From Wikipedia, the free encyclopedia

A spanning tree (blue heavy edges) of a grid graph.

In the mathematical field of graph theory, a spanning tree T of a connected, undirected graph G is a tree composed of all the vertices and some (or perhaps all) of the edges of G. Informally, a spanning tree of G is a selection of edges of G that form a tree spanning every vertex. That is, every vertex lies in the tree, but no cycles (or loops) are formed. On the other hand, every bridge of G must belong to T.

A spanning tree of a connected graph G can also be defined as a maximal set of edges of G that contains no cycle, or as a minimal set of edges that connect all vertices.

In certain fields of graph theory it is often useful to find a minimum spanning tree of a weighted graph. Other optimization problems on spanning trees have also been studied, including the maximum spanning tree, the minimum tree that spans at least k vertices, the minimum spanning tree with at most k edges per vertex, the spanning tree with the largest number of leaves (closely related to the smallest connected dominating set), the spanning tree with the fewest leaves (closely related to the Hamiltonian path problem), the minimum diameter spanning tree, and the minimum dilation spanning tree.

A spanning forest is a type of subgraph that generalises the concept of a spanning tree. However there are two definitions in common use. One is that a spanning forest is a subgraph that consists of a spanning tree in each connected component of a graph. (Equivalently, it is a maximal cycle-free subgraph.) This definition is common in computer science and optimisation. It is also the definition used when discussing minimum spanning forests, the generalization to disconnected graphs of minimum spanning trees. Another definition, common in graph theory, is that a spanning forest is any subgraph that is both a forest (contains no cycles) and spanning (includes every vertex).

The number t(G) of spanning trees of a connected graph is an important invariant. In some cases, it is easy to calculate t(G) directly. It is also widely used in data structures in different computer languages. For example, if G is itself a tree, then t(G)=1, while if G is the cycle graph C_n with n vertices, then t(G)=n. For any graph G, the number t(G) can be calculated using Kirchhoff's matrix-tree theorem (follow the link for an explicit example using the theorem).

Cayley's formula is a formula for the number of spanning trees in the complete graph K_n with n vertices. The formula states that t(K_n) = n^{n − 2}. Another way of stating Cayley's formula is that there are exactly n^{n − 2} labelled trees with n vertices. Cayley's formula can be proved using Kirchhoff's matrix-tree theorem or via the Prüfer code.

If G is the complete bipartite graph K_p,q, then t(G) = p^{q − 1}q^{p − 1}, while if G is the n-dimensional hypercube graph Q_n, then . These formulas are also consequences of the matrix-tree theorem.

If G is a multigraph and e is an edge of G, then the number t(G) of spanning trees of G satisfies the deletion-contraction recurrence t(G)=t(G-e)+t(G/e), where G-e is the multigraph obtained by deleting e and G/e is the contraction of G by e, where multiple edges arising from this contraction are not deleted.

A spanning tree chosen randomly from among all the spanning trees with equal probability is called a uniform spanning tree (UST). This model has been extensively researched in probability and mathematical physics.

• Eppstein, David (1999). "Spanning trees and spanners". Handbook of Computational Geometry: 425–461, Elsevier. 

• Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5.  A2.1: ND2, pg.206.

• Wu, Bang Ye; Chao, Kun-Mao (2004). Spanning Trees and Optimization Problems. CRC Press. ISBN 1584884363.