I think some aspect of the answer is still missing that is not shared by other topicA-vs-topicB-type-questions: Graphs can (sometimes) be drawn. Even if no graph theoretic technique ever solves a ring theoretic question and no ring theory ever improves our understanding of graph theory, the fact that we can draw or otherwise visualise certain graphs can be a big boost to understanding in and of itself.
Maybe there is some structure in the ring in question that is purely ring theoretical, useful for the problem at hand, but somewhat difficult to discover. The very fact that one can draw such a structure as a graph makes it visible and intuitively graspable. Human brains can find visual patterns in a fraction of a second. Finding patterns in complex algebraic structures is (many) order of magnitude slower and harder for human brains. This alone can be huge a benefit of associating graphs to algebraic (or other non-trivial) objects.
And there does not need to be any interaction between ring and graph theory for this benefit. The very fact that one can do an inductive but purely algebraic argument, say by induction over the vertices of the graph, is often enough. You just had to draw the graph to see what the right ordering for the induction is (maybe you inductively delete leaves from a tree or something common like that), nothing more.