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  "body": "\n## Definition\n\nLet $d \\ge 3$ be an integer and let $\\varepsilon \u003e 0$.\nFor a finite graph $G$ with at least $2$ vertices, let $h(G)$ be its [[Cheeger constant|Y7S6W12N64]].\nIt is the minimum of $\\abs{\\partial S} / \\abs{S}$ over the sets $S$ of vertices with $1 \\le \\abs{S} \\le \\abs{V(G)}/2$.\nHere $\\partial S$ is the set of edges with exactly one end in $S$.\nA finite graph $G$ is an $\\varepsilon$-expander if $h(G) \\ge \\varepsilon$.\nA family of expander graphs of degree $d$ is a sequence $(G_n)_{n \\ge 1}$ of finite $d$-[[regular graphs|0KA4S8R39M]] such that:\n\n1. The number of vertices of $G_n$ tends to infinity as $n \\to \\infty$.\n2. There is $\\varepsilon \u003e 0$ such that $h(G_n) \\ge \\varepsilon$ for every $n$.\n\n## Remarks\n\nThe complete graph $K_4$ is a $2$-expander: a set $S$ of $1$ or $2$ vertices has $\\abs{S}(4 - \\abs{S})$ edges leaving it, so the ratio is $4 - \\abs{S} \\ge 2$.\nThe cycles $C_n$ do not form a family of expanders: they are $2$-regular, and an arc $S$ of $\\floor{n/2}$ consecutive vertices has $\\abs{\\partial S} = 2$, so $h(C_n) \\le 2/\\floor{n/2} \\to 0$, and clause 2 fails.\nFamilies of expanders of every degree $d \\ge 3$ exist.\nFor example, for each $d \\ge 3$ there is $\\varepsilon \u003e 0$ such that a random $d$-regular graph on $n$ vertices is an $\\varepsilon$-expander with probability tending to $1$ as $n \\to \\infty$.\nAn equivalent spectral form uses the second largest eigenvalue $\\lambda_2$ of the adjacency matrix.\nThe discrete Cheeger inequalities say $\\frac{d - \\lambda_2}{2} \\le h(G) \\le \\sqrt{2d(d - \\lambda_2)}$ for a $d$-regular graph $G$.\nSo clause 2 holds if and only if the spectral gap $d - \\lambda_2(G_n)$ is bounded away from $0$.\n",
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