Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 7 2 2022 12 01 Inverse problem for the Forgotten and the hyper Zagreb indices of trees 203 209 14266 10.22049/cco.2021.27034.1182 EN Joseph Varghese Kureethara Christ University Anjusha Asok Department of Mathematics, Christ University, Bangalore, India Ismail Naci Cangul Department of Mathematics Uludag University, Gorukle 16059 Bursa-Turkey Journal Article 2020 11 29 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. https://comb-opt.azaruniv.ac.ir/article_14266_96c5e70ed539dcd220803b9fb53ba7d2.pdf