Networks Graph Theory 2Fa075

1. The problem is to understand what networks are in graph theory. 2. In graph theory, a network is a graph in which edges have weights or capacities. 3. These weights can represent distances, costs, or capacities depending on the context. 4. A network is often represented as $$G = (V, E, w)$$ where $$V$$ is the set of vertices, $$E$$ is the set of edges, and $$w$$ is a weight function assigning a number to each edge. 5. Important rules include that weights are usually non-negative and that networks can be directed or undirected. 6. Networks are used to model real-world problems like transportation, communication, and flow problems. 7. For example, in a flow network, the capacity constraints and flow conservation laws apply. 8. Understanding networks involves studying paths, connectivity, flows, and cuts within the graph. 9. Algorithms like Dijkstra's for shortest paths or Ford-Fulkerson for maximum flow are fundamental in network analysis. 10. Thus, networks in graph theory extend basic graphs by adding meaningful weights to edges to model complex systems.