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  "body": "\n## Statement\n\nLet $G$ be a finite [[multigraph|0C62PZXX02]] without loops, with vertices $v_1, \\dots, v_n$ and $n \\ge 2$.\nLet $D$ be the diagonal matrix of the vertex degrees.\nLet $A$ be the matrix with $A_{ii} = 0$ and $A_{ij}$ the number of edges between $v_i$ and $v_j$ for $i \\ne j$.\nLet $L = D - A$ be the [[Laplacian matrix|TN6M6DVZRP]] of $G$.\nFor each $i$, let $L_i$ be the matrix obtained from $L$ by deleting row $i$ and column $i$.\nThen the number of [[spanning trees|0S4ZEZM7KE]] of $G$ equals $\\det L_i$, for every $i \\in \\set{1, \\dots, n}$.\n\n## Remarks\n\nOrient each edge and let $B$ be the $n \\times m$ incidence matrix, with entries $+1$ and $-1$ at the head and tail of each edge; then $L = B B^T$.\nDelete row $i$ of $B$ to get $B_i$, so that $L_i = B_i B_i^T$.\nBy the [[Cauchy–Binet formula|08WE2W7DYS]], $\\det L_i$ is the sum of $(\\det N)^2$ over the $(n-1) \\times (n-1)$ submatrices $N$ of $B_i$.\nSuch a minor is $\\pm 1$ if its $n - 1$ edges form a spanning tree and $0$ otherwise, which gives the count.\n\nExample: for the triangle $K_3$, every vertex has degree $2$ and\n$$L_3 = \\begin{pmatrix} 2 \u0026 -1 \\\\ -1 \u0026 2 \\end{pmatrix}, \\qquad \\det L_3 = 3,$$\nwhich matches the $3$ spanning trees obtained by removing one edge.\nFor the complete graph $K_n$ the theorem gives $n^{n-2}$, which is [[Cayley's formula|0M2SSGSYPE]].\nIf $G$ is not connected, both sides are $0$.\n\n## Explanations\n\n### L1 human\n\nCounting spanning trees by hand is hard even for small graphs.\nThe theorem gives a recipe.\nWrite the degree of each vertex on the diagonal and $-1$ for each edge off the diagonal.\nCross out one row and the matching column, and take the [[determinant|00KDZRW4C0]].\nThe number you get is the number of spanning trees, whichever row you crossed out.\n\n### L2 human\n\nAn equivalent form uses the eigenvalues $0 = \\mu_1 \\le \\mu_2 \\le \\dots \\le \\mu_n$ of $L$: the number of spanning trees is $\\frac{1}{n} \\mu_2 \\cdots \\mu_n$.\nThis links spectral graph theory to enumeration.\nThe theorem goes back to Kirchhoff's study of electrical networks.\nIt has a weighted version, where each edge carries a variable and the determinant sums the weights of the spanning trees, and a version for directed graphs that counts arborescences.\n",
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