Remarks on the Graovac–Ghorbani index of bipartite graphs

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 (GGGG) index, which yielded promising results when compared to analogous descriptors. In this paper, we investigate the structure of graphs that maximize and minimize the GGGG index. Specifically, we show that amongst all bipartite graphs, the minimum GGGG index is attained by a complete bipartite graph, while the maximum GGGG 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 GGGG index of paths. In order to obtain our results, we introduce a normalized version of the GGGG index and call it the normalized Graovac–Ghorbani (NGGNGG) index. Finally, we discuss some related open questions as a potential extension of our work.