Theorem. The determinant as a rooted-forest sum [fdip-005V]
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  • November 1, 2026

A rooted spanning forest on \(V\) is an acyclic set \(F\subseteq E\) together with exactly one chosen vertex in each component, including isolated vertices. Write \(R\) for these roots and \(r=|R|\). Loops cannot occur, and distinct parallel labels count as distinct choices. With the data of Definition [fdip-005R], \[\det (L_w+m^2I) =\sum _{(F,R)}\left (\prod _{e\in F}w_e\right )m^{2|R|}.\] Normalize these nonnegative terms to obtain a probability law on rooted spanning forests. It satisfies \[\begin {aligned} \mathbb P(e\in F)&=w_ed_e^TGd_e,\\ \mathbb E[|R|]&=m^2\operatorname {tr}G,\\ \mathbb E[|F|]&=n-m^2\operatorname {tr}G. \end {aligned}\] If the positive-weight support is connected, with weighted spanning-tree sum \(\tau _w\), then \[\det (L_w+m^2I)=n\tau _wm^2+O(m^4) \qquad (m\downarrow 0).\] Thus the spanning-tree coefficient appears after division by \(m^2\); the unscaled determinant tends to zero.

Proof. 

  • November 1, 2026

Form the \(n\times (|E|+n)\) matrix with columns \(\sqrt {w_e}d_e\) for edges and \(m\delta _v\) for vertices. Call it \(A\); then \(AA^T=L_w+m^2I\). Cauchy-Binet expresses its determinant as the sum of the squared \(n\times n\) minors. A chosen minor consists of edge columns \(F\) and vertex columns \(R\), with \(|F|+|R|=n\). A cycle in \(F\) makes its incidence columns dependent (this includes a loop and the two-edge cycle of parallel edges), so only forests contribute.

Suppose \(F\) is a forest with components \(C_i\). Their incidence columns span the coordinate-sum-zero subspace in each \(\mathbb R^{C_i}\), of dimension \(|C_i|-1\). This follows by successively removing leaves. A component with no chosen vertex column therefore prevents full rank. There are as many chosen vertex columns as components, because \(|R|=n-|F|\); hence full rank requires exactly one in each component. Conversely, for a tree with one root, the matrix of its unweighted incidence columns and that root's coordinate vector has determinant \(\pm 1\): remove a nonroot leaf and expand along its row, then finish with the root's \(1\times 1\) matrix. Applying this to all components proves that precisely the stated forests contribute, with squared minor \((\prod _{e\in F}w_e)m^{2|R|}\). Zero-weight edge columns vanish, consistent with deletion.

Put \(t=m^2\) and call this polynomial \(D(w,t)\). Termwise differentiation gives \(w_e\partial _eD=\sum _{F\ni e,R}(\prod _{f\in F}w_f)t^{|R|}\) and \(t\partial _tD=\sum _{F,R}|R|(\prod _{f\in F}w_f)t^{|R|}\). Divide by \(D>0\) and use the determinant derivative from Theorem [fdip-005S], first in direction \(d_ed_e^T\), then in direction \(I\). The edge identity holds also at \(w_e=0\), when both sides vanish. Finally every forest has \(|F|+|R|=n\). For connected positive support, the coefficient of \(t\) comes from spanning trees; each has \(n\) choices of root. All other terms have at least two roots and hence order \(t^2\).

Equivalently, add a new boundary vertex \(*\) with an edge of weight \(m^2\) to each old vertex. Attach \(*\) to the chosen root of each forest component; this is a bijection to spanning trees of the augmented graph. Its Dirichlet Laplacian at \(*\) is \(L_w+m^2I\). This is the trivial-connection case of Kenyon's boundary theorem: cycle factors vanish, leaving trees rooted at the boundary [kenyon2011spanning, Theorem 7, Section 8, p. 15]. The conductance extension is given in [kenyon2011spanning, Section 7.1, Theorem 5, p. 13]. The proof above includes the needed weights and zero-weight faces. Chaiken's all-minors matrix-tree theorem enumerates directed forests by the minors of a Laplacian whose columns sum to zero [chaiken1982minors, pp. 319-329]. The expansion proved above is the undirected weighted case, obtained from Cauchy-Binet. It does not reproduce Chaiken's directed sign rule.

TauCeti's Matrix.det_mul_transpose_eq_sum_plucker_sq gives the Cauchy-Binet identity for a matrix over a commutative ring with rows indexed by \(\mathrm {Fin}(n)\) and a finite ordered column set. Reindexing the present real matrix supplies those hypotheses. Its Plücker coordinates are the maximal minors. This supports the sum-of-squares step; the classification of nonzero forest minors and the probability consequences are proved here.