There are cases where we want to allow the number of edges to drop so low that the graph breaks into multiple pieces, for example, when we talk about connected components but more often then not, we're thinking about a connected graph with no parallel edges and then we can pin down the number of edges m to be somewhere between the linear and the number of nodes, linear in N and quadratic in N. Now I'm not going to give you a super formal definition of what a sparse or a dense graph is, and people are a little loose with this terminology in practice. 