## 3 Results and Discussion
### General Depiction of the Mode of Graphene Plasmonic Trimers
In the highest energy E' state, the dipoles of the nanodisks on the base show symmetric distribution with respect to the longitudinal direction, so the nonzero net dipole appears along this direction. In the other E' case, the vertical dipole disappears in the vertex atom of the trimer and the dipoles of the other base atoms arrange symmetrically with respect to the vertical direction. Consequently, it shows the nonzero net dipole along the longitudinal direction. Additionally, there is another nondegenerate mode related to the irreducible representation of A\({}_{2}\)' symmetry located at the lower energy state. Interestingly, in this case, the dipole of the vertex atom disappears, similar to the first case of the bonding E' mode. However, the net dipole of the whole trimer system in antibonding A\({}_{2}\)' mode is along the longitudinal direction and in the bonding E' mode is along the vertical direction. As for the doubly degenerate bonding E' mode with lower energy, it shows the net dipole along the longitudinal direction. Moreover, among the lower energy state bonding modes, the nondegenerate A\({}_{1}\)' is also located at the lowest position, and the dipole of the total plasmonic trimer is centered symmetrically. As a result, the total dipole of the whole plasmonic trimer is zero. As the vertex angle \(\theta\) increases gradually, the \(\mathbf{D_{3h}}\) symmetry reduces to lower symmetry \(\mathbf{C_{2v}}\), and the plasmonic modes interact with each other, which finally results in lifting the degeneracy of the higher symmetric modes, and new energy states appear, shown in the center columns of Figure 3. The structures of the new plasmonic modes A\({}_{1}\) and B\({}_{2}\) are gradually changed from those of the \(\mathbf{D_{3h}}\) modes as the vertex angle \(\theta\) increases. However, the direction of the dipole moment of the vertex nanodisk does not change when the vertex angle increases, while the counterparts in the other two nanodisks on the base rotate synchronously counterclockwise and clockwise. It is noticeable that for the B\({}_{2}\) modes, the nonzero net dipole moment is along the longitudinal direction, while for the A\({}_{1}\) mode, the nonzero net dipole moment is along the vertical direction. When the vertex angle further increases to \(\pi\), the graphene nanodisks become a chain and the trimer molecule shows \(\mathbf{D}_{\text{s}\text{o}h}\) symmetry. The dipole distribution of this symmetry group is illustrated in the right column of Figure 3. The antibonding E\({}^{\prime}\)-A\({}_{1}\) path evolves to the \(\prod_{\text{i}\text{u}}\) mode, where the dipole moment of the center nanodisk is shown upwards and those of the nanodisks on both the sides show reversed vertical direction. The antibonding E\({}^{\prime}\)-B\({}_{2}\) path evolves to \(\Sigma\text{u}^{+}\) mode, and the net dipole moment is three times of that of the center nanodisk, since all the directions of the three nanodisks are identical along the longitudinal direction. The mode of the antibonding A\({}_{2}\)'-B\({}_{2}\) path becomes \(\prod_{\text{g}}\), where the dipole moment of the vertex disks stays at zero and the total is zero. The mode of the bonding E\({}^{\prime}\)-A\({}_{1}\) path becomes \(\Sigma_{\text{g+}}\). The total dipole moment of this mode is also zero, because the dipole of the center nanodisk disappears, and those of the nanodisks on the two sizes are along the reversed longitudinal direction. The bonding E\({}^{\prime}\)-B\({}_{2}\) path transforms into \(\Sigma_{\text{u+}}\) mode and the mode evolution of bonding \(\Sigma_{\text{u}}\)\({}^{+}\) irreducible representation of the \(\mathbf{D}_{\text{s}\text{o}h}\) symmetry group, where the dipole of the center nanodisk points left, while the other two nanodisks in the two sides point right.