  Inspired by complete graph theory, we demonstrate that a metallic claw
"meta-atom" structure can carry a high number of nearly degenerate resonant
modes. A photonic meta-crystal composing of a lattice of such meta-atoms
exhibits a large number of flat bands that are squeezed into a narrow frequency
window, and these flat bands can be designed to locate in a wide complete 3D
bandgap. The degeneracy dimension (Nf) of the flat bands is determined by the
number of branches (Nb) of the metallic claw with Nf=Nb-3, which is
geometrically related to the complete graph theory. Different from those flat
bands emerging from special lattice arrangements (e.g., Kagome lattice), the
isolated flat bands here are insensitive to lattice perturbations. The proposed
mechanism offers a new platform for realizing various dispersion-less phenomena
and a new paradigm to realize high density of states and spectra compressing.
