Minimum atom-bond sum-connectivity index of trees with a fixed order and/or number of pendent vertices

Let d(u) be the degree of a vertex u of a graph G. The atom-bond sum-connectivity (ABS) index of a graph G is the sum of the numbers (1 - 2(d(v) + d(w))(-1))(1/2) over all edges vw of G. This paper gives the characterization of the graph possessing the minimum ABS index in the class of all trees of a fixed number of pendent vertices; the star is the unique extremal graph in the mentioned class of graphs. The problem of determining graphs possessing the minimum ABS index in the class of all trees with n vertices and p pendent vertices is also addressed; such extremal trees have the maximum degree 3 when n >= 3p - 2 >= 7, and the balanced double star is the unique such extremal tree for the case p = n -2.