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  "body": "\n## Definition\n\nLet $\\mathcal{P}$ be a class of finite simple [[graphs|00C0BR7D7H]] that is closed under [[isomorphism|0F7QKGJS6B]].\nFor an integer $n \\ge 1$:\n\n1. The labelled count $\\ell_{\\mathcal{P}}(n)$ is the number of graphs in $\\mathcal{P}$ with vertex set $\\set{1, \\ldots, n}$.\n2. The unlabelled count $u_{\\mathcal{P}}(n)$ is the number of isomorphism classes of graphs in $\\mathcal{P}$ with $n$ vertices.\n\n*Graph enumeration* is the study of the functions $\\ell_{\\mathcal{P}}$ and $u_{\\mathcal{P}}$: exact formulas, generating functions and asymptotic behaviour as $n \\to \\infty$.\n\n## Remarks\n\nExample: for the class of all graphs, $\\ell(n) = 2^{\\binom{n}{2}}$, since each of the $\\binom{n}{2}$ pairs is an edge or not.\nFor $n = 3$ this gives $8$, while $u(3) = 4$, one class for each number of edges $0, 1, 2, 3$.\n\nNon-example: the class of graphs in which $1$ and $2$ are adjacent is not closed under isomorphism, so it has no unlabelled count.\n\nAlways $u_{\\mathcal{P}}(n) \\le \\ell_{\\mathcal{P}}(n) \\le n! \\, u_{\\mathcal{P}}(n)$, because an isomorphism class contains at most $n!$ graphs on $\\set{1, \\ldots, n}$.\nFor all graphs, $u(n)$ is asymptotic to $2^{\\binom{n}{2}} / n!$, because almost all graphs have no nontrivial automorphism.\n",
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