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

Abstract

<abstract><p>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\ge 3p-2\ge7 $, and the balanced double star is the unique such extremal tree for the case $ p = n-2 $.</p></abstract>