  The aim of the present paper is to introduce the notion of first order
(supersymmetric) Dirac operators on discrete and metric (``quantum'') graphs.
In order to cover all self-adjoint boundary conditions for the associated
metric graph Laplacian, we develop systematically a new type of discrete graph
operators acting on a decorated graph. The decoration at each vertex of
degree-d is given by a subspace of $\C^d$, generalising the fact that a
function on the standard vertex space has only a scalar value.
  We develop the notion of exterior derivative, differential forms, Dirac and
Laplace operators in the discrete and metric case, using a supersymmetric
framework. We calculate the (supersymmetric) index of the discrete Dirac
operator generalising the standard index formula involving the Euler
characteristic of a graph. Finally, we show that the corresponding index for
the metric Dirac operator agrees with the discrete one.
