Understanding Bipartite Graphs | Definition, Properties, and Applications

Bipartie

In mathematics, a bipartite graph is a type of graph that consists of two distinct sets of vertices, denoted as U and V, with the property that every edge in the graph connects a vertex in U to a vertex in V

In mathematics, a bipartite graph is a type of graph that consists of two distinct sets of vertices, denoted as U and V, with the property that every edge in the graph connects a vertex in U to a vertex in V. In other words, there are no edges connecting vertices within the same set.

Formally, a graph G = (U, V, E) is bipartite if and only if its vertex set V can be partitioned into two non-empty disjoint sets U and V, such that every edge in E connects a vertex in U to a vertex in V.

One practical example of a bipartite graph is a friendship network in a social media platform. Let’s say we have a set of users in U and a set of their friends (also users) in V. An edge is drawn between each user and their friends. Since users are not connected to each other directly, the graph can be considered bipartite.

One important property of bipartite graphs is that they do not contain any odd cycles. This means that it is not possible to find a cycle of length 3, 5, 7, etc., in a bipartite graph.

Bipartite graphs have applications in various fields such as computer science, biology, and economics. They can be used to model relationships between different entities, such as matching problems, assignment problems, and network flows.

To determine whether a given graph is bipartite or not, you can use various algorithms such as depth-first search or breadth-first search. These algorithms check if it is possible to assign vertices to two sets such that no adjacent vertices belong to the same set. If such an assignment is possible, the graph is bipartite; otherwise, it is not.

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