{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Nonexistence of a distance-regular graph with intersection array $\\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\\}$\n", "\n", "We will show that a distance-regular graph with intersection array $\\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\\}$ does not exist. The existence of such a graph would give a counterexample to a conjecture by MacLean and Terwilliger, see [Bipartite distance-regular graphs: The $Q$-polynomial property and pseudo primitive idempotents](http://dx.doi.org/10.1016/j.disc.2014.04.025) by M. Lang." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%display latex\n", "import drg" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Such a graph would be bipartite with $3500$ vertices." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$ " ], "text/plain": [ "True" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}3500$ " ], "text/plain": [ "3500" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "p = drg.DRGParameters([55, 54, 50, 35, 10], [1, 5, 20, 45, 55])\n", "show(p.is_bipartite())\n", "show(p.order())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that all intersection numbers are nonnegative integers." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { 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ZwkOH8+yat/Hx8iDI35ecA4dZ+s7/ERX+vw9JD0+ewL8y36JX9274dfUi/pW1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\n4jSzm42bmxuFJWUqazFbUWk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ERMKxbIiISDiWDRERCceyISIi4Vg2REQkHMuGiIiEY9kQEZFwLBsiIhKOZUNE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\njmVDRETCsWyIiEi4/wcewpIzN9FsHgA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V43288cii+Zg+KQCFJ8sAAD82N0N/4BO8/ORjmB08DUGTJ2L3plU4cvwkCk+U9VnPnjMe\niQQXYWlJCYImTRC88OPpO7Bw1s8wJ2SGxe+//ZdP4T0/BtOWrsCGnXq0tLYJXjv4pokoKS4WvP1I\ncbWM+8uzP7UNjXhs63a8tXktXAT8MpNixlcTdsvN2H/4S5y/WA8A+NRwDGe+PYe7btUAAAz/LEdH\nZyfmzvzfz2ey7ziMV3vhi69PWVzTXjMeiRyEbtjQ0ACfqZMEbfvOXz9DyelyFOVYPu734F2R8PVR\n40ZPFY6XV2Ldjmyc+fYc3nthk6D11So35JeeQLGd3Ylqa2pwp4WM+8tTiF89/zLiFy9E0OSJqP6+\ntt/tpZZxfzJWJ+CxrdsxbtFDcFAooJDLsWv9k5g1PRAAUFPfAKWDA8a49txbV6vcUVPfYHFNtcoN\nh06etv4fQTYnuAiNRiOclI79bvfdhYt4+tU38Mn2F+DoYHn5X98zv/u/AwP84OOhwp1PrEfl+Rr4\n3+jT7204KR1RU1sDjUYjdPwRwdFB0SdjIXn257W8D/Gv//wH65YtAQCY0f8TBaSUsRCv5f0JX50o\nw/70ZIxXe+NQ6T+wMm0HbvT0uOpeutlsvuLxWCelI9rajFbPQrYn+JGlVCrRZmzvdzvDP8txsekH\nhGgTu4+1dJpMOFT6NXb8YT9aD+3vc8e4NXAyzGYzyr87L6gI24zt8FH74MN9+4SOPyIsio7uk/FA\n8uztM8NxfPn1P+F8e3SPy2dqn8DSuyKxe9PqPteRUsb9aW0zYtMbufjwpc2ICgsBAEyd4IeS099g\n29vvY07IDPh4qGDs6MCPzc099govNDZBfYV3s2kztsNJ4DF3EpfgIlSpVKipb+x3u3kzg3B8784e\nl2mf24ab/cZjXewSiw/akrJvIJPJcIOHStAstQ1N8FarERwcLGz4EULt49Mn44Hk2dtrq+Px/IqH\nu////MV6RD29Ce8+v+GKZ6illHF/2js60N7Rgd5RK+RymEwmAIDmpxPhoFAg/2gp7rtjFgDg9Nnv\ncLb2IsKm3mxx3dqGJri7C7vPk7gEF+GMoCAUHf6s3+1cXZwxxd+3z2Wq60fjZr/xqDj3Pd7+5FMs\nCJ8JjzFjcKy8Aqu378LsoFswdYKfoFmKT5cjdPYcoaOPGJYy7i9PoOtESE19I858dw5msxnHyysx\n+rrrMF7tBfcxozHW26vn9Z2dYTabEfATH9zoZfklXlLKGOh6+kz5d+e797orztXg2JkKqMaMxji1\nF2YH3YK1mb+Fs1IJ3xu88Vnxcbz1cT5eeeoxAMAYV1f8KvourH5tF9zHjMLo667Dky/vxKxpUxAa\naPmXjb1mPBIJLkKNRoPc3By0thkFP4Xmksv3WpSODsg/WoLX8v6E5pZWjFN74hdzbsPG5b8UtFZL\naxtOVFQjfpV9HbsChGfcey/w9T9+hJTs30Emk0Emk+GOhLUAgN0bVyF2wTxBa1xOihkXnTqNOY+v\n685wTcabAIDY+fOwe9MqvPPcBqzfuRvLklPR8OO/4OujxgvxWjx674LuNV558lEo5HI8sOE3aGtv\nR9StGmSuedziHPac8Ugk+CV2R48eRWhoKD7LSsPtQbeIPdcV/b34OCJXrkVhYSFmzpw5bHOIgRmL\njxmTJYKfR6jRaODv54fsfQfFnKdf2fsPIsDf3+7OZgLMeCgwY7JEcBHK5XLEJyTg3fxDqGv6QcyZ\nruhiYxPy8g8jPiHBLl+ozozFx4zJEqt+ClqtFjK5HFnvHxBrnqvK+uAAZHI5tFrx31Z9uDBj8TFj\n6s2qIvT09ERiYiJe3PMuTlWdFWsmi05VncXWPXlITEy06zezZMbiY8bU24j6zJIf2k0oKS21+896\nYMbiY8Z0OasPULi4uECfk4PCE2VI2rVHjJn6SNqVi6JTp6HPyZHEnYcZi48Z0+UG9LnG48aNg6ur\nKzanvwJnJyUi/vvCczG89FYekrP3IjU1FTExMaLdzrWGGYuPGdMlA/6A9/DwcJhMJui2bUer0YjI\n4Gk2PQPW0dGJja/nIDl7L5KSkrBx40abrT1SMGPxMWMCBlGEABAZGQlXV1ckv7wdfyk0YNa0KfBy\nu37QQ52qOotFa5PxXv4hpKWlSfrOw4zFx4zJ6pMllhQUFEC7fDmqq6vxbOwSJCxeCC93N6vXudjY\nhKwPDmDrnjz4+voiJzcXYWFhgx3PLjBj8TFj6bJJEQJdZ+F0Oh0yMjJgNpmwZO5tiIuOQuiUyXBx\ndrry9VrbUHiyDNn7DyIv/zBkcjkSExORkpLCA8q9MGPxMWNpslkRXlJfXw+9Xo+dWVmoqKyEQqFA\nYIAvgm+aCLXKrevNKI3tqG1oQvHpcpyoqEZnZycC/P0Rn5AArVbL51f1gxmLjxlLi82L8BKTyQSD\nwdD9VVJcjAu1taiprYGP2gfeajWCgoOh0Wi6v/hyI+swY/ExY2kQrQgtKS4uhkajgcFgsLs3/LxW\nMGPxMWP7w19dRCR5LEIikjwWIRFJHouQiCSPRUhEksciJCLJYxESkeSxCIlI8liERCR5LEIikjwW\nIRFJHouQiCSPRUhEksciJCLJYxESkeSxCIlI8liERCR5LEIikjwWIRFJHouQiCSPRUhEksciJCLJ\nYxESkeSxCIlI8liERCR5LEIikjwWIRFJHouQiCSPRUhEksciJCLJYxESkeSxCIlI8liERCR5LEIi\nkjwWIRFJHouQiCSPRUhEksciJCLJYxESkeSxCIlI8liERCR5LEIikjwWIRFJHouQiCSPRUhEksci\nJCLJYxESkeSxCIlI8liERCR5LEIikjwWIRFJnsxsNpvFWNhkMsFgMKCoqAgGgwGlJSWoralB7YVa\nqL3VUPv4YEZQEDQaDUJCQqDRaCCXs5etwYzFx4zFZynjhoYGGI1GKJVKqFQq0TO2eRHW1dVBr9dj\nZ1YWKquq4OCgQGCAH4ImTYCPhzuclI5oM7ajpr4RJWe+wYmKKnR0dCLA3x8r4uOh1Wrh6elpy5Hs\nDjMWHzMW37WUsc2KsKWlBTqdDhkZGTCbTIiZezvi7olC6JTJcHZSXvF6rW1GFJ4sQ/a+g3g3/xBk\ncjkSExORkpICFxcXW4xmN5ix+Jix+K7FjG1ShAUFBdAuX47q6mqsj41B/OK74eXuZvU6FxubkPXB\nAWzdkwdfX1/oc3IQHh4+2PHsAjMWHzMW37Wa8aD/0E5LS0NERATcnRQozs2ELm7pgP5hAODl7obN\ncQ+hODcTbkoFIiIikJ6ePtgRRzxmLD5mLL5rOWPFli1btgz0yklJSUhKSsKzsTHYu3ktvFXuAx7k\ncl5u12P5gjthbG/H5vRXYTKZEBkZaZO1RxpmLD5mLL5rPeMBF2FaWhqSkpKwdWUcdHFLbX4WRy6X\nY97MIDg7KaHbth2jRo2S3J8XzFh8zFh8IyHjAR0jLCgoQEREBNYtW4IX4rXWXt1q67N2I3Xvezhy\n5AjCwsJEv71rATMWHzMW30jJ2OoibGlpwYzp0+HupMDnr2+DQqGwelhrdXR0ImLFajQZO1F67Jjd\nn4VjxuJjxuIbSRlbvY+q0+lQXV0N/cbVQ/IPAwAHBwX0m1ahuroaOp1uSG5zODFj8TFj8Y2kjK3a\nI6yrq8PYsWOxftkS6OKWDmjQwUjO3outb72Hc+fOwcPDY8hvfygwY/ExY/GNtIyt2iPU6/WA2YyE\n+xcOeMDBSFi8EGaTqWsOO8WMxceMxTfSMha8R2gymTBxwgRE/HQCcjc/M6ghByM2ORUFZZU4U15u\nd6/pZMbiY8biG4kZC/4JGAwGVFZVIe6eqB6XHy79Goue2Yyx0UuhCJ+PfYe/6PH95pZWPJ6+A+MX\nPQTXOxZh6oOP4o0/ftRjmzajESvTMuEVtQRj5t6HBzY8jwsNTRbniIuOQkVlJQwGg9DRR4xLGc+c\nctMVM+3o6MS6HdmY/tAKjJ5zL8ZGL8XylHR8X1ffZ72PjnyFsF8/Bdc7FsHj57/A/c+m9Pj+t7UX\nsXB1EkZFLsINd/8SazN/C5PJJImMe9+PAxY/DEX4/D5fiduy+qyx4OlNFu/rV8rTEilm3NuLue/g\n1l89gevnLobPgv+HxetScPrsdz22saYberMmY8FFWFRUBAcHBUKnTO5xeXNLK6ZPmoDMNSshk8n6\nXO/p7W/gr4XF+F3yszj1zpt4MuY+JG7LwoHPv+re5qlX38CfCwrxhxc24e9ZaTh/sR6/2PCcxTlC\np0yGQqGwyzvQpYzHq72vmOl/2lpx7Mw30MU9hOLcHfhgqw5lZ7/DvWuTe2z3/qef4+GUdPwq+i4c\n37sTR3a9gl/+/H9PNDWZTLh7VRI6OjvxxZuvIidpNXL//Ffo3twjiYx734+P6jPw/YHfd399sv0F\nyGQyLJl7e4/tXvn9B1DI5X1+LlfL0xIpZtzb58dO4PEH7sGX2a/ir6+9iPaODtz15Aa0tLZ1b2NN\nN/RmTcYOglZEV8sHBvj1eVF0VFgIosJCAACW/sr+8h+nELtgHm6bMRUA8Mii+dj14Z9ReLIMCyNu\nxY/NzdAf+AS/T3kWs4OnAQB2b1qFKb98FIUnyhAa2DNMF2cnBAb42uUd6FLG99wehntu73oOVO9M\nx7i64uCrL/S4LGN1An7266fw3YWLGOvthc7OTjz96utIf+IRLL/7593b/dRvXPd//+UrA/5Z/S3+\nb8dL8HS7HrdM9EfKI7FYv1OPLXHL7D7j3vdjj+vH9Pj//blfYcJPbui+3wLAsTMV2J73IQqzX8MN\nC3/ZY/v+8nRw6HnWVAr346u9gQIAfPRyz0LTJ62GesH/g6HsDCKmT7W6G3qzJmPBe4SlJSUImjRB\n6Obdwm65GfsPf4nzF7v+dPvUcAxnvj2Hu27VAAAM/yxHR2cn5s6c0X2dyb7jMF7thS++PmVxzeCb\nJqKkuNjqWa51A8246V//hkwGuI0aBQAoLivH+boGAIDm4ZX4SfSDuHtVEk5WVndf58uv/4lbJvjB\n0+367svu+pkGP/y7GScqqyWdcXtHB97+5FP8Kvqu7staWtuwdPNWZK5eCW9V39fH9penJVLO2JKm\nfzVDJpNBNWY0gIF1Q29CMxa8R9jQ0ACfqZOEbt4tY3UCHtu6HeMWPQQHhQIKuRy71j+JWdMDAQA1\n9Q1QOjhgjKtrj+upVe6oqW+wuKZa5Yb80hMotrM7UW1NDe60MuM2oxHrd+rx4J2RGHVd15NHK87X\nwGw2IyX7d3j5ycfg6+ONbW+/jzsS1uJ0XjbcRo9CTX0D1L1e76l27/r/ru9JN+M/flaAH/7djIcX\nzOu+7Ontb2DWtEAsjLjV4nX6y3P6pIA+15Fyxr2ZzWY8/erriJgWiCn+vgAG1g29qVVuOHTydL/b\nCS5Co9EIJ6Wj0M27vZb3J3x1ogz705MxXu2NQ6X/wMq0HbjR0wNzQmZc8Xpms9niMUcAcFI6oqa2\nBhqNxup5rmWODgqrMu7o6MSSjb+BTAbseObx7ssvHaDfuPyXuHd212sud29ahXGLHsJ7/3cYjyya\nf9V1ZTKZpDPWH/gL5oeFwMdDBQDYd/gLfGo4hpI9OwZ0m7wf9y8hLRMnq87i8Ovb+t32at3Qm5PS\nEW1txn63E1yESqUSbcZ2oZsD6HojxU1v5OLDlzZ3H0ecOsEPJae/wba338eckBnw8VDB2NGBH5ub\nezT/hcamPr9hL2kztsNH7YMP9+2zap5r3aLoaMEZXyrBb2vrkJ+5tXtvEABu8Ox6AN/sN777MqWj\nIwJuvAFnay4AAHw8VCg61fM3ZW1jI4Cu37hSzfhszQX8ragEf9y6ufuyTw3HUXH+e7jNu7/Htvev\nfw63z7gF+Zkv9ZunJVLNuLfH03fg4y+O4tDOdNzo9b8nPw+kG3prM7bDqZ9jlYAVRahSqVBT3yh0\ncwBdx1raOzrQu7wVcnn3XovmpxPhoFAg/2gp7rtjFgDg9NnvcLb2IsKm3mxx3dqGJnir1QgODrZq\nnmud2sdHUMaXSrDi/Pf4v8xUuP/3mMolmsmT4OToiLKz3yF82hQAXT+Lqu9r4XuDNwAgbOrNeDH3\nHdQ1/dB9XOuTr4px/ShXTPEfL9mMdx/4C9Tu7lgQPrP7svWxMXhkUc+ngtyydAVefWoFFs7q+lO5\nvzwtkWoOaB9EAAAaz0lEQVTGl3s8fQf2Hf4Cn2WlYbyPd4/vDaQbeqttaIK7u6rf7QQX4YygIBQd\n/qzP5c0trSj/7nz32c2KczU4dqYCqjGjMU7thdlBt2Bt5m/hrFTC9wZvfFZ8HG99nI9XnnoMQNdZ\n0F9F34XVr+2C+5hRGH3ddXjy5Z2YNW3KFc8KFZ8uR+jsOUJHHzEuZXy1TG/0VOEXG55D6ZkK7E9L\nRntnB2obuu50qjGj4ejggNGu1+Gx++7Glt++hbHenvD18Ubq3vcgkwEPzOl6OsjPbw3GFP/xiE1O\nw9aVv8L3dQ3Q7dqDlfdHw9HBwe4ztsRsNiP3o7/i4bvv7PEEXG+Vm8UTJOPUXvC9QQ2g/zwtkWLG\nl0tIy8Q7f/0Mf0rdAlcX5+778fWurnB2Ug6oG3oTmrHgItRoNMjNzUFrm7HHafGiU6cx5/F1kMlk\nkMlkWJPxJgAgdv487N60Cu88twHrd+7GsuRUNPz4L/j6qPFCvBaP3ruge41XnnwUCrkcD2z4Ddra\n2xF1qwaZax7vMwPQdfbuREU14lfZ13EV4H8ZFxw/iainN1rMdHPcUuz//CvIZDIEPbwSwP+Omfxf\n5ku4PegWAEB64q/h6KDAwylpaGkz4tbAycjPfAnXj+r6E0Mul2N/egoSUjMw69FVcHV2xsN334nk\nR5ZJIuPe92MA+NvREnx7oQ7ay55ydCW9j1FdLU9LpJrx5d7440eQyWSIXLm2x+W7N65C7H9PVFnT\nDb1Zk7Hgl9gdPXoUoaGh+CwrrfvBNhz+XnwckSvXorCwEDNnzuz/CiMIMxYfMxbfSMxY8PMINRoN\n/P38kL3v4KAHHIzs/QcR4O9vd2faAGY8FJix+EZixoKLUC6XIz4hAe/mH0Jd0w+DGnCgLjY2IS//\nMOITEuzuheoAMx4KzFh8IzFjq34KWq0WMrkcWe8fGPCAg5H1wQHI5HJoteK/5fdwYcbiY8biG2kZ\nW1WEnp6eSExMxIt73sWpqrMDGnCgTlWdxdY9eUhMTLTbN7MEmPFQYMbiG2kZj6jPLPmh3YSS0lJ+\n1oMImDEztrWRlLHVByhcXFygz8lB4YkyJO2y/BZDtpa0KxdFp05Dn5Nj93cegBkPBWYsvpGU8YA+\n13jcuHFwdXXF5vRX4OykRMR/30BBDC+9lYfk7L1ITU1FTEyMaLdzrWHG4mPG4hspGQ/4A97Dw8Nh\nMpmg27YdrUYjIoOn2fQMWEdHJza+noPk7L1ISkrCxo0bbbb2SMGMxceMxTcSMh5wEQJAZGQkXF1d\nkfzydvyl0IBZ06bA67L3YxuoU1VnsWhtMt7LP4S0tDRJ3nkuYcbiY8biu9YztvpkiSUFBQXQLl+O\n6upqPBu7BAmLF8LLve9rM/tzsbEJWR8cwNY9efD19UVObq5Vn1Zvz5ix+Jix+K7VjG1ShEDXGSKd\nToeMjAyYTSYsmXsb4qKjEDplMlycna58vdY2FJ4sQ/b+g8jLPwyZXI7ExESkpKRI4oCyNZix+Jix\n+K7FjG1WhJfU19dDr9djZ1YWKioroVAoEBjgi+CbJkKtcut6o0RjO2obmlB8uhwnKqrR2dmJAH9/\nxCckQKvV2vXzq2yBGYuPGYvvWsrY5kV4iclkgsFg6P4qKS7Ghdpa1NTWwEftA2+1GkHBwdBoNN1f\n9vhyIzExY/ExY/FZyrixsQFtbUY4OSnh7q4SPWPRitCS4uJiaDQaGAwGu3szymsFMxYfM7Y//NVF\nRJLHIiQiyWMREpHksQiJSPJYhEQkeSxCIpI8FiERSR6LkIgkj0VIRJLHIiQiyWMREpHksQiJSPJY\nhEQkeSxCIpI8FiERSR6LkIgkj0VIRJLHIiQiyWMREpHksQiJSPJYhEQkeSxCIpI8FiERSR6LkIgk\nj0VIRJLHIiQiyWMREpHksQiJSPJYhEQkeSxCIpI8FiERSR6LkIgkj0VIRJLHIiQiyWMREpHksQiJ\nSPJYhEQkeSxCIpI8FiERSR6LkIgkj0VIRJLHIiQiyWMREpHksQiJSPJYhEQkeSxCIpI8FiERSR6L\nkIgkj0VIRJLHIiQiyWMREpHkycxms1mMhU0mEwwGA4qKimAwGFBaUoLamhrUXqiF2lsNtY8PZgQF\nQaPRICQkBBqNBnI5e9kazFh8zFgabF6EdXV10Ov12JmVhcqqKjg4KBAY4IegSRPg4+EOJ6Uj2ozt\nqKlvRMmZb3CiogodHZ0I8PfHivh4aLVaeHp62nIku8OMxceMpcVmRdjS0gKdToeMjAyYTSbEzL0d\ncfdEIXTKZDg7Ka94vdY2IwpPliF730G8m38IMrkciYmJSElJgYuLiy1GsxvMWHzMWJpsUoQFBQXQ\nLl+O6upqrI+NQfziu+Hl7mb1Ohcbm5D1wQFs3ZMHX19f6HNyEB4ePtjx7AIzFh8zlq5BH8xIS0tD\nREQE3J0UKM7NhC5u6YDuPADg5e6GzXEPoTg3E25KBSIiIpCenj7YEUc8Ziw+Zixtii1btmwZ6JWT\nkpKQlJSEZ2NjsHfzWnir3G0ylJfb9Vi+4E4Y29uxOf1VmEwmREZG2mTtkYYZi48Z04CLMC0tDUlJ\nSdi6Mg66uKU2P1Mml8sxb2YQnJ2U0G3bjlGjRknuzwtmLD5mTMAAjxEWFBQgIiIC65YtwQvxWjHm\n6mF91m6k7n0PR44cQVhYmOi3dy1gxuJjxnSJ1UXY0tKCGdOnw91Jgc9f3waFQiHWbN06OjoRsWI1\nmoydKD12zO7PwjFj8TFjupzVfwfodDpUV1dDv3H1kNx5AMDBQQH9plWorq6GTqcbktscTsxYfMyY\nLmfVHmFdXR3Gjh2L9cuWQBe3VMy5LErO3outb72Hc+fOwcPDY8hvfygwY/ExY+rNqj1CvV4PmM1I\nuH+hWPNcVcLihTCbTF1z2ClmLD5mTL0J3iM0mUyYOGECIn46AbmbnxF7riuKTU5FQVklzpSX291r\nOpmx+JgxWSL4J2AwGFBZVYW4e6KsuoEXc9+BInw+Vm1/o/uyyIRnoAif3/3lMGsBEtIyBa0XFx2F\nispKGAwGq+YYCYRkbCnPN//0MeasXAu3eYuhCJ+PH5ubr3h9Y3s7gmIToAifj+PlFRa3kWLGh0u/\nxqJnNmNs9FIowudj3+Eveny/uaUVj6fvwPhFD8H1jkWY+uCjeOOPH/XYps1oxMq0THhFLcGYuffh\ngQ3P40JDk8U57DnjkUhwERYVFcHBQYHQKZMFL370ZBl+u+8gpk8M6HG5TCbDI4vmo+aj3+P7A7/H\n+f1vI3VlnKA1Q6dMhkKhsMs7UH8ZXynPlrY2RIXNxIblv4RMJrvqbazNzMZYL8+rbifFjJtbWjF9\n0gRkrllpMZunt7+BvxYW43fJz+LUO2/iyZj7kLgtCwc+/6p7m6defQN/LijEH17YhL9npeH8xXr8\nYsNzFuew54xHIqv2CAMD/K76wvPL/fs/LViWnIo31z8Ft9Gufb5/nbMTvNzd4K3q+hp1nbCnErg4\nOyEwwNcu70BXy/hqeT6x5F6sfegB3NrPL6mPvziKvx0tRlriI7jaEREpZhwVFoKUR2Nx7+xwi9l8\n+Y9TiF0wD7fNmIrxPt54ZNF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ICBQKhelkS29P4BNlY55NZNPWh7RabRuANq1W25c3axcoG/P6OptH49cPIRag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\nw9UFG6MWIzcjBS5iBkqlEsn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"text/plain": [ "Graphics object consisting of 28 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "p.show_distancePartitions(vertex_size=650)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Krein parameters are also nonnegative. Many Krein parameters are zero - in particular, the graph would be $Q$-polynomial for the natural ordering of the eigenvalues." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{aligned}0: &\\ \\left(\\begin{array}{rrrrrr}\n", "1 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 132 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 1617 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 1617 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 132 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 1\n", "\\end{array}\\right) \\\\\n", "1: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 1 & 0 & 0 & 0 & 0 \\\\\n", "1 & \\frac{50}{3} & \\frac{343}{3} & 0 & 0 & 0 \\\\\n", "0 & \\frac{343}{3} & \\frac{2450}{3} & 686 & 0 & 0 \\\\\n", "0 & 