Fractons on graphs and complexity

We introduce two exotic lattice models on a general spatial graph. The first one is matter theory of compact Lifshitz scalar field, while the second certain rank-2 $U(1)$ gauge fractons. Both are defined via discrete Laplacian operator unveil an intriguing correspondence between physical observables these and graph quantities. For instance, ground state degeneracy equals number spanning trees graph, which common measure complexity in ("GSD = complexity"). global symmetry identified as Jacobian group In theory, superselection sectors fractons one-to-one with divisor classes theory. particular, under mild assumptions fracton immobility proven using graph-theoretic Abel-Jacobi map.