Graph embedding tensor: unifying topological description, symmetry detection, and structure generation for two-dimensional carbon allotropes

We introduce the embedding tensor Φ, a rank-3, purely integer tensor that elevates faces and edges to the same footing as vertices to provide a unique, coordinate-free representation of any three-connected two-dimensional carbon lattice. Defined by the incidence of vertices, edges, and polygonal faces, Φ obeys simple summation rules derived from Euler characteristics. Casting Φ into a flag graph enables exact, tolerance-free identification of wallpaper symmetries. Building on this algebraic framework, we develop an iterative add-dimer search that generates all structures with NF faces from all crystals with NF − 1 faces, while automatically discarding duplicates via tensor isomorphism and spotting non-primitive cells through symmetry checks. Exploiting symmetry keeps the combinatorial growth of dimer insertions tractable even for NF > 5, transforming an otherwise exponential search into a practically feasible approach for high-throughput exploration. Once candidate topologies are enumerated, approximate real-space coordinates and lattice vectors can be reconstructed analytically from Φ and sparse crossing matrices, providing initial geometries for electronic or vibrational calculations. The method delivers an end-to-end pipeline from exhaustive, symmetry-aware enumeration to metadata tagging and coordinate generation, while requiring only integer arithmetic.