The atom-bond connectivity (ABC) index is a well-known degree-based molecular structure descriptor with a variety of chemical applications. In 2010 Graovac and Ghorbani introduced a distance-based analog of this index, the Graovac-Ghorbani (GG) index, which yielded promising results when compared to analogous descriptors. In this paper, we investigate the structure of graphs that maximize and minimize the GG index. Specifically, we show that amongst all bipartite graphs, the minimum GG index is attained by a complete bipartite graph, while the maximum GG index is attained by a path or a cycle-like graph; the structure of the resulting graph depends on the number of vertices. Through the course of the research, we also derive an asymptotic estimate of the GG index of paths. In order to obtain our results, we introduce a normalized version of the GG index and call it the normalized Graovac-Ghorbani (NGG) index. Finally, we discuss some related open questions as a potential extension of our work.
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, Yang, Wang and Zhang [33]. 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. * Supported by the ARRS Program P1-0383.
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