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  "body": "\n## Summary\n\nSpectral graph theory studies a graph through the [[eigenvalues|0BQAA2BK3Q]] and eigenvectors of matrices attached to it, mainly the [[adjacency matrix|08B9CX109G]] and the [[Laplacian matrix|TN6M6DVZRP]].\nThese eigenvalues are invariants of the graph up to isomorphism, and many of them have combinatorial meaning.\nFor example, a finite graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric about $0$.\nIts number of connected components equals the multiplicity of the Laplacian eigenvalue $0$.\nThe [[Cheeger constant|Y7S6W12N64]] is controlled by the second Laplacian eigenvalue, which links the subject to [[expander graphs|DXJXVFV79H]].\n\n## Explanations\n\n### L1 human\n\nA graph can be turned into a matrix, and a matrix has eigenvalues.\nSpectral graph theory asks what these numbers tell us about the graph.\nThey detect, for example, whether the graph is connected, whether it is bipartite, and how hard it is to cut it into two large pieces.\nThey do not tell everything: the star with four leaves and the $4$-cycle together with an isolated vertex have the same adjacency eigenvalues $2, 0, 0, 0, -2$, but they are different graphs.\n",
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