Some new bounds on the modified first Zagreb index

Document Type : Original paper

Authors

1 Faculty of Electronic Engineering, University of Nis, Nis, Serbia

2 Faculty of Electronic Engineering

3 University of Hail, Saudi Arabia

Abstract

Let $G$ be a graph containing no isolated vertices. For the graph $G$, its modified first Zagreb index is defined as the sum of reciprocals of squares of vertex degrees of $G$. This article provides some new bounds on the modified first Zagreb index of $G$ in terms of some other well-known graph invariants of $G$. From the obtained bounds, several known results follow directly.

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References

[1] A. Ali, I. Gutman, E.I. Milovanović, and I.Z. Milovanović,  Sum of powers of the degrees of graphs: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 80 (2018), no. 1, 5–84.
[2] K.C. Das, K. Xu, and J. Wang, On inverse degree and topological indices of graphs, Filomat 30 (2016), no. 8, 2111–2120.
[3] S. Fajtlowicz, On conjectures of Graffiti-II, Congr. Numer. 60 (1987), 187–197.
[4] I. Gutman, Degree–based topological indices, Croat. Chem. Acta 86 (2013), no. 4, 351–361.
[5] I. Gutman, K.C. Das, B. Furtula, E.I. Milovanović, and I.Z. Milovanović,  Generalizations of Szõkefalvi-Nagy and Chebyshev inequalities with applications in spectral graph theory, Appl. Math. Comput. 313 (2017), 235–244.
[6] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total ϕelectron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
[7] J. Hao, Theorems about Zagreb indices and modified Zagreb indices, MATCH Commun. Math. Comput. Chem. 65 (2011), no. 3, 659–670.
[8] J.L.W. Jensen, Sur les fonctions convexes et les in´egalit´es entre les valeurs moyennes, Acta Math. 30 (1906), no. 1, 175–193.
[9] X. Li and H. Zhao, Trees with the first three smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50 (2004), 57–62.
[10] C. Liu, J. Li, and Y. Pan, On extremal modified Zagreb indices of trees, MATCH Commun. Math. Comput. Chem. 85 (2021), no. 2, 349–366.
[11] A. Miličević and S. Nikolić, On variable Zagreb indices, Croat. Chem. Acta 77 (2004), no. 1-2, 97–101.
[12] P. Milošević, I.Z. Milovanović, E.I. Milovanović, and M. Matejić,  Some inequalities for general zeroth-order Randić index, Filomat 33 (2019), no. 16, 5249–5258.
[13] I.Z. Milovanović, M. Matejić, and E.I. Milovanović, ˇ A note on the general zeroth–order Randi´c coindex of graphs, Contrib. Math. 1 (2020), no. 1, 17–21.
[14] B. Mitić, E.I. Milovanović, M. Matejić, and I.Z. Milovanović,  Some properties of the inverse degree index and coindex of trees, Filomat (in press).
[15] D.S. Mitrinović, J.E. Pečarić, and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht–Boston–London, 1993.
[16] D.S. Mitrinović and P. M. Vasić, Analytic Inequalities, Springer Verlag, BerlinHeidelberg-New York, 1970.
[17] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124.
[18] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley VCH, Weinheim, 2000.
[19] R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2010), no. 2, 359–372.
Communications in Combinatorics and Optimization
Volume 8, Issue 1
March 2023
Pages 13-21

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