Exploring the solution space of low-energy transfer trajectories in the multi-body gravitational environment is a challenging task. The highly nonlinear dynamics and many degrees of freedom of control inputs lead to a vast variety and number of possible solutions. The present paper develops a graph-based framework for computing low-energy transfer trajectories in a multi-objective fashion. Periapsis states obtained from the newly introduced special apsis conditions, with which pivotal dynamical objects in the low-energy regime such as the zero velocity surface and periodic orbits associate, and their connectivity calculated in the fixed-time-of-arrival method represent the graphs. Initial guess solutions are obtained as shortest paths in the graphs that are optimized to minimize the fuel cost. The framework is shown to be effectively applicable to distinct phase-space regions in a unified manner demonstrating its versatility in the complex dynamical environment. The present paper finds new Pareto solutions in the halo-to-halo transfer problem as well as the high-energy excursion technique that enhances the flyby effect in the multi-flyby transfer problem as by-products of the applications.