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Topological indices are graph invariants determined by the distance or degree of vertices of the molecular graph. Topological indices have been used effectively in chemical graph theory in explaining the structures and predicting certain physicochemical properties of chemical compounds. In this research, we introduce the first, second, and forgotten downhill Zagreb indices and calculate those topological indices for some standard families of graphs and the join of graphs. Also, the downhill topological indices for the firefly graph, book graph, and stacked book graph are established. Finally, the downhill indices of Graphene and honeycomb network are obtained.
- A. Alwardi, A. Alqesmah, R. Rangarajan, I.N. Cangul, Entire Zagreb indices of graphs, Discrete Math. Algorithm. Appl. 10 (2018), 1850037.
- J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, Berlin, Germany, 2008.
- 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.
- B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015), 1184-1190.
- M. Ghorbani, M.A. Hosseinzadeh, The third version of zagreb index, Discrete Math. Algorithm. Appl. 05 (2013), 1350039.
- I. Gutman, K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83-92.
- I. Gutman, N. Trinajstic, Graph theory and molecular orbitals, Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535-538.
- J. Deering, Uphill & Downhill Domination in Graphs and Related Graph Parameters. Thesis, East Tennessee State University, 2013.
- F. Harary, Graph theory, Addison-Wesley, Reading Mass, 1969.
- S.T. Hedetniemi, T.W. Haynes, J.D. Jamieson, W.B. Jamieson, Downhill domination in graphs, Discuss. Math., Graph Theory. 34 (2014), 603-612.
- M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009), 804-811.
- J. Kok, N.K. Sudev, U. Mary, On chromatic Zagreb indices of certain graphs, Discrete Math. Algorithm. Appl. 09 (2017), 1750014.
- S. Nikoli ´c, G. Kovaˇcevi ´c, A. Miliˇcevi ´c, N. Trinajsti ´c, The Zagreb indices 30 years after, Croatica Chemica Acta, 76 (2003), 113-124.
- A. Saleh, A. Aqeel, I. N. Cangul, On the entire ABC index of graphs, Proc. Jangjeon Math. Soc. 23 (2020), 39-51.
- Anwar Saleh, Najat Muthana, Wafa Al-Shammakh, Hanaa Alashwali, Monotone chromatic number of graphs, Int. J. Anal. Appl. 18 (2020), 1108-1122.
- B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52 (2004), 113-118.
- B. Zhou, I. Gutman, Relations between Wiener, hyper-Wiener and Zagreb indices, Chem. Phys. Lett. 394 (2004), 93-95.
- B. Zhou, I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem. 54 (2005), 233-239.
- S. Wang, B. Wei, Multiplicative Zagreb Indices of Cacti, Discrete Math. Algorithm. Appl. 8 (2016), 1650040.