Some inequalities for the forgotten topological index

Abstract

Let \(G= (V,E)\) be a simple connected graph with vertex set \(V=\{1,2,\ldots,n\}\) and edge set \(E=\{e_1, e_2,\ldots,e_m\}\). Let \(d_i\) be the degree of its vertex \(i\) and \(d(e_i)\) the degree of its edge \(e_i\). We consider the recently introduced degree–based graph invariants: the forgotten index \(F = \sum_{i\in V} d_i^3\), the hyper-Zagreb index \(HM = \sum_{i\sim j} (d_i + d_j)^2\), and the reformulated first Zagreb index \(EM_1 = \sum_{e_i\in E} d(e_i)^2\). A number of lower and upper bounds for \(F\), \(HM\), and \(EM_1\) are established, and the equality cases determined.