Problem 33 - Discrete Mathematics with Applications

Question

Find the number of edges in the bipartite graph \(K_{m, n}\).

Step-by-Step Solution

Answer

The number of edges in the complete bipartite graph \(K_{m,n}\) is \(m\times n\).

Step 1: Determine the partition sets of the vertices

In a complete bipartite graph \(K_{m,n}\), the graph can be divided into two disjoint sets of vertices with \(m\) and \(n\) vertices in each set, respectively.

Step 2: Multiply the number of vertices in each partition for obtaining the total number of edges

Since there is an edge between every vertex in the first set and every vertex in the second set, the number of edges in the bipartite graph will be equal to the product of the number of vertices in each partition. In other words, the total number of edges is \(m\times n\).

Step 3: The number of edges in the bipartite graph

Thus, the number of edges in the complete bipartite graph \(K_{m,n}\) is \(m\times n\).

Key Concepts

Complete Bipartite Graph

A complete bipartite graph, denoted as K_m,n, is a type of graph that is especially interesting in the area of graph theory. This graph is 'complete' in the sense that it has all possible edges between two distinct sets of vertices, without any edges within each set. The two sets of vertices are commonly referred to as partitions, and they are separate from each other.

In a K_m,n, one set contains m vertices and the other contains n vertices. Here, each vertex in the first set is connected to every vertex in the second set by exactly one edge, and vice versa. This property ensures that there are no edges that connect vertices within the same set. The number of edges such a graph would have is simply the product of the number of vertices in each partition: m times n, giving us a straightforward formula to work with.

Graph Theory

Graph theory is a significant branch of mathematics and computer science that deals with graphs, which are mathematical structures used to model pairwise relations between objects. These objects are represented by vertices (also known as nodes), and the relations between pairs of objects are depicted as edges (also known as links).

Graphs come in numerous types, including undirected, directed, weighted, unweighted, simple, and complete, to name a few. Each type of graph serves different purposes and conveys different types of information, which can be explored in various fields such as sociology, biology, computer networking, and of course, mathematics. Graph theory allows us to solve problems related to connectivity, network flow, routing, and many more, making it an invaluable tool for analyzing complex systems.

Vertices Partition

A vertices partition in a graph refers to a division of the set of all vertices into distinct, non-overlapping subsets. For a graph to be bipartite, it must be possible to partition the vertices into two such subsets in a way where edges only exist between vertices belonging to different partitions.

In a complete bipartite graph K_m,n, the partition consists of two subsets with m and n vertices, leading to all vertices being connected to all the vertices of the other subset, but none within their own. This partitioning is a fundamental concept in understanding the structure and behavior of bipartite graphs. For students, mastering the idea of vertices partition is crucial for solving problems related to bipartite graphs and their applications.