On the Graovac-Ghorbani index for bicyclic graphs with no pendant vertices

Let $G=(V,E)$ be a simple undirected and connected graph on $n$ vertices. The Graovac–Ghorbani index of a graph $G$ is defined as $$AB{C}_{GG}(G)=\sum _{uv\in E(G)}\sqrt{\frac{n_u+n_v-2}{n_u{n}_v}},$$ where $n_u$ is the number of vertices closer to vertex $u$ than vertex $v$ of the edge $uv \in E(G)$and$n_{v}$ is defined analogously. It is well-known that all bicyclic graphs with no pendant vertices are composed by three families of graphs, which we denote by ${\mathcal{B}}_n=B_1(n)\cup B_2(n)\cup B_3(n)\mathrm{.}$ In this paper, we give an lower bound to the $AB{C}_{GG}$ index for all graphs in $B_1(n)$ and prove it is sharp by presenting its extremal graphs. Additionally, we conjecture a sharp lower bound to the $AB{C}_{GG}$ index for all graphs in ${\mathcal{B}}_n\mathrm{.}$