  We study unusual gapped topological phases where they admit $\mathbb{Z}_N$
fractional excitations in the same manner as topologically ordered phases, yet
their ground state degeneracy depends on the local geometry of the system.
Placing such phases on 2D lattice, composed of an arbitrary connected graph and
1D line, we find that the fusion rules of quasiparticle excitations are
described by the Laplacian of the graph and that the number of superselection
sectors is related to the kernel of the Laplacian. Based on this analysis, we
further show that the ground state degeneracy is given by $\bigl[N\times
\prod_{i}\text{gcd}(N, p_i)\bigr]^2$, where $p_i$'s are invariant factors of
the Laplacian that are greater than one and gcd stands for the greatest common
divisor. We also discuss braiding statistics between quasiparticle excitations.
