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  "body": "\n## Definition\n\nLet $G$ be a finite simple [[graph|0JD5C82QMB]] with vertices $v_1, \\dots, v_n$.\nLet $D$ be the diagonal matrix with $D_{ii} = \\deg(v_i)$, the [[degree|0T0F6CCS9W]] of $v_i$, and let $A$ be the [[adjacency matrix|0NZ8MDBN6Q]] of $G$.\nThe Laplacian matrix of $G$ is the $n \\times n$ matrix $L = D - A$.\nSo $L_{ii} = \\deg(v_i)$, $L_{ij} = -1$ if $i \\ne j$ and $v_i v_j$ is an edge, and $L_{ij} = 0$ otherwise.\n\n## Remarks\n\nFor every real vector $x$, $x^{T} L x = \\sum (x_i - x_j)^2$, the sum running over the edges $v_i v_j$.\nHence $L$ is [[symmetric|0N28QN4CYA]] and [[positive semidefinite|0880F9EQCP]], the all-ones vector lies in its kernel, and the multiplicity of the eigenvalue $0$ equals the number of connected components of $G$.\nExample: for the path $v_1 v_2 v_3$,\n$$L = \\begin{pmatrix} 1 \u0026 -1 \u0026 0 \\\\ -1 \u0026 2 \u0026 -1 \\\\ 0 \u0026 -1 \u0026 1 \\end{pmatrix},$$\nwhose eigenvalues are $0$, $1$ and $3$, with eigenvectors $(1,1,1)$, $(1,0,-1)$ and $(1,-2,1)$.\nNon-example: the matrix $D + A$ has the right diagonal but the wrong sign off the diagonal.\nIt is the signless Laplacian, and it has the same spectrum as $L$ when $G$ is bipartite, but not in general.\nThe normalized Laplacian $D^{-1/2} L D^{-1/2}$, defined when no vertex is isolated, is also common.\nMathlib defines the Laplacian matrix as `SimpleGraph.lapMatrix`.\n",
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