The incidence matrix A has m = 6 rows and n = 4 columns. Each row corresponds
to an edge in the graph, and each column corresponds to a node. We do have to
number the edges and nodes, and also choose directions for the arrows, in order to
construct A. But the numbering and edge directions are arbitrary. Flows can travel
both ways, and a different choice of arrows will not change the reality of the model.
The entries −1 and 1 in each row of A give a record of the corresponding edge:
The first edge leaves node 1 and goes to node 2.
The first row has −1 in column 1 and +1 in column 2.
Row 5 is typical. Edge 5 leaves node 2 (by −1 in column 2), and it enters node 4
(+1 in column 4). We chose arrows from lower-numbered nodes to higher-numbered
nodes, for simplicity. Then the −1 comes before the +1 in each row. In all cases, you
can write down A immediately by looking at the graph. The graph and the matrix
have the same information.