  Motivated by recent interests 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
new geometric context -- graph theory. By placing the phases on a 2D lattice
consisting of two arbitrary connected graphs, $G_x\boxtimes G_y$, 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.
