What is the method to find the maximum flow in a network?

To determine the maximum flow in a network, one effective method involves finding the minimum capacity along each possible path from the source to the sink. This is because the flow in a network is constrained by the edge with the least capacity within that path. By identifying the minimum edge capacity for each route from the starting point to the endpoint, you can ascertain how much flow can actually pass through the network without exceeding any edge's capacity. Summing these minimum capacities across all possible paths allows for calculating the total possible maximum flow from the source to the sink.

This method is grounded in the Max-Flow Min-Cut Theorem, which states that the maximum flow in a network is equal to the capacity of the minimum cut. This interaction between path capacity and overall flow helps ensure that every aspect of the network's flow potential is maximized without violating capacity constraints.

The other methods described do not appropriately reflect the fundamental principles of network flow analysis. For instance, calculating the total capacity of all vertices does not consider how flows interact along paths. Similarly, merely adding maximum capacities from the source to the sink does not acknowledge the limitations imposed by the individual edge capacities. Counting edges in a path does not provide essential information on flow capacities, as it overlooks the vital aspect