  In this paper we associate an abelian category to a finite directed graph and
prove the categories arising from two graphs are equivalent if the incidence
matrices of the graphs are shift equivalent. The abelian category is the
quotient of the category of graded vector space representations of the quiver
obtained by making the graded representations that are the sum of their finite
dimensional submodules isomorphic to zero.
  Actually, the main result in this paper is that the abelian categories are
equivalent if the incidence matrices are strong shift equivalent. That result
is combined with an earlier result of the author to prove that if the incidence
matrices are shift equivalent, then the associated abelian categories are
equivalent.
  Given William's Theorem that subshifts of finite type associated to two
directed graphs are conjugate if and only if the graphs are strong shift
equivalent, our main result can be reformulated as follows: if the subshifts
associated to two directed graphs are conjugate, then the categories associated
to those graphs are equivalent.
