On the maximum Graovac-Pisanski index of bicyclic graphs

For a simple graph $G=(V(G),E(G))$, the Graovac-Pisanski index of $G$ is defined as

$$GP(G)=\frac{|V(G)|}{2|\mathrm{A}\mathrm{u}\mathrm{t}(G)|}\sum _{u\in V(G)}\sum _{\alpha \in \mathrm{A}\mathrm{u}\mathrm{t}(G)}{d}_G(u,\alpha (u)),$$

where $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ is the automorphism group of $G$ and $d_G(u,v)$ is the length of a shortest path between the two vertices $u$ and $v$ in $G$. Obviously, $GP(G)=0$ if $G$ has no nontrivial automorphisms. Let $B_n^{3,3}$ be the graph consisting of two disjoint 3-cycles with a path of length $n-5$ joining them. In this article, we prove that among all those $n$-vertex bicyclic graphs in which every edge lies on at most one cycle, $B_n^{3,3}$ has the maximum Graovac-Pisanski index.