  We study the class of weighted locally gentle quivers. This naturally extends
the class of gentle quivers and gentle algebras, which have been intensively
studied in the representation theory of finite-dimensional algebras, to a wider
class of potentially infinite-dimensional algebras. Weights on the arrows of
these quivers lead to gradings on the corresponding algebras. For the natural
grading by path lengths, any locally gentle algebra is a Koszul algebra.
  Our main result is a general combinatorial formula for the determinant of the
weighted Cartan matrix of a weighted locally gentle quiver. This determinant is
invariant under graded derived equivalences of the corresponding algebras. We
show that this weighted Cartan determinant is a rational function which is
completely determined by the combinatorics of the quiver, more precisely by the
number and the weight of certain oriented cycles. This leads to combinatorial
invariants of the graded derived categories of graded locally gentle algebras.
