  We compare the spectral properties of two kinds of linear operators
characterizing the (classical) geodesic flow and its quantization on connected
locally finite graphs without dead ends. The first kind are transfer operators
acting on vector spaces associated with the set of non backtracking paths in
the graphs. The second kind of operators are averaging operators acting on
vector spaces associated with the space of vertices of the graph. The choice of
vector spaces reflects regularity properties. Our main results are
correspondences between classical and quantum spectral objects as well as some
automatic regularity properties for eigenfunctions of transfer operators.
