This research presents a map- ping process of arbitrary LQNs to graphs, which provides a powerful tool for analyz- ing and designing LQNs to generate mul- tipartite entanglement. We also introduce the perfect matching diagram (PM dia- gram), which is a refined directed graph that includes all the essential information on the entanglement generation by an LQN. The PM diagram furnishes rigorous crite- ria for the entanglement of an LQN and solid guidelines for designing suitable LQNs for the genuine entanglement. Based on the structure of PM diagrams, we compose LQNs for fundamental N-partite genuinely entangled states.