  We propose path integral description for quantum mechanical systems on
compact graphs consisting of N segments of the same length. Provided the bulk
Hamiltonian is segment-independent, scale-invariant boundary conditions given
by self-adjoint extension of a Hamiltonian operator turn out to be in
one-to-one correspondence with N \times N matrix-valued weight factors on the
path integral side. We show that these weight factors are given by
N-dimensional unitary representations of the infinite dihedral group.
