  We study Bose-Hubbard models in a family of diamond necklace lattices with
$n$ central sites. The single-particle spectrum of these models presents
compact localized states (CLSs) that occupy the up and down sites of each
diamond. By performing an appropriate basis rotation, the fragmentation of the
many-boson Hilbert space becomes apparent in the adjacency graph of the
Hamiltonian, showing disconnected sub-sectors with a wide range of dimensions.
The models present a conserved quantity related to the occupation of the
single-particle CLSs that uniquely identifies the different sub-sectors of the
many-boson Hilbert space. Due to the fragmentation of the Hilbert space, the
distribution of entanglement entropies of the system presents a nested-dome
structure. We find weak thermalization through sub-sector-restricted
entanglement evolution and a wide range of entanglement entropy scalings from
area-law to logarithmic growth. Additionally, we observe how the
distinguishability between the different domes increases with the number of
central sites and we explain the mechanism behind this fact by analyzing the
graph structure of the Hamiltonian.
