Inverse problem for the Forgotten and the hyper Zagreb indices of trees

Document Type : Original paper


1 Christ University

2 Department of Mathematics, Christ University, Bangalore, India

3 Department of Mathematics Uludag University, Gorukle 16059 Bursa-Turkey


Let $G=(E(G),V(G))$ be a (molecular) graph with vertex set $V(G)$ and edge set $E(G)$. The forgotten Zagreb index and the hyper Zagreb index of G are defined by $F(G) = \sum_{u \in V(G)} d(u)^{3}$ and $HM(G) = \sum_{uv \in E(G)}(d(u)+d(v))^{2}$ where $d(u)$ and d(v) are the degrees of the vertices $u$ and $v$ in $G$, respectively. A recent problem called the inverse problem deals with the numerical realizations of topological indices. We see that there exist trees for all even positive integers with $F(G)>88$ and with $HM(G)>158$. Along with the result, we show that there exist no trees with $F(G) < 90$ and $HM(G) < 160$ with some exceptional even positive integers and hence characterize the forgotten Zagreb index and the hyper Zagreb index for trees.


Main Subjects

[1] H. Aram and N. Dehgardi, Reformulated F-index of graph operations, Commun. Comb. Optim. 2 (2017), no. 2, 87–98.
[2] M. Eliasi and A. Ghalavand, On trees and the multiplicative sum Zagreb index, Commun. Comb. Optim. 1 (2016), no. 2, 137–148.
[3] W. Gao, M.K. Siddiqui, N.A. Rehman, and M.H. Muhammad, Topological characterization of dendrimer, benzenoid, and nanocone, Zeitschrift für Naturforschung C 74 (2018), no. 1-2, 35–43.
[4] W. Gao, W. Wang, and M.R. Farahani, Topological indices study of molecular structure in anticancer drugs, J. Chemistry 2016 (2016), Article ID 3216327.
[5] I. Gutman, M. Togan, A. Yurttas, A.S. Cevik, and I.N. Cangul, Inverse problem for sigma index, MATCH Commun. Math. Comput. Chem. 79 (2018), no. 2, 491–508.
[6] X. Li, Z. Li, and L. Wang, The inverse problems for some topological indices in combinatorial chemistry, J. Comput. Biology 10 (2003), no. 1, 47–55.
[7] T. Réti, R. Sharafdini, A. Drégelyi-Kiss, and H. Haghbin, Graph irregularity indices used as molecular descriptors in QSPR studies, MATCH Commun. Math. Comput. Chem. 79 (2018), 509–524.
[8] D. Vukičević and M. Gašperov, Bond additive modeling 1. Adriatic indices, Croatica Chemica Acta 83 (2010), no. 3, 243–260.
[9] A. Yurtas, M. Togan, V. Lokesha, I.N. Cangul, and I. Gutman, Inverse problem for Zagreb indices, J. Math. Chemistry 57 (2019), no. 2, 609–615.