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Recall the definition of the arc-node incidence matrix of a network.
A number of topological properties of a network with \( m \) nodes and \( n \) edges can be inferred from those of its node-arc incidence matrix \( A \), and of the reduced incidence matrix \( \tilde{A} \), which is obtained from \( A \) 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 \( \tilde{A} \) is equal to \( m-1 \).
