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Recently, beyond diagonal reconfigurable intelligent surface (BD-RIS) hasbeen proposed to generalize conventional RIS. BD-RIS has a scattering matrixthat is not restricted to being diagonal and thus brings a performanceimprovement over conventional RIS. While different BD-RIS architectures havebeen proposed, it still remains an open problem to develop a systematicapproach to design BD-RIS architectures achieving the optimal trade-off betweenperformance and circuit complexity. In this work, we propose novel modeling,architecture design, and optimization for BD-RIS based on graph theory. Thisgraph theoretical modeling allows us to develop two new efficient BD-RISarchitectures, denoted as tree-connected and forest-connected RIS.Tree-connected RIS, whose corresponding graph is a tree, is proven to be theleast complex BD-RIS architecture able to achieve the performance upper boundin multiple-input single-output (MISO) systems. Besides, forest-connected RISallows us to strike a balance between performance and complexity, furtherdecreasing the complexity over tree-connected RIS. To optimize tree-connectedRIS, we derive a closed-form global optimal solution, while forest-connectedRIS is optimized through a low-complexity iterative algorithm. Numericalresults confirm that tree-connected (resp. forest-connected) RIS achieves thesame performance as fully-connected (resp. group-connected) RIS, while reducingthe complexity by up to 16.4 times.

Beyond Diagonal Reconfigurable Intelligent Surfaces Utilizing Graph Theory: Modeling, Architecture Design, and Optimization