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  "body": "\n## Statement\n\nAs noted above, the multiplicities of the eigenvalues are given by\n\n$$M_{\\pm} = \\frac{1}{2}\\left[(v - 1) \\pm \\frac{2k + (v - 1)(\\lambda - \\mu)}{\\sqrt{(\\lambda - \\mu)^2 + 4(k - \\mu)}}\\right]$$\n\nwhich must be integers.\n\nIn 1960, Alan Hoffman and Robert Singleton examined those expressions when applied on Moore graphs, which are strongly regular graphs that have *λ* = 0 and *μ* = 1. Such graphs are free of triangles (otherwise *λ* would exceed zero) and quadrilaterals (otherwise *μ* would exceed 1), hence they have a girth (smallest cycle length) of 5. Substituting the values of *λ* and *μ* in the equation $(v - k - 1)\\mu = k(k - \\lambda - 1)$, it can be seen that $v = k^2 + 1$, and the eigenvalue multiplicities reduce to\n\n$$M_{\\pm} = \\frac{1}{2}\\left[k^2 \\pm \\frac{2k - k^2}{\\sqrt{4k - 3}}\\right]$$\n\nFor the multiplicities to be integers, the quantity $\\frac{2k - k^2}{\\sqrt{4k - 3}}$ must be rational, therefore either the numerator $2k - k^2$ is zero or the denominator $\\sqrt{4k - 3}$ is an integer.\n\nIf the numerator $2k - k^2$ is zero, the possibilities are:\n\n- *k* = 0 and *v* = 1 yields a trivial graph with one vertex and no edges, and\n- *k* = 2 and *v* = 5 yields the 5-vertex [[cycle graph|0QQ0F12962]] $C_5$, usually drawn as a regular pentagon.\n\nIf the denominator $\\sqrt{4k - 3}$ is an integer *t*, then $4k - 3$ is a perfect square $t^2$, so $k = \\frac{t^2 + 3}{4}$. Substituting:\n\n$$\\begin{aligned} M_{\\pm} \u0026= \\frac{1}{2} \\left[\\left(\\frac{t^2 + 3}{4}\\right)^2 \\pm \\frac{\\frac{t^2 + 3}{2} - \\left(\\frac{t^2 + 3}{4}\\right)^2}{t}\\right], \\end{aligned}$$\n\nso\n\n$$\\begin{aligned} 32 M_{\\pm} \u0026= (t^2 + 3)^2 \\pm \\frac{8(t^2 + 3) - (t^2 + 3)^2}{t} \\\\ \u0026= t^4 + 6t^2 + 9 \\pm \\frac{- t^4 + 2t^2 + 15}{t} \\\\ \u0026= t^4 + 6t^2 + 9 \\pm \\left(-t^3 + 2t + \\frac{15}{t}\\right). \\end{aligned}$$\n\nSince both sides are integers, $\\frac{15}{t}$ must be an integer, therefore *t* is a factor of 15, namely $t \\in \\{\\pm 1, \\pm 3, \\pm 5, \\pm 15\\}$, therefore $k \\in \\{1, 3, 7, 57\\}$. In turn:\n\n- *k* = 1 and *v* = 2 yields a trivial graph of two vertices joined by an edge,\n- *k* = 3 and *v* = 10 yields the [[Petersen graph|0NR95CTFCZ]],\n- *k* = 7 and *v* = 50 yields the Hoffman–Singleton graph, discovered by Hoffman and Singleton in the course of this analysis, and\n- *k* = 57 and *v* = 3250 famously predicts a graph that has neither been discovered since 1960, nor has its existence been disproven.\n\nThe Hoffman-Singleton theorem states that there are no girth-5 Moore graphs except the ones listed above.\n",
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