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A weighted graph assigns a weight to each edge, which could represent cost, distance, or any metric. Used in optimizations, such as finding the shortest path.

An undirected graph has edges that do not have a direction associated with them. It is used in networking scenarios or to represent mutual relationships.

A directed graph has all edges directed from one vertex to another. Used for representing one-way relationships, such as web navigation or citation networks.

A tree is an acyclic connected graph. It is used to model hierarchy and relationships, as in organizational charts or file systems.

A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets $U$ and $V$ such that every edge connects a vertex in $U$ to one in $V$. Used for modeling relationships between two distinct classes of objects.