Math 2.0

Combinatorics Definition

Minimum spanning tree

Builds on Spanning tree

Definition

Let GG be a finite connected graph with edge set EE, and let w:E→ℝw : E \to \mathbb{R} be a weight function. The weight of a subgraph is the sum of the weights of its edges. A minimum spanning tree of (G,w)(G, w) is a spanning tree of GG whose weight is the smallest among all spanning trees of GG.

Remarks

A minimum spanning tree exists, because a finite graph has finitely many spanning trees. It is unique when all weights are distinct. Kruskal's algorithm and Prim's algorithm find a minimum spanning tree with greedy choices.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
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 treedefinitionMinimum spanning tree, definition, no evidence levelMinimum spanning 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
  12. Minimum spanning tree definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Minimum_spanning_tree, lead section
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