Gapped lineon and fracton models on graphs

We introduce a Z₍ stabilizer code that can be defined on any spatial lattice of the form C₋ₙ, where is a general graph. We also present the low-energy limit of this stabilizer code as a Euclidean lattice action, which we refer to as the anisotropic Z₍ Laplacian model. It is gapped, robust (i. e. , stable under small deformations), and has lineons. Its ground-state degeneracy (GSD) is expressed in terms of a ``mod N-reduction'' of the Jacobian group of the graph. In the special case when space is an LLₙ cubic lattice, the logarithm of the GSD depends on L in an erratic way and grows no faster than O (L). We also discuss another gapped model, the Z₍ Laplacian model, which can be defined on any graph. It has fractons and a similarly strange GSD.