Recently, beyond diagonal reconfigurable intelligent surface (BD-RIS) has been proposed to generalize conventional RIS. BD-RIS has a scattering matrix that is not restricted to being diagonal and thus brings a performance improvement over conventional RIS. While different BD-RIS architectures have been proposed, it still remains an open problem to develop a systematic approach to design BD-RIS architectures achieving the optimal trade-off between performance and circuit complexity. In this work, we propose novel modeling, architecture design, and optimization for BD-RIS based on graph theory. This graph theoretical modeling allows us to develop two new efficient BD-RIS architectures, denoted as tree-connected and forest-connected RIS. Tree-connected RIS, whose corresponding graph is a tree, is proven to be the least complex BD-RIS architecture able to achieve the performance upper bound in multiple-input single-output (MISO) systems. Besides, forest-connected RIS allows us to strike a balance between performance and complexity, further decreasing the complexity over tree-connected RIS. To optimize tree-connected RIS, we derive a closed-form global optimal solution, while forest-connected RIS is optimized through a low-complexity iterative algorithm. Numerical results confirm that tree-connected (resp. forest-connected) RIS achieves the same performance as fully-connected (resp. group-connected) RIS, while reducing the complexity by up to 16.4 times.

翻译：最近，超对角可重构智能表面（BD-RIS）被提出以推广传统RIS。BD-RIS的散射矩阵不限于对角阵，因此相比传统RIS带来了性能提升。尽管已有多种BD-RIS架构被提出，但如何系统性地设计能在性能与电路复杂度之间实现最优权衡的BD-RIS架构仍是一个开放问题。本文基于图论提出了BD-RIS的新颖建模、架构设计与优化方法。这种图论建模使我们能够开发两种新型高效BD-RIS架构，分别称为树连接RIS与森林连接RIS。树连接RIS——其对应图为树——被证明是在多输入单输出（MISO）系统中能达到性能上界的最小复杂度BD-RIS架构。此外，森林连接RIS可平衡性能与复杂度，进一步降低树连接RIS的复杂度。为优化树连接RIS，我们推导了闭式全局最优解；而森林连接RIS则通过低复杂度迭代算法进行优化。数值结果证实：树连接RIS（森林连接RIS）实现了与全连接RIS（组连接RIS）相同的性能，同时将复杂度降低了高达16.4倍。