Math 2.0

Combinatorics Theorem

Unique paths in trees

Builds on PathTree

Statement

Let GG be a graph. Then GG is a tree if and only if GG has at least one vertex and these conditions hold. For any two vertices uu and vv of GG, there is exactly one path from uu to vv.

Remarks

Existence of a path between any two vertices is connectedness. Uniqueness of the path is equivalent to acyclicity: two different paths between the same vertices contain a cycle. In a tree, the unique path from uu to vv is a shortest walk, so its length is the distance d(u,v)d(u, v).

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionWalk, definition, no evidence levelWalkdefinitionPath, definition, no evidence levelPathdefinitionReachability, definition, no evidence levelReachabilitydefinitionTrail, definition, no evidence levelTraildefinitionConnected graph, definition, no evidence levelConnected graphdefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionAcyclic graph, definition, no evidence levelAcyclic graphdefinitionTree, definition, no evidence levelTreedefinitionUnique paths in trees, statement, no evidence levelUnique paths in treesstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Graph definition
  2. Walk definition
  3. Path definition
  4. Reachability definition
  5. Trail definition
  6. Connected graph definition
  7. Circuit definition
  8. Cycle definition
  9. Acyclic graph definition
  10. Tree definition
  11. Unique paths in trees statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Tree_(graph_theory), Properties
About this page
Tag096B7Q5742
URIm2://096B7Q5742
Kindstatement, theorem
WritingW0 stub
MSC05C05
StatusThis copy of the site has no build data. Run m2 status 096B7Q5742.
UsesPath, Tree

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.