**Abstract:** I propose a path integral description for quantum mechanical systems
on compact graphs that consist of N segments of the same length.
Provided the bulk Hamiltonian is segment-independent, N \times N
hermitian unitary matrices that specify scale-invariant boundary
conditions on the operator formalism side are turned out to be in
one-to-one correspondence with N \times N matrix-valued weight factors
on the path integral side. I show that these weight factors are
generally given by N-dimensional unitary representations of the
infinite dihedral group.