Math 2.0

Combinatorics Definition

Spanning tree

Builds on SubgraphTree

Definition

Let GG be a graph. A spanning tree of GG is a subgraph TT of GG that contains every vertex of GG and is a tree.

Remarks

A graph has a spanning tree if and only if it is connected. The node every connected graph has a spanning tree gives one direction. A spanning tree of a finite connected graph with nn vertices has n−1n - 1 edges. The complete graph KnK_n has nn−2n^{n-2} spanning trees, by Cayley's formula. A minimum spanning tree has the least total weight.

Reading path

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

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Graph, definition, no evidence levelGraphdefinitionSubgraph, definition, no evidence levelSubgraphdefinitionWalk, definition, no evidence levelWalkdefinitionReachability, 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 levelTreedefinitionSpanning tree, definition, no evidence levelSpanning treedefinition
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. Subgraph definition
  3. Walk 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. Spanning tree definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Spanning_tree, Definitions
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DefinesSpanning tree
UsesSubgraph, Tree

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