{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Nonexistence of a distance-regular graph with intersection array $\\{135, 128, 16; 1, 16, 120\\}$\n", "\n", "We will show that a distance-regular graph with intersection array $\\{135, 128, 16; 1, 16, 120\\}$ does not exist." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%display latex\n", "import drg" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Such a graph would have $1360$ vertices." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "1360" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p = drg.DRGParameters([135, 128, 16], [1, 16, 120])\n", "p.order()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that all intersection numbers are nonnegative integers." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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iIkLDwjBx0iSYzWbExMTAbDaLigZDQ64aTHDkmmOGhlzlbnAczXBTUxM6Ozvh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\nJTRXs9lssFgsPV8nqqrQ3NyEjo5OmEz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"text/plain": [ "Graphics object consisting of 38 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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HgNpQUPmP9yHlc0BtKKj8xuuQCiGgNhRU/uJtSIUUUBsKKj/xMqRCDKgNBZV/eBdSIQfU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\nQZBUioei7scPH1Pi5dK3odm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DrSNpXEzUhF+W94fgaXLExcyhkBLBYB3S5qYmxM+f68ta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\nCQlwFFJCAhyFlJAA9/8DNzVkAnZBRbEAAAAASUVORK5CYII=\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": {}, "source": [ "The Krein parameters are also nonnegative. The graph would not be $Q$-polynomial, but we still have $q^3_{33} = 0$." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "0: [ 1 0 0 0]\n", " [ 0 306 0 0]\n", " [ 0 0 900 0]\n", " [ 0 0 0 153]\n", "\n", "1: [ 0 1 0 0]\n", " [ 1 68 220 17]\n", " [ 0 220 1700/3 340/3]\n", " [ 0 17 340/3 68/3]\n", "\n", "2: [ 0 0 1 0]\n", " [ 0 374/5 578/3 578/15]\n", " [ 1 578/3 610 289/3]\n", " [ 0 578/15 289/3 272/15]\n", "\n", "3: [ 0 0 0 1]\n", " [ 0 34 680/3 136/3]\n", " [ 0 680/3 1700/3 320/3]\n", " [ 1 136/3 320/3 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 mutually adjacent vertices $u, v, w$. Note that we have $a_1 = 6$, 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": [ "" ], "text/plain": [ "0: [0 0 0 0]\n", " [0 1 0 0]\n", " [0 0 0 0]\n", " [0 0 0 0]\n", "\n", "1: [ 0 1 0 0]\n", " [ 1 alpha -alpha + 5 0]\n", " [ 0 -alpha + 5 alpha + 123 0]\n", " [ 0 0 0 0]\n", "\n", "2: [ 0 0 0 0]\n", " [ 0 -alpha + 5 alpha + 123 0]\n", " [ 0 alpha + 123 19/8*alpha + 4641/8 -27/8*alpha + 967/8]\n", " [ 0 0 -27/8*alpha + 967/8 27/8*alpha + 57/8]\n", "\n", "3: [ 0 0 0 0]\n", " [ 0 0 0 0]\n", " [ 0 0 -27/8*alpha + 967/8 27/8*alpha + 57/8]\n", " [ 0 0 27/8*alpha + 57/8 -27/8*alpha + 71/8]" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "alpha = var(\"alpha\")\n", "S111 = p.tripleEquations(1, 1, 1, params={alpha: (1, 1, 1)})\n", "S111" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The number of vertices at distance $3$ from all of $u, v, w$ should be a nonnegative integer. Nonnegativity is only achieved when $\\alpha = 0, 1, 2$, however integrality is not achieved in any of these cases." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "[71/8, 11/2, 17/8, -5/4]" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "[S111[3, 3, 3].subs(alpha == x) for x in [0, 1, 2, 3]]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We thus conclude that a graph with intersection array $\\{135, 128, 16; 1, 16, 120\\}$ **does not exist**." ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.9.beta2", "language": "sage", "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": "2.7.15" } }, "nbformat": 4, "nbformat_minor": 2 }