Remarks on the Graovac-Ghorbani index of bipartite graphs

TL;DR

This paper studies the Graovac-Ghorbani index for bipartite graphs, identifying extremal structures and deriving asymptotic estimates, with implications for chemical graph theory.

Contribution

It characterizes graphs that maximize and minimize the GG index among bipartite graphs and introduces a normalized version for further analysis.

Findings

Complete bipartite graphs minimize the GG index.

Paths and cycle-like graphs maximize the GG index.

Asymptotic estimate of the GG index for paths.

Abstract

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.