  In this paper we study path integral for a single spinless particle on a star
graph with N edges, whose vertex is known to be described by U(N) family of
boundary conditions. After carefully studying the free particle case, both at
the critical and off-critical levels, we propose a new path integral
formulation that correctly captures all the scale-invariant subfamily of
boundary conditions realized at fixed points of boundary renormalization group
flow. Our proposal is based on the folding trick, which maps a scalar-valued
wave function on star graph to an N-component vector-valued wave function on
half-line. All the parameters of scale-invariant subfamily of boundary
conditions are encoded into the momentum independent weight factors, which
appear to be associated with the two distinct path classes on half-line that
form the cyclic group Z_2. We show that, when bulk interactions are
edge-independent, these weight factors are generally given by an N-dimensional
unitary representation of Z_2. Generalization to momentum dependent weight
factors and applications to worldline formalism are briefly discussed.
