On structural properties of trees with minimal atom-bond connectivity index IV: Solving a conjecture about the pendent paths of length three

The atom-bond connectivity (ABC) index is one of the most investigated degreebased molecular structure descriptors with a variety of chemical applications. It is known that among all connected graphs, the trees minimize the ABC index. However, a full characterization of trees with a minimal ABC index is still an open problem. By now, one of the proved properties is that a tree with a minimal ABC index may have, at most, one pendent path of length 3, with the conjecture that it cannot be a case if the order of a tree is larger than 1178. Here, we provide an affirmative answer of a strengthened version of that conjecture, showing that a tree with minimal ABC index cannot contain a pendent path of length 3 if its order is larger than 415. 1 Preliminaries and related results Let G = (V,E) be a simple undirected graph of order n = |V | and size m = |E|. For v ∈ V (G), the degree of v, denoted by dG(v), is the number of edges incident to v. When it is clear from the context we will write d(v), which will always assume dG(v). The atom-bond connectivity index of G is defined as ABC(G) = ∑ uv∈E(G) √ (d(u) + d(v)− 2) d(u)d(v) = ∑ uv∈E(G) f(d(u), d(v)). (1) The ABC index was introduced in 1998 by Estrada, Torres, Rodŕıguez and Gutman [16] and is one of the most investigated degree-based molecular structure descriptors. More about the (degree-based) molecular structure descriptors can be found in [19,25,35] and in the references 1 ar X iv :1 70 6. 08 58 7v 1 [ cs .D M ] 2 6 Ju n 20 17