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  "body": "\n## Summary\n\nThe Laplacian matrix of a finite graph is the degree matrix minus the [[adjacency matrix|08B9CX109G]].\nIt is symmetric and positive semidefinite, and its eigenvalues carry information about connectivity: the number of connected components is the multiplicity of the eigenvalue $0$.\nThrough [[Kirchhoff's matrix tree theorem|S1PG1N2VEX]] it counts spanning trees, and its second smallest eigenvalue controls how well the graph is connected.\nIt is the discrete analogue of the Laplace operator of analysis.\n\n## Explanations\n\n### L1 human\n\nPut a number on each vertex of a graph.\nThe Laplacian compares each number with the numbers at the neighbouring vertices: it adds up the differences.\nIf the numbers are all equal on a connected graph, the Laplacian gives zero everywhere, just as a constant function has zero second derivative.\nFor the path with three vertices, the Laplacian is a $3 \\times 3$ matrix with the degrees $1, 2, 1$ on the diagonal and $-1$ for each edge.\n",
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