Inverse problem for the forgotten and the hyper zagreb indices of trees

Abstract

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) = Sigma(u is an element of V) (G) d(u)(3) and HM(G) = Sigma(uv is an element of 2E)(G) (d(u) + d(v))(2) whered(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.