12 May 2026 Tue 3 pm

We study localization and flat-band formation in lattices and graphs generated by repeated edge inflation of square, honeycomb, and triangular parent lattices. Replacing each bond with a finite tight-binding chain gives rise to several distinct classes of flat bands: chain-induced flat bands at the eigenenergies of the inserted chains, symmetry-protected zero-energy flat bands in bipartite edge-inflated lattices, and nearly flat junction bands near the spectral edges for sufficiently long chains. We analyze these mechanisms for ordered Lieb-$L$, super$^{L}$honeycomb, and super$^{L}$triangular lattices, and examine their response to bond disorder, site disorder, random magnetic flux, and randomness in the inflation process itself.

While bond and site disorder broaden most flat bands, the zero-energy chiral band and the junction-induced flat bands remain robust under certain perturbations. Remarkably, substantial flat-band features also persist in randomly edge-inflated graphs, even in the absence of translational symmetry. In particular, the number of zero-energy states is found to be well approximated by the matching deficiency $N - 2\nu(G)$, indicating that local tree-like structure continues to control the low-energy nullity.

Finally, we consider random regular graphs (RRGs), in which each vertex has a fixed degree while edges are connected randomly. Although such graphs do not exhibit flat bands, edge inflation generates a rich pattern of flat-band structures. These results identify edge-inflated lattices and graphs as a broad class of systems in which geometry alone gives rise to robust localization in both ordered and disordered settings.

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