{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "# Nonexistence of a distance-regular graph with intersection array $\\{234, 165, 12; 1, 30, 198\\}$\n", "\n", "We will show that a distance-regular graph with intersection array $\\{234, 165, 12; 1, 30, 198\\}$ does not exist." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "import drg" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Such a graph would have $1600$ vertices." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "1600" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p = drg.DRGParameters([234, 165, 12], [1, 30, 198])\n", "p.order()" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "We see that all intersection numbers are nonnegative integers." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "image/png": 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LTH5AE8OHY2L4cKOvd/fsio1L3rZoHMdPl6C5uVmSv+SVIi3eKBUuaNZ+2+6y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\nxtPTExU3aoQcC2cqq2vR3cc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kHByGhsRSQnAkO+rMZjMMBgNKS0thMBhw7OhR1NbUwMtTi5kxMQgMCsKE0FDo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\nHIxtu/e6dB3b9uzFiJAQHqRkF84xdYe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9osbExACAXsuFasqtTCaDQqGAvb09nJ2dDcqtPpCk96CvqC0tLeoFo+sbGsDn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\nAAAASUVORK5CYII=\n", "text/plain": [ "Graphics object consisting of 32 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "p.show_distancePartitions(vertex_size = 650)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "The Krein parameters are also nonnegative. We have $q^1_{22} = q^2_{12} = q^2_{21} = 0$, so the graph would be $Q$-polynomial with respect to the ordering of the eigenvalues $0, 2, 3, 1$." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/plain": [ "0: [ 1 0 0 0]\n", " [ 0 78 0 0]\n", " [ 0 0 234 0]\n", " [ 0 0 0 1287]\n", "\n", "1: [ 0 1 0 0]\n", " [ 1 8 36 33]\n", " [ 0 36 0 198]\n", " [ 0 33 198 1056]\n", "\n", "2: [ 0 0 1 0]\n", " [ 0 12 0 66]\n", " [ 1 0 68 165]\n", " [ 0 66 165 1056]\n", "\n", "3: [ 0 0 0 1]\n", " [ 0 2 12 64]\n", " [ 0 12 30 192]\n", " [ 1 64 192 1030]" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p.kreinParameters()" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "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": { "collapsed": true, "deletable": true, "editable": true }, "outputs": [], "source": [ "p.check_feasible(skip = [\"sporadic\"])" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "We now compute the triple intersection numbers with respect to three vertices $u, v, w$ at mutual distances $3$. Note that we have $p^3_{33} = 8$, so such triples must exist. The parameter $a$ will denote the number of vertices at distance $3$ from all of $u, v, w$." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/plain": [ "0: [0 0 0 0]\n", " [0 0 0 0]\n", " [0 0 0 0]\n", " [0 0 0 1]\n", "\n", "1: [ 0 0 0 0]\n", " [ 0 0 0 0]\n", " [ 0 0 3*a + 186 -3*a + 12]\n", " [ 0 0 -3*a + 12 3*a + 24]\n", "\n", "2: [ 0 0 0 0]\n", " [ 0 0 3*a + 186 -3*a + 12]\n", " [ 0 3*a + 186 -10*a + 832 7*a + 38]\n", " [ 0 -3*a + 12 7*a + 38 -4*a - 17]\n", "\n", "3: [ 0 0 0 1]\n", " [ 0 0 -3*a + 12 3*a + 24]\n", " [ 0 -3*a + 12 7*a + 38 -4*a - 17]\n", " [ 1 3*a + 24 -4*a - 17 a]" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S333 = p.tripleEquations(3, 3, 3, params = {\"a\": (3, 3, 3)})\n", "S333" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "We now note that since $a$ must be nonnegative, the number of vertices at distances $2, 3, 3$ from $u, v, w$ must be negative - contradiction" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "-4*a - 17\n", "a\n" ] } ], "source": [ "print(S333[2, 3, 3])\n", "print(S333[3, 3, 3])" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "We thus conclude that a graph with intersection array $\\{234, 165, 12; 1, 30, 198\\}$ **does not exist**." ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 7.6", "language": "", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.13" } }, "nbformat": 4, "nbformat_minor": 2 }