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Learn about cycles in simple graphs, their properties, and why they cannot be of length 0, 1, or 2. Explore the concept of trees, unique paths in trees, and a proof of the uniqueness of paths between vertices in a tree.
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Cycles in Simple Graphs A cycle in a simple graph is a sequence of vertices v0, …, vn for some n>0, where v0, ….vn-1 are distinct, v0=vn, and {vi,vi+1} is an edge for 0≤i<n. The length of the cycle is n. Fact: There can be no cycles of length 0, 1, or 2. (Why?)
Trees A tree is a connected acyclic graph. (These are a kind of undirected graph, so the definition is different from the directed trees we talked about earlier.)
In a tree there is a unique path between any pair of vertices Proof. If there were two vertices with no path between them, the graph would not be connected. Suppose there were two vertices with two distinct paths between them. By the WOP there would be a shortest possible path in the graph for which there is a distinct path between the same pair of endpoints. The second path must have no vertices in common with the first except the endpoints (otherwise first path would not be the shortest). Then following one path forward and the other back creates a cycle, contradiction. So there cannot be more or less than one path between any pair of vertices. QED.
