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 sho...
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