First degree-based entropy of graphs

  • A. Ghalavand
  • M. Eliasi
  • A. R. AshrafiEmail author
Original Research


The first degree-based entropy of a connected graph G is defined as: \(I_1(G)=\log (\sum _{v_i\in V(G)}\deg (v_i))-\sum _{v_j\in V(G)}\frac{\deg (v_j)\log \deg (v_j)}{\sum _{v_i\in V(G)}\deg ( v_i)}\). In this paper, we apply majorization technique to extend some known results about the maximum and minimum values of the first degree-based entropy of trees, unicyclic and bicyclic graphs.


Entropy Tree Degree sequence Unicyclic graph Bicyclic graph 



We are indebted to the referees for his/her suggestions and helpful remarks leaded us to improve this paper. The research of the third author is partially supported by the University of Kashan under Grant No. 572760/245.


  1. 1.
    Aczel, J., Daróczy, Z.: On Measures of Information and Their Characterizations. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Arndt, C.: Information Measures. Springer, Berlin (2004)Google Scholar
  3. 3.
    Cao, S., Dehmer, M., Shi, Y.: Extremality of degree-based graph entropies. Inf. Sci. 278, 22–33 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, Z., Dehmer, M., Emmert-Streib, F., Shi, Y.: Entropy bounds for dendrimers. Appl. Math. Comput. 242, 462–472 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, Z., Dehmer, M., Shi, Y.: A note on distance-based graph entropies. Entropy 16, 5416–5427 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Daroczy, Z., Jarai, A.: On the measurable solutions of functional equation arising in information theory. Acta Math. Acad. Sci. Hung. 34, 105–116 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Das, K., Shi, Y.: Some properties on entropies of graphs. MATCH Commun. Math. Comput. Chem. 78(2), 259–272 (2017)MathSciNetGoogle Scholar
  8. 8.
    Dehmer, M., Emmert-Streib, F., Chen, Z., Li, X., Shi, Y.: Mathematical Foundations and Applications of Graph Entropy. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (2016)CrossRefzbMATHGoogle Scholar
  9. 9.
    Dehmer, M., Mowshowitz, A.: Generalized graph entropies. Complexity 17(2), 45–50 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dehmer, M., Li, X., Shi, Y.: Connections between generalized graph entropies and graph energy. Complexity 21(1), 35–41 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dehmer, M.: Information processing in complex networks: graph entropy and information functionals. Appl. Math. Comput. 201, 82–94 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dehmer, M., Mowshowitz, A.: A history of graph entropy measures. Inf. Sci. 1, 57–78 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dragomir, S., Goh, C.: Some bounds on entropy measures in information theory. Appl. Math. Lett. 10, 23–28 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Eliasi, M., Ghalavand, A.: Extermal trees with respect to some versions of Zagreb indices via majorization. Iran. J. Math. Chem. 8(4), 391–401 (2017)zbMATHGoogle Scholar
  15. 15.
    Ilić, A., Dehmer, M.: On the distance based graph entropies. Appl. Math. Comput. 269, 647–650 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ilić, A.: On the extremal values of general degree-based graph entropies. Inf. Sci. 370–371, 424–427 (2016)MathSciNetGoogle Scholar
  17. 17.
    Marshall, A.W., Olkin, I.: Inequalities, Theory of Majorization. Academic Press, New York (1979)zbMATHGoogle Scholar
  18. 18.
    Rashevsky, N.: Life, information theory and topology. Bull. Math. Biophys. 17, 229–235 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Trucco, E.: A note on the information content of graphs. Bull. Math. Biol. 18, 129–135 (1965)MathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Faculty of Mathematical ScienceUniversity of KashanKashanIslamic Republic of Iran
  2. 2.Department of MathematicsKhansar Faculty of Mathematics and Computer ScienceKhansarIslamic Republic of Iran

Personalised recommendations