  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 $$ABC_{GG}(G)= \sum_{uv
\in E(G)} \sqrt{\frac{n_{u}+n_{v}-2} {n_{u} n_{v}}},$$ 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 $\mathcal{B}_{n} = B_1(n) \cup B_2(n) \cup B_3(n).$ In this paper,
we give an lower bound to the $ABC_{GG}$ 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 $ABC_{GG}$ index for all graphs in
$\mathcal{B}_{n}.$
