## 6 Conclusions  
This research investigates a graph structure registration algorithm that utilizes point cloud data and introduces a novel coarse-to-fine graph structure registration network, referred to as EKNet. First, the feature construction module captures the geometric features and complex network features of the graph structure. Next, EKNet utilizes these features for network training. Following this, the network engages in node feature matching and enhances the registration outcomes by applying geometric consistency discrimination. The final step involves the precise computation of rotation and translation matrices through Singular Value Decomposition (SVD) to achieve refined registration.  
The experimental findings indicate that the EKNet method excels in managing graph structure registration tasks. The root mean square error (RMSE) of our registration method is 0.03 at an overlap ratio of 20%, and reaches 0.002 at an overlap ratio of 60%. In the presence of noise, the EKNet algorithm demonstrated remarkable precision, achieving a Mean Squared Error (MSE) of rotation at \(1.10837\times 10^{-5}\), which is an order of magnitude better than the suboptimal method. EKNet effectively addresses several critical challenges associated with point cloud registration tasks, including incomplete overlap, significant initial discrepancies, and noise interference, which frequently result in registration failures or inaccurate matching outcomes when employing conventional techniques. In the presence of noise, it can still maintain 0.003. In comparison to traditional Graph Convolutional Networks (GCNs), our approach realized a 27.28% enhancement in registration speed and an 80.66% improvement in registration accuracy over the suboptimal method. Our method is more stable under different initial translation distances and rotation errors. EKNet achieves high efficiency, robustness, and accuracy in the registration of sparse graph structures within building scenes, all without necessitating iterative processes.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **RMSE(R)** & **RMSE(t)** & **MAE(R)** & **MAE(t)** & **Times(s)** & **Contribution(\%)** \\ \hline Full Models & 0.003 & 0.021 & 0.002 & 0.014 & 18.83 & - \\ w/o FCM & 0.038 & 0.321 & 0.020 & 0.178 & 46.83 & 25.0\% \\ w/o FLNM & 0.080 & 0.589 & 0.038 & 0.291 & 31.21 & 48.5\% \\ w/o NFMM & 0.015 & 0.128 & 0.009 & 0.083 & 14.25 & 9.4\% \\ w/o GCDM & 0.024 & 0.193 & 0.017 & 0.143 & 24.01 & 17.1\% \\ \hline \hline \end{tabular}
\end{table}
Table 7: Ablation study results.  
**Author Contributions:** C.Q.: Conceptualization, investigation, methodology, writing--original draft, software. H.D.: Contributed equally with the first author. X.N.: Data curation, software. D.W.: writing--review and editing. B.W.: Investigation. H.C.: Writing--review and editing. J.H.: Writing--review and editing, supervision. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable  
**Data Availability Statement:** Data used in Section 5 can be found in the source code repositories at the following link: [[https://github.com/llqkgithub/GPCR](https://github.com/llqkgithub/GPCR)]([https://github.com/llqkgithub/GPCR](https://github.com/llqkgithub/GPCR)) (accessed on 24 April 2025).  
**Conflicts of Interest:** The authors declare no conflicts of interest.