Estimating the second and third geometric-arithmetic indices

Abstract

Arithmetic-geometric indices are graph invariants defined as the sum of terms \(\sqrt{Q_u\,Q_v} / [(Q_u + Q_v)/2]\) over all edges uv of the graph, where Qu is some quantity associated with the vertex u. If Qu is the number of vertices (resp. edges) lying closer to u than to v, then one speaks of the second (resp. third) geometric-arithmetic index, GA2 and GA3 . We obtain inequalities between GA2 and GA3 for trees, revealing that the main parameters determining their relation are the number of vertices and the number of pendent vertices.