Bounds for Zagreb indices

Abstract

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). Let \(d_i\) be the degree of the vertex \(v_i \in V(G)\). The first and second Zagreb indices, \(M_1 = \sum_{v_i \in V(G)} d_i^2\) and \(M_2 = \sum_{v_iv_j \in E(G)} d_i\,d_j\) are the oldest and most thoroughly investigated vertex degree-based molecular structure descriptors. An unusually large number of lower and upper bounds for \(M_1\) and \(M_2\) have been established. We provide a survey of the most significant estimates of this kind, attempting to cover the existing literature up to the end of year 2016.