What characterizes a weighted graph?

A weighted graph is characterized by each edge having a numerical value that represents an attribute of the edge, typically indicating cost or capacity. This numerical value allows for the representation of various practical scenarios such as distances between nodes, costs to traverse edges, or maximum flow rates. This feature is crucial in many algorithms used for graph analysis, such as Dijkstra's or Prim's algorithm, where these weights enable the determination of optimal paths or minimum spanning trees within the graph.

In contrast, while it is true that all edges in a weighted graph may have values representing relationships, it is the specific numerical nature of those values indicating cost or capacity that solidifies its classification as a weighted graph. The definition excludes other graph characteristics like cycle presence or restrictions on node connectivity, as those pertain to different types of graphs, such as directed, undirected, acyclic, or specific connectivity conditions.