On the Uphill Zagreb Indices of Graphs

Main Article Content

Anwar Saleh
Sara Bazhear
Najat Muthana


One of the tools, to research and investigation the structural dependence of various properties and some activities of chemical structures and networks is the topological indices of graphs. In this research work, we introduce novel indices of graphs which they based on the uphill degree of the vertices termed as uphill Zagreb topological indices. Exact formulae of these new indices for some important and famous families of graphs are established.

Article Details


  1. A. Alwardi, A. Alqesmah, R. Rangarajan, I.N. Cangul, Entire Zagreb indices of graphs, Discrete Math. Algorithm. Appl. 10 (2018), 1850037. https://doi.org/10.1142/S1793830918500374.
  2. J.A. Aruldoss and G. Gurulakshmi, The dominator coloring of central and middle graph of some special graphs, Int. J. Math. Appl. 4 (2016), 67–73.
  3. J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, Berlin, 2008.
  4. J. Braun, A. Kerber, M. Meringer, C. Rucker, Similarity of molecular descriptors: the equivalence of Zagreb indices and walk counts, MATCH Commun. Math. Comput. Chem. 54 (2005), 163-176.
  5. A.E. Brondani, C.S. Oliveira, F.A.M. França, L. de Lima, Aα-spectrum of a firefly graph, Electron. Notes Theor. Computer Sci. 346 (2019), 209–219. https://doi.org/10.1016/j.entcs.2019.08.019.
  6. W.C. Chen, H.I. Lu, Y.N. Yeh, Operations of interlaced trees and graceful trees, Southeast Asian Bull. Math. 21 (1997), 337-348.
  7. J. Deering, Uphill and downhill domination in graphs and related graph parameters, Thesis, East Tennessee State University, Johnson, (2013).
  8. M.V. Diudea, Nanomolecules and nanostructures-poynomial and indices, Univ. Kragujevac, 2010.
  9. M.V. Diudea, M.S. Florescu, P.V. Khadikar, Molecular topology and its applications, Eficon, Bucarest, 2006.
  10. B. Furtula, I. Gutman, A forgotten topological index, J Math Chem. 53 (2015), 1184–1190. https://doi.org/10.1007/s10910-015-0480-z.
  11. W. Gao, The Randić energy of generalized double sun, Czech. Math. J. (2021), 1–28. https://doi.org/10.21136/CMJ.2021.0463-20.
  12. I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83-92.
  13. I. Gutman, B. Furula (Eds.), Novel molecular structure descriptors-theory and applications I, Univ. Kragujevac, Kragujevac, 2010 .
  14. I. Gutman, B. Furula (Eds.), Novel molecular structure descriptors-theory and applications II, Univ. Kragujevac, Kragujevac, 2010.
  15. I. Gutman, B. Furula (Eds.), Distance in Molecular Graphs, Univ. Kragujevac, Kragujevac, 2012.
  16. I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535–538. https://doi.org/10.1016/0009-2614(72)85099-1.
  17. F. Harary, Graph theory, Addison-Wesley, Reading Mass, 1969.
  18. M. Karelson, Molecular descriptors in QSAR-QSPR, Wiley, New York, 2000.
  19. M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, The first and second Zagreb indices of some graph operations, Discr. Appl. Math. 157 (2009), 804–811. https://doi.org/10.1016/j.dam.2008.06.015.
  20. J. Kok, N.K. Sudev, U. Mary, On chromatic Zagreb indices of certain graphs, Discrete Math. Algorithm. Appl. 9 (2017), 1750014. https://doi.org/10.1142/S1793830917500148.
  21. S. Nikolic´, G. Kovacˇevic´, A. Milicˇevic´, N. Trinajstic´, The Zagreb indices 30 years after, Croat. Chem. Acta, 76 (2003), 113-124.
  22. A. Saleh, A. Aqeel and I. N. Cangul, On the entire ABC index of graphs, Proc. Jangjeon Math. Soc. 23 (2020), 39-51.
  23. A. Saleh, N. Muthana, W. Al-Shammakh, H. Alashwali, Monotone chromatic number of graphs, Int. J. Anal. Appl. 18 (2020), 1108-1122. https://doi.org/10.28924/2291-8639-18-2020-1108.
  24. R. Todeschini, V. Consonni, Handbook of molecular descriptors, Wiley-VCH, Weinheim, 2000.
  25. R. Todeschini, V. Consonni, Molecular descriptors for chemoinformatics, Wiley-VCH, Weinheim, 2009.
  26. S. Wang, B. Wei, Multiplicative Zagreb indices of cacti, Discrete Math. Algorithm. Appl. 8 (2016), 1650040. https://doi.org/10.1142/S1793830916500403.
  27. B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52 (2004), 113-118.
  28. B. Zhou, I. Gutman, Relations between Wiener, hyper-Wiener and Zagreb indices, Chem. Phys. Lett. 394 (2004), 93–95. https://doi.org/10.1016/j.cplett.2004.06.117.
  29. B. Zhou, I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem. 54 (2005), 233-239.