On a family of trees with minimal atom-bond connectivity index

Let G = ( V , E ) be a graph, d i the degree of the vertex i , and i j the edge incident to the vertices i and j . The atom-bond connectivity index (or, simply, ABC index) is defined as A B C ( G ) = ? i j ? E ( d i + d j - 2 ) / ( d i d j ) . While this vertex-degree-based graph invariant is relatively well-known in chemistry, only recently a significant number of results emerged among the mathematical community. Though, several important problems remained open. One of them is the characterization of the tree(s) with minimal ABC index. In this paper, we will present some structural properties of one family of trees containing a pendent path of length 3 which would minimize the ABC index, mainly including: it contains no the so-called B k with k ? 4 , and contains at most two B 2 's.