The matrix tree theorem, also called Kirchhoff's matrix-tree theorem (Buekenhout and Parker 1998), states that the number of nonidentical spanning
trees of a graph 
is equal to any cofactor of its
Laplacian matrix (Skiena 1990, p. 235).
Matrix Tree Theorem
See also
Laplacian Matrix, Spanning TreeExplore with Wolfram|Alpha
References
Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension
." Disc. Math. 186, 69-94, 1998.Chaiken,
S. "A Combinatorial Proof of the All-Minors Matrix Tree Theorem." SIAM
J. Alg. Disc. Methods 3, 319-329, 1982.Cvetković, D. M.;
Doob, M.; and Sachs, H. Spectra
of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, p. 38,
1998.Kirchhoff, G. "Über die Auflösung der Gleichungen,
auf welche man bei der untersuchung der linearen verteilung galvanischer Ströme
geführt wird." Ann. Phys. Chem. 72, 497-508, 1847.Skiena,
S. Implementing
Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading,
MA: Addison-Wesley, p. 235, 1990.Referenced on Wolfram|Alpha
Matrix Tree TheoremCite this as:
Weisstein, Eric W. "Matrix Tree Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MatrixTreeTheorem.html