On a digraph, define a sink to be a vertex with out-degree zero and a source to be a vertex with in-degree zero. Let be a digraph with non-negative weights and with exactly one sink and exactly one source. A flow on is an assignment of values to each edge of satisfying certain rules:
For any edge , we must have (where is the weight of ).
For any vertex , excluding the source and the sink, let be the set of edges incident to and let be the set of edges incident from . Then we must have
Let be the edges incident from the source, and let be the set of edges incident to the sink. If is a flow, then
We will refer to this quantity as the amount of flow.
We may interpret a flow as a means of transmitting something through a network. Suppose we think of the edges in a graph as pipes, with the weights corresponding with the capacities of the pipes; we are pouring water into the system through the source and draining it through the sink. Then the first rule requires that we do not pump more water through a pipe than is possible, and the second rule requires that any water entering a junction of pipes must leave. Under this interpretation, the maximum amount of flow corresponds to the maximum amount of water we could pump through this network.
Instead of water in pipes, one may think of electric charge in a network of conductors. Rule (2) above is one of Kirchoff’s two laws for such networks; the other says that the sum of the voltage drops around any circuit is zero.
|Date of creation||2013-03-22 13:00:50|
|Last modified on||2013-03-22 13:00:50|
|Last modified by||bgins (4516)|