On bicyclic graphs with maximal Graovac-Ghorbani index

TL;DR

This paper determines the maximum Graovac-Ghorbani index for bicyclic graphs and characterizes the extremal graphs, confirming a conjecture related to bounds of this index.

Contribution

It provides the first exact determination of the maximal Graovac-Ghorbani index for bicyclic graphs and characterizes the extremal structures, solving an open conjecture.

Findings

Identified the extremal bicyclic graphs with maximal Graovac-Ghorbani index.

Proved the conjectured bounds are tight and achieved by specific graphs.

Confirmed the conjecture posed by Pacheco et al. regarding the index.

Abstract

Graovac-Ghorbani index is a new version of the atom-bond connectivity index. D. Pacheco et al. [D. Pacheco, L. de Lima, C. S. Oliveira, On the Graovac-Ghorbani Index for Bicyclic Graph with No Pendent Vertices, MATCH Commun. Math. Comput. Chem. 86 (2021) 429-448] conjectured a sharp lower and upper bounds to the Graovac-Ghorbani index for all bicyclic graphs. Motivated by their nice work, in this paper we determine the maximal Graovac-Ghorbani index of bicyclic graphs and characterize the corresponding extremal graphs, which solves one of their Conjectures.