  The indistinguishability of quantum particles is widely used as a resource
for the generation of entanglement. Linear quantum networks (LQNs), in which
identical particles linearly evolve to arrive at multimode detectors, exploit
the indistinguishability to generate various multipartite entangled states by
the proper control of transformation operators. However, it is challenging to
devise a suitable LQN that carries a specific entangled state or compute the
possible entangled state in a given LQN as the particle and mode number
increase. This research presents a mapping process of arbitrary LQNs to graphs,
which provides a powerful tool for analyzing and designing LQNs to generate
multipartite entanglement. We also introduce the perfect matching diagram (PM
diagram), which is a refined directed graph that includes all the essential
information on the entanglement generation by an LQN. The PM diagram furnishes
rigorous criteria 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.
