Some forbidden combinations of branches in minimal-ABC trees

The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studied vertex-degree-based graph invariants in chemical graph theory. For a given graph G, the ABC index is defined as ∑uv∈Ed(u)+d(v)−2d(u)d(v), where d(u) is the degree of vertex u in G and E denotes the set of edges of G. In this paper, we present some new structural properties of trees with a minimal ABC index (also refer to as a minimal-ABC tree), which is a step further towards understanding their complete characterization. We show that a minimal-ABC tree cannot simultaneously contain a B4-branch and B1 or B2-branches.