Inequalities involving the harmonic-arithmetic index

Let G be a simple graph with vertex set \(V=\{v_{1},v_{2},\ldots ,v_{n}\}\). The notion \(i\sim j\) is used to indicate that the vertices \(v_{i}\) and \(v_{j}\) of G are adjacent. For a vertex \(v_{i}\in V\), let \(d_{i}\) be the degree of \(v_{i}\). The harmonic-arithmetic (HA) index of G is defined as \(HA(G) =\sum _{i\sim j} 4d_id_j(d_i+d_j)^{-2}\). In this paper, a considerable number of inequalities involving the HA index and other topological indices are derived. For every obtained inequality, all the graphs that satisfy the equality case are also characterized.