0 & 686 & \\frac{2450}{3} & \\frac{343}{3} & 0 \\\\\n", "0 & 0 & 0 & \\frac{343}{3} & \\frac{50}{3} & 1 \\\\\n", "0 & 0 & 0 & 0 & 1 & 0\n", "\\end{array}\\right) \\\\\n", "2: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 1 & 0 & 0 & 0 \\\\\n", "0 & \\frac{28}{3} & \\frac{200}{3} & 56 & 0 & 0 \\\\\n", "1 & \\frac{200}{3} & \\frac{2380}{3} & 700 & 56 & 0 \\\\\n", "0 & 56 & 700 & \\frac{2380}{3} & \\frac{200}{3} & 1 \\\\\n", "0 & 0 & 56 & \\frac{200}{3} & \\frac{28}{3} & 0 \\\\\n", "0 & 0 & 0 & 1 & 0 & 0\n", "\\end{array}\\right) \\\\\n", "3: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 56 & \\frac{200}{3} & \\frac{28}{3} & 0 \\\\\n", "0 & 56 & 700 & \\frac{2380}{3} & \\frac{200}{3} & 1 \\\\\n", "1 & \\frac{200}{3} & \\frac{2380}{3} & 700 & 56 & 0 \\\\\n", "0 & \\frac{28}{3} & \\frac{200}{3} & 56 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 & 0 & 0\n", "\\end{array}\\right) \\\\\n", "4: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & \\frac{343}{3} & \\frac{50}{3} & 1 \\\\\n", "0 & 0 & 686 & \\frac{2450}{3} & \\frac{343}{3} & 0 \\\\\n", "0 & \\frac{343}{3} & \\frac{2450}{3} & 686 & 0 & 0 \\\\\n", "1 & \\frac{50}{3} & \\frac{343}{3} & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 & 0 & 0\n", "\\end{array}\\right) \\\\\n", "5: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 0 & 0 & 0 & 1 \\\\\n", "0 & 0 & 0 & 0 & 132 & 0 \\\\\n", "0 & 0 & 0 & 1617 & 0 & 0 \\\\\n", "0 & 0 & 1617 & 0 & 0 & 0 \\\\\n", "0 & 132 & 0 & 0 & 0 & 0 \\\\\n", "1 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right) \\\\\\end{aligned}$ " ], "text/plain": [ "0: [ 1 0 0 0 0 0]\n", " [ 0 132 0 0 0 0]\n", " [ 0 0 1617 0 0 0]\n", " [ 0 0 0 1617 0 0]\n", " [ 0 0 0 0 132 0]\n", " [ 0 0 0 0 0 1]\n", "\n", "1: [ 0 1 0 0 0 0]\n", " [ 1 50/3 343/3 0 0 0]\n", " [ 0 343/3 2450/3 686 0 0]\n", " [ 0 0 686 2450/3 343/3 0]\n", " [ 0 0 0 343/3 50/3 1]\n", " [ 0 0 0 0 1 0]\n", "\n", "2: [ 0 0 1 0 0 0]\n", " [ 0 28/3 200/3 56 0 0]\n", " [ 1 200/3 2380/3 700 56 0]\n", " [ 0 56 700 2380/3 200/3 1]\n", " [ 0 0 56 200/3 28/3 0]\n", " [ 0 0 0 1 0 0]\n", "\n", "3: [ 0 0 0 1 0 0]\n", " [ 0 0 56 200/3 28/3 0]\n", " [ 0 56 700 2380/3 200/3 1]\n", " [ 1 200/3 2380/3 700 56 0]\n", " [ 0 28/3 200/3 56 0 0]\n", " [ 0 0 1 0 0 0]\n", "\n", "4: [ 0 0 0 0 1 0]\n", " [ 0 0 0 343/3 50/3 1]\n", " [ 0 0 686 2450/3 343/3 0]\n", " [ 0 343/3 2450/3 686 0 0]\n", " [ 1 50/3 343/3 0 0 0]\n", " [ 0 1 0 0 0 0]\n", "\n", "5: [ 0 0 0 0 0 1]\n", " [ 0 0 0 0 132 0]\n", " [ 0 0 0 1617 0 0]\n", " [ 0 0 1617 0 0 0]\n", " [ 0 132 0 0 0 0]\n", " [ 1 0 0 0 0 0]" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p.kreinParameters()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We check the remaining known feasibility conditions. We skip the sporadic nonexistence check since the intersection array is already included." