Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21287220221201Inverse problem for the Forgotten and the hyper Zagreb indices of trees2032091426610.22049/cco.2021.27034.1182ENJoseph VargheseKureetharaChrist University0000-0001-5030-3948AnjushaAsokDepartment of Mathematics, Christ University, Bangalore, India0000-0002-8255-9179Ismail NaciCangulDepartment of Mathematics
Uludag University,
Gorukle 16059 Bursa-Turkey0000 0002 0700 5774Journal Article20201129Let $G=(E(G),V(G))$ 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.http://comb-opt.azaruniv.ac.ir/article_14266_96c5e70ed539dcd220803b9fb53ba7d2.pdf