dETL

detL (read as determinant of the Laplacian) is the determinant of the Laplacian matrix L of a finite graph G. For an undirected graph with vertex set V of size n, L = D − A, where D is the diagonal matrix of degrees and A is the adjacency matrix.

One defining property is that L has at least one zero eigenvalue; the multiplicity of zero as an eigenvalue equals the number of connected components c of G. Consequently, for any nonempty graph, det L = 0. In particular, a connected graph yields exactly one zero eigenvalue, hence det L = 0.

The determinant of L's cofactors relates to spanning trees via Kirchhoff's Matrix-Tree Theorem: for any i, det(L with i-th row and i-th column removed) equals the number of spanning trees of G.

Applications and notes: det L is generally not used to extract structural counts; cofactors are used instead. For directed graphs and more general Laplacians, determinants can behave differently and must be considered with care. In continuous settings, the Laplacian operator has a spectrum and its determinant is defined in special ways (zeta-regularized determinant).

See also: Laplacian matrix, Matrix-Tree Theorem, Kirchhoff's theorem, spanning trees.