Minimizing Degree-based Topological Indices for Trees with Given Number of Pendent Vertices

We derive sharp lower bounds for the first and the second Zagreb indices (M-1 and M-2 respectively) for trees and chemical trees with the given number of pendent vertices and find optimal trees. M-1 is minimized by a tree with all internal vertices having degree 4, while M-2 is minimized by a tree where each "stem" vertex is incident to 3 or 4 pendent vertices and one internal vertex, while the rest internal vertices are incident to 3 other internal vertices. The technique is shown to generalize to the weighted first Zagreb index, the zeroth order general Randic index, as long as to many other degree-based indices.