## 5 Conclusions  
This paper proposes a beam-hopping-based joint beam scheduling and resource allocation framework, which leverages beam hopping technology to perform global scheduling of beam positions across time slots, enabling efficient multi-dimensional optimization in time, space, frequency, and power domains. The overall resource allocation problem is decomposed into three subproblems: beam-footprint partitioning, beam-to-slot assignment, and joint frequency-power allocation. First, in the beam partitioning stage, a multi-stage optimization algorithm is introduced to maximize the number of users served by each beam through ILP-based modeling and iterative refinement. Second, for beam-to-slot assignment, a combination of greedy and simulated annealing algorithms is employed to optimize beam allocation across time slots, aiming to reduce inter-beam interference and improve resource utilization efficiency. Finally, in the frequency and power allocation stage, the Hungarian matching algorithm and water-filling method are used to efficiently allocate sub-bands and beam power, with the objective of maximizing user throughput. Simulation results demonstrate that the proposed dimension-wise optimization approach significantly improves resource utilization and throughput with relatively low computational complexity. Compared to traditional joint iterative optimization methods, the proposed method avoids the heavy computational burden caused by multi-dimensional iterative coupling. Under the simulation settings used in this study, the system achieves a resource utilization rate of 99.5%, with a total user throughput improvement ranging from 7.2% to 20%. The proposed framework provides a flexible and scalable solution for dynamic resource management in LEO satellite systems. By decoupling the complex joint optimization into well-structured subproblems and integrating both exact and heuristic algorithms, it significantly enhances system performance in terms of throughput and resource efficiency. Importantly, this framework also offers a promising direction for addressing the coexistence challenges in heterogeneous satellite networks, such as LEO-GEO integration, by enabling coordinated and interference-aware scheduling across multiple dimensions. As such, it lays a solid foundation for the development of intelligent, adaptive, and high-capacity satellite systems in future space-air-ground integrated networks. However, the current scheduling framework is primarily based on offline computation and assumes static system states within each BH cycle. In highly dynamic orbital environments or under burst-like traffic demands, the responsiveness and timeliness of scheduling decisions become critical. Future research could incorporate real-time scheduling mechanisms that address computational latency and rescheduling strategies in response to dynamic user access and satellite mobility. For example, task priority and link-state awareness could be integrated to design fast-response dynamic scheduling strategies suited for more complex operational scenarios. While this paper has demonstrated the effectiveness of the proposed method in medium-scale systems, larger-scale satellite constellations and higher user densities will significantly increase resource coupling and optimization dimensions. To this end, future work may consider developing distributed or hierarchical scheduling frameworks tailored for ultra-large-scale constellations--for instance, by enabling inter-layer coordination across different orbits to achieve globally efficient resource allocation. In addition, incorporating intelligent methods such as reinforcement learning or graph neural networks to enable context-aware scheduling strategies presents a promising direction for further exploration. Conceptualization, J.Z., W.L. and Y.L. ; Methodology, J.Z., W.L. and S.L. ; Software, J.Z. and H.W. ; Writing--original draft, J.Z. ; Writing--review and editing, H.W., Y.L. and J.Z. ; Funding acquisition, W.L. and S.L.