Eigenfunctions and the integrated density of states on Archimedean tilings

We study existence and absence of $\ell^2$-eigenfunctions the combinatorial Laplacian on 11 Archimedean tilings Euclidean plane by regular convex polygons. show that exactly two these (namely $(3.6)^2$ Kagome tiling $(3.12^2)$ tiling) have $\ell^2$-eigenfunctions. These eigenfunctions are infinitely degenerate constituted explicitly described which supported a finite number vertices underlying graph hexagons $12$-gons in tilings, respectively). Furthermore, we provide an explicit expression for Integrated Density States (IDS) terms eigenvalues Floquet matrices deduce integral formulas IDS $(4^4)$, $(3^6)$, $(6^3)$, $(3.6)^2$, tilings. Our method proof can be applied to other $\mathbb{Z}^d$-periodic graphs as well.