Is anybody working on using a neural network to build and update a symbolic graph, and vice versa? Or at least using a symbolic graph as an input to a NLP neural network, so the network could learn to rely on the symbolic graph when it is useful?
Yes. ConceptNet [1] and distributional word embeddings go really well together, and can compare word meanings better than either one alone. Here's the preprint of the AAAI 2017 paper [2].