Remarks on maximum atom-bond connectivity index with given graph parameters

The atom-bond connectivity (ABC) index is a degree-based molecular structure descriptor that can be used for modelling thermodynamic properties of organic chemical compounds. Motivated by its applicable potential, a series of investigations have been carried out in the past several years. In this note we first consider graphs with given edge-connectivity that attain the maximum ABC index. In particular, we give an affirmative answer to the conjecture about the structure of graphs with edge-connectivity equal to one that maximize the ABC index, which was recently raised by Zhang et al. (2016). In addition, we provide supporting evidence for another conjecture posed by the same authors which concerns graphs that maximize the ABC index among all graphs with chromatic number equal to some fixed χ≥3. Specifically, we confirm this conjecture in the case where the order of the graph is divisible by χ.