A fully connected graph, with an edge between every two vertices, conveys the same amount of information as a graph with no edges at all. The important graphs are the ones where some things are not connected to some other things.

A graph with weighted vertices provides more information than both the above graphs. Also, "connected" in graph theory usually means "there is a path from every node to every other node", not "there are edges between every node". Of course when talking about real things connected to each other, it is usually more interesting to note in what way they are connected, rather than observe that there exists a connection -- and that does require narrowing one's focus.