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  "body": "\n## Definition\n\nLet $v, k, \\lambda, \\mu$ be non-negative integers.\nA finite [[graph|0JD5C82QMB]] $G$ is strongly regular with parameters $(v, k, \\lambda, \\mu)$, written $\\mathrm{srg}(v, k, \\lambda, \\mu)$, if:\n\n1. $G$ has $v$ vertices and is $k$-[[regular|0KA4S8R39M]].\n2. Every two adjacent vertices have exactly $\\lambda$ common [[neighbours|01T3TMGTTE]].\n3. Every two distinct non-adjacent vertices have exactly $\\mu$ common neighbours.\n\n## Remarks\n\nMany authors also require that $G$ is neither complete nor edgeless, so that both conditions 2 and 3 apply to some pair.\nCounting the edges between the neighbours of a vertex $x$ and the vertices not adjacent to $x$ gives $k(k - \\lambda - 1) = (v - k - 1)\\mu$.\nExample: the [[Petersen graph|0NR95CTFCZ]] is an $\\mathrm{srg}(10, 3, 0, 1)$: it is $3$-regular, it has no triangles, so $\\lambda = 0$, and two non-adjacent vertices have exactly one common neighbour.\nCheck of the identity: $3 \\cdot (3 - 0 - 1) = 6 = (10 - 3 - 1) \\cdot 1$.\nNon-example: the [[cycle graph|0QQ0F12962]] $C_6$ is $2$-regular, but two vertices at distance $2$ have one common neighbour while two opposite vertices have none, so clause 3 fails.\nThe [[complement|0RNMP6VCR2]] of an $\\mathrm{srg}(v, k, \\lambda, \\mu)$ is an $\\mathrm{srg}(v, v - k - 1, v - 2 - 2k + \\mu, v - 2k + \\lambda)$.\n",
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