Problem 50 - Discrete Mathematics with Applications

Question

Find a spanning tree for each complete bipartite graph. $$K_{2,2}$$

Step-by-Step Solution

Answer

A spanning tree for the complete bipartite graph \(K_{2,2}\) with sets A(a1, a2) and B(b1, b2) can be obtained by selecting any three edges from the original graph. For example, we can choose edges (a1, b1), (a1, b2), and (a2, b1). The obtained spanning tree will include all vertices and have no cycles, as seen in the following structure: ``` a1 --- b1 | | a2 --- b2 ```

Step 1: Identify the sets A and B in \(K_{2,2}\)

In the complete bipartite graph \(K_{2,2}\), it is given that there are two vertices in each set. Let's label the vertices in set A as a1 and a2, set B as b1 and b2.

Step 2: Draw the graph \(K_{2,2}\) and identify the edges

In the graph \(K_{2,2}\), we have the following edges: (a1, b1), (a1, b2), (a2, b1), and (a2, b2). These edges represent all possible connections between the vertices in sets A and B.

Step 3: Select a spanning tree

A spanning tree must include all vertices from the graph and must not contain any cycles. We can choose any three edges from the original graph to construct a spanning tree for \(K_{2,2}\), as there is no restriction on which edges to choose. Let's choose edges (a1, b1), (a1, b2), and (a2, b1) as the edges of our spanning tree.

Step 4: Draw the spanning tree

The spanning tree containing the edges (a1, b1), (a1, b2), and (a2, b1) can be represented with the following structure: ``` a1 --- b1 | | a2 --- b2 ``` In this tree, all vertices of the graph are included, and there are no cycles. Therefore, it is a spanning tree for the complete bipartite graph \(K_{2,2}\).

Key Concepts

Spanning Tree

In graph theory, a spanning tree is a crucial concept used to simplify complex networks. A spanning tree is a subgraph that includes all the vertices of the original graph but contains the minimal number of edges necessary to maintain connectivity. It ensures there are no cycles, meaning there are no closed loops within the graph.
Consider a graph with \(n\) vertices. A spanning tree will have exactly \(n-1\) edges, allowing one less edge for a total connection than the number of vertices, which prevents any cycles from forming. Spanning trees are particularly useful for finding the minimum connections needed to preserve the structure of a graph.

• All vertices must be included: Every vertex from the original graph is present in the spanning tree.
• No cycles: There is no closed trail in the spanning tree that would allow you to revisit any node without doubling back on your path.
• Tree property: A tree on \(n\) vertices always contains \(n-1\) edges.

Complete Bipartite Graph

A complete bipartite graph is a special type of bipartite graph where every vertex in the first set is connected to every vertex in the second set. It is represented by the notation \(K_{m,n}\), where \(m\) and \(n\) refer to the number of vertices in each subset of the graph.
For example, in \(K_{2,2}\) there are two vertices in each set, resulting in the following edges:

• Each vertex from set A connects to both vertices in set B.
• This creates a mesh-like connection between the sets.

Complete bipartite graphs have unique properties that make them useful in various graph theory problems and applications, such as matching problems and network design.
These graphs can represent systems where two groups are completely interconnected, showcasing clear and simple relationships.

• Two disjoint sets: Groups of vertices with no inter-group edges.
• Complete connectivity: Every vertex in one set is connected to every vertex in the other set.
• Applications: Useful in scheduling and organizing events or resources.

Vertices

Vertices are the fundamental units of graphs in graph theory. They can be thought of as nodes connecting different paths in a network. Each vertex can connect to one or multiple other vertices via edges, which are the lines connecting them.
The role of vertices in graph theory is crucial:

• Points of connection: These form the foundation of graphs by acting as points that edges connect.
• Characteristics: In a given graph, vertices can vary in degree, indicating the number of connections (edges) it has.
• Labelling: Often vertices are labelled alphabetically (e.g., a1, a2) for identification, aiding in understanding graph configuration.

Vertices help in understanding the structure and connectivity of graphs and are essential for solving many graph-related problems such as finding paths, circuits, and spanning trees.In a complete bipartite graph like \(K_{2,2}\), vertices are divided into two sets, each fully connected to the opposite set.