Spanning trees connect vertices minimum edges from "summary" of Introduction to Graph Theory by Douglas Brent West
A spanning tree is a connected subgraph of a graph that includes all of the vertices of the original graph. In other words, a spanning tree is a way of connecting all the vertices of a graph without creating any cycles. One important property of a spanning tree is that it contains the minimum number of edges necessary to connect all of the vertices. This means that a spanning tree is a tree (a connected acyclic graph) that spans all of the vertices of the original graph. To understand why spanning trees connect vertices with the minimum number of edges, consider the following. If a graph has \( n \) vertices, then a tree on \( n \) vertices has exactly \( n-1 \) edges. This is because a tree is a minimally connected graph - adding an edge to a tree would create a cycle. Therefore, if we want to connect all of the vertices of a graph with the fewest number of edges possible, we should aim to create a spanning tree. Finding a spanning tree of a graph is a fundamental problem in graph theory, with many real-world applications. For example, in a computer network, a spanning tree can be used to ensure that all computers are connected without creating loops in the network. In a transportation network, a spanning tree can be used to find the most efficient way to connect a set of locations. There are many algorithms for finding spanning trees of graphs, each with different properties and complexities. Some algorithms, such as Prim's algorithm and Kruskal's algorithm, are commonly used to find minimum spanning trees, which are spanning trees that have the smallest possible total edge weight. By finding a spanning tree of a graph, we can connect all of the vertices with the minimum number of edges necessary, making it a useful and important concept in graph theory.
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