Graph Theory Questions Long
In graph theory, a weighted graph is a type of graph where each edge is assigned a numerical value or weight. This weight represents some kind of measurement or cost associated with traversing that edge. The weights can be positive or negative, and they can also be zero.
The concept of a weighted graph is used to model real-world scenarios where the edges between vertices have different levels of importance, distance, or cost. For example, in a transportation network, the weights could represent the distance between two locations, the time it takes to travel between them, or the cost of transportation. In a social network, the weights could represent the strength of relationships between individuals.
Weighted graphs are often represented using a matrix or a list of edges, where each edge is associated with its weight. The weight can be stored as a separate value or as part of the edge itself. The weights can be used to calculate various properties of the graph, such as the shortest path between two vertices or the minimum spanning tree.
There are several algorithms and techniques that are specifically designed for working with weighted graphs. Some of the most commonly used algorithms include Dijkstra's algorithm for finding the shortest path, Prim's algorithm for finding the minimum spanning tree, and the Bellman-Ford algorithm for finding the shortest path with negative weights.
In summary, a weighted graph is a graph where each edge is assigned a numerical value or weight. It is used to model real-world scenarios where the edges have different levels of importance, distance, or cost. The weights can be positive, negative, or zero, and they are used to calculate various properties of the graph.