Matrix tree theorem

Matrix tree theorem provides a connection between the number of spanning trees of a graph and graph's Laplacian matrix.

A degree matrix $$D$$ of an undirected graph is a square matrix, which contains the sequence of degrees of graph's vertices on its diagonal, and zeroes elsewhere.

A Laplacian of a graph is defined as: $$L = D - A$$, where $$A$$ is the adjacency matrix, and $$D$$ is the degree matrix.

Matrix tree theorem states, the determinant of a matrix, obtained by deleting the $$k$$-th row and column from the graph's Laplacian is independent from the choice of $$k$$, and is equal to the number of spanning trees of the graph.