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.