Graphs are not data structures. They are mathematical representation of relations. Yes, you can think and theoretize about problems using graphs, and there is a large body of theory about it. But when you need to implement an algorithm, you are choosing data structures to best represent the problem, not graphs. There are many data structures that represent general graphs, and even more for special kinds of graphs.
In your question, you mix these two things. The same theoretical solution may be in terms of graph, but practical solutions may use different data structures to represent the graph.