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "p.check_feasible(skip=[\"sporadic\"])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We now compute the triple intersection numbers with respect to three vertices $u, v, w$ at mutual distances $2$. Note that we have $p^2_{22} = 243$, so such triples must exist. The parameter $\\alpha$ will denote the number of vertices adjacent to all of $u, v, w$." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{aligned}0: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right) \\\\\n", "1: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & \\alpha & 0 & -\\alpha + 5 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & -\\alpha + 5 & 0 & \\alpha + 45 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right) \\\\\n", "2: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 1 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "1 & 0 & 20 \\, \\alpha + 92 & 0 & -20 \\, \\alpha + 150 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & -20 \\, \\alpha + 150 & 0 & 20 \\, \\alpha + 200 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right) \\\\\n", "3: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & -\\alpha + 5 & 0 & \\alpha + 45 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & \\alpha + 45 & 0 & 11 \\, \\alpha + 1055 & 0 & -12 \\, \\alpha + 160 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & -12 \\, \\alpha + 160 & 0 & 12 \\, \\alpha + 15\n", "\\end{array}\\right) \\\\\n", "4: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & -20 \\, \\alpha + 150 & 0 & 20 \\, \\alpha + 200 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 20 \\, \\alpha + 200 & 0 & -20 \\, \\alpha + 605 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0\n", "\\end{array}\\right) \\\\\n", "5: &\\ \\left(\\begin{array}{rrrrrr}\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & -12 \\, \\alpha + 160 & 0 & 12 \\, \\alpha + 15 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 12 \\, \\alpha + 15 & 0 & -12 \\, \\alpha + 20\n", "\\end{array}\\right) \\\\\\end{aligned}$ " ], "text/plain": [ "0: [0 0 0 0 0 0]\n", " [0 0 0 0 0 0]\n", " [0 0 1 0 0 0]\n", " [0 0 0 0 0 0]\n", " [0 0 0 0 0 0]\n", " [0 0 0 0 0 0]\n", "\n", "1: [ 0 0 0 0 0 0]\n", " [ 0 alpha 0 -alpha + 5 0 0]\n", " [ 0 0 0 0 0 0]\n", " [ 0 -alpha + 5 0 alpha + 45 0 0]\n", " [ 0 0 0 0 0 0]\n", " [ 0 0 