Kirchhoff's theorem

Kirchhoff's theorem, also called the matrix-tree theorem, states that a cofactor of a graph Laplacian gives the number of spanning trees of a graph.

Graph Laplacian: A matrix encoding a graph’s vertex degrees and adjacency, usually defined as the degree matrix minus the adjacency matrix. Its principal cofactors contain the spanning-tree count.
Graph (discrete mathematics): A structure consisting of vertices connected by edges. The theorem applies to finite graphs and their spanning trees.
Counting Spanning Trees: The problem of determining how many spanning trees a graph contains. The theorem gives a general determinant-based solution to this counting problem.
Gustav Kirchhoff: A German physicist whose work established foundational laws of electrical circuits and spectral analysis. Kirchhoff introduced the graph-counting result while studying electrical networks.
Laplacian matrix-tree theorem: The matrix-tree theorem expressed using a graph’s Laplacian matrix. This is the standard Laplacian formulation of Kirchhoff’s result.
Tree (graph theory): A connected graph containing no cycles. A spanning tree is a tree that includes every vertex of the original graph.
Kirchhoff's Circuit Laws: Rules stating that electric current and voltage obey conservation laws in electrical circuits. Despite the shared name, these circuit laws are distinct from Kirchhoff’s graph-counting theorem.
Arthur Cayley: A nineteenth-century British mathematician who made major contributions to algebra and graph enumeration. Cayley’s formula is a celebrated special case of the tree-counting framework.
Cofactor (linear algebra): A signed determinant of a matrix minor, used in determinant expansions and matrix inverses. Deleting one Laplacian row and column produces the determinant used by the theorem.
Adjacency Matrix: A matrix whose entries record which pairs of graph vertices are connected by edges. It helps define the Laplacian whose cofactor counts trees.

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Laplacian matrixRelated: The Laplacian determinant minors count spanning trees.