Notes on trees with minimal atom-bond connectivity index

Abstract

If G = (V, E) is a molecular graph, and d(u) is the degree of its vertex u, then the atom-bond connectivity index of G is \(ABC = \sum_{uv\in E} \sqrt{[d(u) + d(v) − 2]/[d(u) d(v)]}\). This molecular structure descriptor, introduced by Estrada et al. in the late 1990s, found recently interesting applications in the study of the thermodynamic stability of acyclic saturated hydrocarbons, and the strain energy of their cyclic congeners. In connection with this, one needs to know which trees have extremal ABC-values. Whereas it is easy to demonstrate that the star has maximal ABC, characterizing the trees with minimal ABC appears to be a much more difficult task. In this paper we determine a few structural features of the trees with minimal ABC, which brings us a step closer to the complete solution of the problem.