0 0 0 0]\n", "\n", "2: [ 0 0 1 0 0 0]\n", " [ 0 0 0 0 0 0]\n", " [ 1 0 20*alpha + 92 0 -20*alpha + 150 0]\n", " [ 0 0 0 0 0 0]\n", " [ 0 0 -20*alpha + 150 0 20*alpha + 200 0]\n", " [ 0 0 0 0 0 0]\n", "\n", "3: [ 0 0 0 0 0 0]\n", " [ 0 -alpha + 5 0 alpha + 45 0 0]\n", " [ 0 0 0 0 0 0]\n", " [ 0 alpha + 45 0 11*alpha + 1055 0 -12*alpha + 160]\n", " [ 0 0 0 0 0 0]\n", " [ 0 0 0 -12*alpha + 160 0 12*alpha + 15]\n", "\n", "4: [ 0 0 0 0 0 0]\n", " [ 0 0 0 0 0 0]\n", " [ 0 0 -20*alpha + 150 0 20*alpha + 200 0]\n", " [ 0 0 0 0 0 0]\n", " [ 0 0 20*alpha + 200 0 -20*alpha + 605 0]\n", " [ 0 0 0 0 0 0]\n", "\n", "5: [ 0 0 0 0 0 0]\n", " [ 0 0 0 0 0 0]\n", " [ 0 0 0 0 0 0]\n", " [ 0 0 0 -12*alpha + 160 0 12*alpha + 15]\n", " [ 0 0 0 0 0 0]\n", " [ 0 0 0 12*alpha + 15 0 -12*alpha + 20]" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S222 = p.tripleEquations(2, 2, 2, params={\"alpha\": (1, 1, 1)})\n", "S222" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The number of vertices at distance $5$ from all of $u, v, w$ is a nonnegative integer, implying that the parameter $\\alpha$ is either $0$ or $1$." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\alpha$ " ], "text/plain": [ "alpha" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}-12 \\, \\alpha + 20$ " ], "text/plain": [ "-12*alpha + 20" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(S222[1, 1, 1])\n", "show(S222[5, 5, 5])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us now consider the set $A$ of common neighbours of $u$ and $v$, and the set $B$ of vertices at distance $2$ from both $u$ and $v$. Their sizes are given by the intersection numbers $p^2_{11}$ and $p^2_{22}$." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}5$ " ], "text/plain": [ "5" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ " $\\newcommand{\\Bold}[1]{\\mathbf{#1}}243$ " ], "text/plain": [ "243" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(p.p[2, 1, 1])\n", "show(p.p[2, 2, 2])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By the above, each vertex in $B$ has at most one neighbour in $A$, so there are at most $243$ edges between $A$ and $B$. However, each vertex in $A$ is adjacent to both $u$ and $v$, and the other $k - 2 = 53$ neighbours are in $B$, amounting to a total of $5 \\cdot 53 = 265$ edges. We have arrived to a contradiction, and we must conclude that a graph with intersection array $\\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\\}$ **does not exist**." ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.2.rc2", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 2 }