Question

Draw a graph that has more than one minimum spanning tree.

Short answer

A graph with vertices A, B, C, D and having all equal edge weights can be created. In this case, more than one Minimum Spanning Tree (MST) can be formed.

Step-by-step solution

Draw Initial Graph

First, start by drawing a simple graph with some vertices. Let's say for instance that you have four vertices: A, B, C and D. Connect these vertices to form a simple square.

Assign Weights

Now assign the same weights to all edges of the graph. To create the edge weights all equal, you may assign every edge a weight of 2.

Review

Analyze and note that in this case more than one minimum spanning tree can be formed. For example, an MST can be obtained by taking the edges AB, AD and BD or BA, BC and AC.

Draw MSTs

Now, draw the minimum spanning trees. The first MST contains the edges AB, AD, BD and the second MST contains the edges BA, BC, AC.

Key concepts

Graph Theory

Graph Theory is a fascinating area of mathematics and computer science that explores the study of graphs, which are structures composed of vertices (also called nodes) and edges that connect these vertices. These graphs can represent a myriad of real-world scenarios, such as social networks, transport networks, and even pathways through a city.

A graph can be either directed, where the edges have a specific direction, or undirected, where they don't. In the core of graph theory is the idea that we can model pairwise relationships between objects using graphs, helping us visualize and solve complex problems.

• Vertices: These are the fundamental units or 'dots' in a graph. Each vertex can connect to one or more other vertices through edges.
• Edges: These are the 'lines' that connect pairs of vertices. They are key in determining how interconnected elements are within a graph.

Understanding these basic components of graph theory is essential because they form the basis for more complex topics, such as spanning trees and minimum spanning trees.

Spanning Tree

In graph theory, a spanning tree is a subgraph that includes all the vertices of the original graph and is a tree. A tree is a special type of graph that is connected and acyclic, meaning it doesn't contain any loops.

Imagine you have a network made of points and lines connecting them. A spanning tree connects all the points (or vertices) with the minimum number of connections needed, ensuring every point is reachable by every other point, without any cycles. This concept is crucial in computer networks because it helps to ensure efficiency and minimize redundancy.

• Connectivity: A spanning tree must be connected, which means there is a path between any pair of vertices.
• Acyclicity: Without cycles, the spanning tree keeps the structure simple and efficient.

In some graphs, there can be more than one spanning tree possible, and finding a minimum spanning tree can help in optimizing certain processes, like network design.

Vertex and Edge Weights

Vertex and edge weights add another layer of complexity and utility to graph theory. In many real-world applications, graphs not only connect vertices but also assign weights to these connections, representing costs, distances, or any quantifiable factor.

Primarily, weights are used in edges and serve as an essential element in finding minimum spanning trees. A minimum spanning tree seeks to connect all vertices in a graph with the least possible total edge weight, which is optimal for cost-saving measures.

• Edge Weights: These are values assigned to the edges of a graph. In a transportation network, for example, an edge weight might represent the fare or distance between two locations.
• Vertex Weights: Though less common than edge weights, vertex weights can be crucial in specialized problems, representing the importance or cost of using a particular node.

The concept of edge weights is integral in determining which connections are "cheapest" or most efficient, allowing algorithms like Kruskal's or Prim's to find a minimum spanning tree effectively.