91Ó°ÊÓ

There are two routes from city \(A\) to city \(D .\) One route passes through city \(B\) and the other route passes through city \(C .\) During the period 7: 00 A.M. to 8: 00 A.M., the average trip times are \(A\) to \(B \quad 30\) minutes \(\begin{array}{ll}A \text { to } C & 15 \text { minutes }\end{array}\) \(\begin{array}{ll}B \text { to } D & 15 \text { minutes }\end{array}\) \(\begin{array}{ll}\text { C to } D & 15 \text { minutes }\end{array}\) The maximum capacities of the routes are \(A\) to \(B\) 1000 vehicles \(\begin{array}{ll}A \text { to } C & 3000 \text { vehicles }\end{array}\) \(B\) to \(D \quad 4000\) vehicles \(C\) to \(D \quad 2000\) vehicles. Represent the flow of traffic from \(A\) to \(D\) during the period 7:00 A.M. to 8: 00 A.M. as a network.

Show that by a suitable ordering of the vertices, the adjacency matrix of a bipartite graph can be written $$ \left(\begin{array}{ll} 0 & A \\ A^{T} & 0 \end{array}\right) $$ where 0 is a matrix consisting only of 0 's and \(A^{T}\) is the transpose of the matrix \(A\).