Strain-Tunable Topological Phase Transitions in Line- and Split-Graph Flat-Band Lattices

In recent years, materials with topological flat bands have attracted significant attention due to their association with extraordinary transport properties and strongly correlated electrons. Yet, generic principles linking lattice architecture, strain, and band topology remain scarce. Here, using a unified graph-theoretic framework we generate entire families of two-dimensional lattices and, using analytical tight-binding calculations, demonstrate that a single mechanical knob -- uniform in-plane strain -- drives universal transitions between trivial insulating, Dirac semimetal, and quantum spin-Hall phases across all lattices. The framework yields several flat band lattices that were hitherto absent or largely unexplored in the literature -- for example, the checkerboard split-graph and triangular-Kagome lattices -- whose strain-driven topological phase diagrams we establish here for the first time. The design rules implied by our studies provide a blueprint for engineering topological states in a wide variety of 2D materials, photonic crystals, and circuit lattices, and are anticipated to accelerate the discovery of strain-programmable quantum matter.