Rank properties of the arc-node incidence matrix

Recall the definition of the arc-node incidence matrix of a network.

A number of topological properties of a network with nodes and  edges can be inferred from those of its node-arc incidence matrix , and of the reduced incidence matrix , which is obtained from by removing its last row. For example, the network is said to be connected if there is a path joining any two nodes. It can be shown that the network is connected if and only if the rank of  is equal to .

See also: Nullspace of a transpose incidence matrix.