Fuzzy Zagreb Indices and Some Bounds for Fuzzy Zagreb Energy

Main Article Content

Mahesh Kale
S. Minirani

Abstract

Topological indices M1/M2 known as first/second Zagreb indices are defined as the sum of the sum/product of degrees of pairs of adjacent vertices of a simple graph. These indices and their properties have been studied in detail under chemical graph theory. In this paper we introduce the concepts of first, second and hyper Zagreb indices of fuzzy graphs. We also study the Zagreb matrices and the associated Zagreb energies of fuzzy graphs. Some bounds for these energies are also obtained.

Article Details

References

  1. I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. total φ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (4) (1972), 535-538.
  2. M. Randic, Characterization of molecular branching, J. Amer. Chem. Soc. 97 (23) (1975), 6609-6615.
  3. L. A. Zadeh, Fuzzy sets, Inform. Control. 8 (3) (1965), 338-353.
  4. A. Rosenfeld, Fuzzy graphs, in: Fuzzy sets and their applications to cognitive and decision processes, Elsevier, 1975, pp. 77-95.
  5. H.-J. Zimmermann, Fuzzy set theory and mathematical programming, in: Fuzzy Sets Theory and Applications, Springer, 1986, pp. 99-114.
  6. M. G. Thomason, Convergence of powers of a fuzzy matrix, J. Math. Anal. Appl. 57 (2) (1977), 476-480.
  7. J. N. Mordeson, P. S. Nair, Fuzzy graphs and fuzzy hypergraphs, Vol. 46, Physica Verlag, Heidelberg, 2012.
  8. M. Sunitha, A. Vijayakumar, Complement of a fuzzy graph, Indian J. Pure Appl. Math. 33 (9) (2002), 1451-1464.
  9. S. Mathew, M. Sunitha, Types of arcs in a fuzzy graph, Inform. Sci. 179 (11) (2009), 1760-1768.
  10. A. Nagoorgani, V. Chandrasekaran, A first look at fuzzy graph theory, Allied Publication Pvt. Ltd, Chennai, (2010).
  11. A. N. Gani, K. Radha, The degree of a vertex in some fuzzy graphs, Int. J. Algorithms Comput. Math 2 (2009), 107-116.
  12. A. Nagoorgani, K. Ponnalagu, A new approach on solving intuitionistic fuzzy linear programming problem, Appl. Math. Sci. 6 (70) (2012), 3467-3474.
  13. N. Anjali, S. Mathew, Energy of a fuzzy graph, Ann. Fuzzy Math. Inform. 6 (3) (2013), 455-465.
  14. S. R. Sharbaf, F. Fayazi, Laplacian energy of a fuzzy graph. Iran. J. Math. Chem. 5 (1) (2014), 1-10
  15. M. Binu, S. Mathew, J. N. Mordeson, Wiener index of a fuzzy graph and application to illegal immigration networks, Fuzzy Sets Syst. 384 (2020), 132-147.
  16. S. R. Islam, S. Maity, M. Pal, Comment on “wiener index of a fuzzy graph and application to illegal immigration networks”, Fuzzy Sets Syst. 384 (2020), 148-151.
  17. S. Mathew, N. Anjali, J. N. Mordeson, Transitive blocks and their applications in fuzzy interconnection networks, Fuzzy Sets Syst. 352 (2018), 142-160.
  18. S. Ali, S. Mathew, J. Mordeson, Hamiltonian fuzzy graphs with application to human trafficking, Inform. Sci. 550 (2021), 268-284.
  19. M. Kale, S. Minirani, On zagreb indices of graphs with a deleted edge, Ann. Pure Appl. Math. 21 (1) (2020), 1-14.
  20. B. Borovicanin, K. C. Das, B. Furtula, I. Gutman, Bounds for zagreb indices, MATCH Commun. Math. Comput. Chem 78 (1) (2017), 17-100.
  21. J. N. Mordeson, S. Mathew, Advanced Topics in Fuzzy Graph Theory, Springer, 2019.