How is a weighted graph defined?

How is a weighted graph defined?

• A. A graph where every vertex has a unique identifier
• B. A graph where edges are assigned numerical values representing costs
• C. A graph that allows only linear connections between vertices
• D. A graph that is undirected and unweighted

Answer: (B)

A weighted graph is defined by the assignment of numerical values to its edges, which represent costs, distances, or other quantitative measures. This means that each edge carries a weight that indicates the "cost" of traversing that edge, which can be particularly useful in optimization problems, such as finding the shortest path or minimum spanning tree. In the context of graph theory and algorithms, these weights allow for more sophisticated analyses and algorithms to be developed, as they enable different types of graph traversal and pathfinding that take these weights into account. For instance, algorithms like Dijkstra's or Prim's rely on these weights to function correctly, emphasizing the significance of this concept. The other options do not define a weighted graph accurately. The uniqueness of vertex identifiers does not relate to weights or costs on edges. Linear connections do not inherently imply any weighting. Lastly, an undirected and unweighted graph would specifically lack the numerical values assigned to edges, thereby not qualifying as a weighted graph.

A weighted graph is defined by the assignment of numerical values to its edges, which represent costs, distances, or other quantitative measures. This means that each edge carries a weight that indicates the "cost" of traversing that edge, which can be particularly useful in optimization problems, such as finding the shortest path or minimum spanning tree.

In the context of graph theory and algorithms, these weights allow for more sophisticated analyses and algorithms to be developed, as they enable different types of graph traversal and pathfinding that take these weights into account. For instance, algorithms like Dijkstra's or Prim's rely on these weights to function correctly, emphasizing the significance of this concept.

The other options do not define a weighted graph accurately. The uniqueness of vertex identifiers does not relate to weights or costs on edges. Linear connections do not inherently imply any weighting. Lastly, an undirected and unweighted graph would specifically lack the numerical values assigned to edges, thereby not qualifying as a weighted graph.