Symmetric higher rank topological phases on generic graphs

Motivated by recent interest in fracton topological phases, we explore the interplay between gapped 2D ${\mathbb{Z}}_N$ topological phases which admit fractional excitations with restricted mobility and geometry of the lattice on which such phases are placed. We investigate the properties of the phases in a geometric context—graph theory. By placing the phases on a 2D lattice consisting of two arbitrary connected graphs, $G_x⊠G_y$, we study the behavior of fractional excitations of the phases. We derive the formula of the ground-state degeneracy of the phases, which depends on invariant factors of the Laplacian.