  We consider graphs made of one-dimensional wires connected at vertices and on
which may live a scalar potential. We are interested in a scattering situation
where the graph is connected to infinite leads. We investigate relations
between the scattering matrix and the continuous part of the local density of
states, the injectivities, emissivities and partial local density of states.
Those latter quantities can be obtained by attaching an extra lead at the point
of interest and by investigating the transport in the limit of zero
transmission into the additional lead. In addition to the continuous part
related to the scattering states, the spectrum of graphs may present a discrete
part related to states that remain uncoupled to the external leads. The theory
is illustrated with the help of a few simple examples.
