The Laplacian matrix of a graph $G$ is $L(G)=D(G)-A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees. According to the Matrix-Tree Theorem, the number of spanning trees in $G$ is equal to any cofactor of an entry of $L(G)$. A rooted forest is a union of disjoint rooted trees. We consider the matrix $W(G)=I+L(G)$ and prove that the $(i,j)$-cofactor of $W(G)$ is equal to the number of spanning rooted forests of $G$, in which the vertices $i$ and $j$ belong to the same tree rooted at $i$. The determinant of $W(G)$ equals the total number of spanning rooted forests, therefore the $(i,j)$-entry of the matrix $W^{-1}(G)$ can be considered as a measure of relative ''forest-accessibility'' of vertex $i$ from $j$ (or $j$ from $i$). These results follow from somewhat more general theorems we prove, which concern weighted multigraphs. The analogous theorems for (multi)digraphs are also established. These results provide a graph-theoretic interpretation for the adjugate to the Laplacian characteristic matrix.

翻译：图 $G$ 的拉普拉斯矩阵为 $L(G)=D(G)-A(G)$，其中 $A(G)$ 是邻接矩阵，$D(G)$ 是顶点度的对角矩阵。根据矩阵树定理，$G$ 中生成树的数量等于 $L(G)$ 任意元素的余子式。有根森林是若干不相交有根树的并集。我们考虑矩阵 $W(G)=I+L(G)$，并证明 $W(G)$ 的 $(i,j)$-余子式等于 $G$ 中满足顶点 $i$ 和 $j$ 属于同一棵以 $i$ 为根树的有根生成森林的数量。$W(G)$ 的行列式等于有根生成森林的总数，因此矩阵 $W^{-1}(G)$ 的第 $(i,j)$ 项可视为顶点 $j$ 相对于 $i$（或 $i$ 相对于 $j$）的“森林可达性”度量。这些结果源于我们证明的若干更一般定理，这些定理涉及加权多重图。针对（多重）有向图的类似定理也被建立。这些结果为拉普拉斯特征矩阵的伴随矩阵提供了图论解释。