Phil. Soc. 36, 201–203 (1940)CrossRefGoogle Scholar
  3. 3.
    Cvetković, D.M., Doob, M., Sachs, H.: Spectra of Graphs-Theory and Application. Academic Press, New York (1980)Google Scholar
  4. 4.
    Chen, A., Chang, A., Shiu, W.C.: Energy Ordering of Unicyclic Graphs. MATCH Commun. Math. Comput. Chem. 55, 95–102 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gutman, I.: Acylclic Systems with Extremal Hückel π-electron energy. Theor. Chim. Acta 45, 79–87 (1977)CrossRefGoogle Scholar
  6. 6.
    Gutman, I., Polansky, O.E.: Mathematical Concepts in Organic Chemistry. Springer, Berlin (1986)zbMATHCrossRefGoogle Scholar
  7. 7.
    Gutman, I., Zhang, F.: On the Ordering of Graphs with Respect to Their Matching Numbers. Discrete Appl. Math. 15, 25–33 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gutman, I.: Acylclic Conjugated Molecules, Trees and Their Energies. J. Math. Chem. 1, 123–143 (1987)MathSciNetGoogle Scholar
  9. 9.
    Gutman, I.: The Energy of a Graph: Old and New Results. In: Betten, A., Kohnert, A., Laue, R., Wassermann, A. (eds.) Algebraic Combinatorics and Applications, pp. 196–211. Springer, Berlin (2001)Google Scholar
  10. 10.
    Gutman, I., Li, X., Zhang, J.: Graph Energy. In: Dehmer, M., Emmert-Streib, F. (eds.) Analysis of Complex Networks: From Biology to Linguistics, pp. 145–174. Wiley-VCH Verlag, Weinheim (2009)Google Scholar
  11. 11.
    Hou, Y.: Unicyclic Graphs with Minimal Energy. J. Math. Chem. 29, 163–168 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hou, Y.: On Trees with the Least Energy and a Given Size of Matching. J. Syst. Sci. Math. Sci. 23, 491–494 (2003)zbMATHGoogle Scholar
  13. 13.
    Huo, B., Li, X., Shi, Y., Wang, L.: Determining the Conjugated Trees with the Third-through the Sixth-minimal Energies. MATCH Commun. Math. Comput. Chem. 65, 521–532 (2011)MathSciNetGoogle Scholar
  14. 14.
    Lin, W., Guo, X., Li, H.: On the Extremal Energies of Trees with a Given Maximum Degree. MATCH Commun. Math. Comput. Chem. 54, 363–378 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Li, S., Li, N.: On Minimal Energies of Acyclic Conjugated Molecules. MATCH Commun. Math. Comput. Chem 61, 341–349 (2009)MathSciNetGoogle Scholar
  16. 16.
    Li, X., Zhang, J., Wang, L.: On Bipartite Graphs with Minimal Energy. Discrete Appl. Math. 157, 869–873 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Zhang, F., Li, H.: On acyclic Conjugated Molecules with Minimal Energies. Discrete Appl. Math. 92, 71–84 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Zhang, F., Lai, Z.: Three Theorems of Comparison of Trees by Their Energy. Science Exploration 3, 12–19 (1983)MathSciNetGoogle Scholar
  19. 19.
    Zhou, B., Li, F.: On Minimal Energies of Trees of a Prescribed Diameter. J. Math. Chem. 39, 465–473 (2006)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yulan Xiao
    • 1
  • Bofeng Huo
    • 1
    • 2
  1. 1.Department of MathematicsQinghai Normal UniversityChina
  2. 2.Ministry of Education and Qinghai ProvinceKey Lab of Tibetan Information Processing (Qinghai Normal University)XiningP.R. China

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