Proving a Conjecture of Gutman Concerning Trees with Minimal ABC Index

The atom-bond connectivity (ABC) index of a graph ( , ) G V E is defined as ( ) [ ( ) ( ) 2] / [ ( ) ( )] uv E ABC G d u d v d u d v , where ( ) d u denotes the degree of vertex u of G . This recently introduced molecular structure descriptor found 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 index. However, the problem of characterizing trees with minimal ABC index appears to be difficult. One approach to the problem is to determine their structural features as much as possible. Gutman et al. [MATCH Comun. Math. Comput. Chem. 67 (2012) 467] conjectured that each pendent vertex of a tree with minimal ABC index belongs to a pendent path of length 2 or 3. We prove this conjecture in the present paper.