{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "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": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "import drg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Such a graph would be bipartite with $3500$ vertices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "3500\n"
     ]
    }
   ],
   "source": [
    "p = drg.DRGParameters([55, 54, 50, 35, 10], [1, 5, 20, 45, 55])\n",
    "print(p.is_bipartite())\n",
    "print(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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VFmvbrt3yeDzakL9D555zji744e+sQ6uWuvaSRN2eMVPZ40arvKJSdz6arZv7XFHtTLRj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\n",
      "text/plain": [
       "Graphics object consisting of 12 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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QY/aoMXvUmC1j6dvlYQMAYWFhsLW1ReK76fjqggJjh/vB5Vfv59FVVypuYvaGRBzJP43U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\nuIBhXG9mcBdKy6B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      "text/plain": [
       "Graphics object consisting of 24 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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ZwkOH8+yat/Hx8iDI35ecA4dZ+s7/ERX+vw9JD0+ewL8y36JX9274dfUi/pW1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\n4jSzm42bmxuFJWUqazFbUWk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      "text/plain": [
       "Graphics object consisting of 32 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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ERMKxbIiISDiWDRERCceyISIi4Vg2REQkHMuGiIiEY9kQEZFwLBsiIhKOZUNE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\nMVtl6mq52vwmqqOjI4ypqTh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AAABJRU5ErkJggg==\n",
      "text/plain": [
       "Graphics object consisting of 35 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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V43288cii+Zg+KQCFJ8sAAD82N0N/4BO8/ORjmB08DUGTJ2L3plU4cvwkCk+U9VnPnjMe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\n+Y9TiF0wD7fNmIrxPt54ZNF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      "text/plain": [
       "Graphics object consisting of 34 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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ICBQKhelkS29P4BNlY55NZNPWh7RabRuANq1W25c3axcoG/P6OptH49cPIRag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\nhLD8f3NW/Yvr6SqbAAAAAElFTkSuQmCC\n",
      "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": {
    "deletable": true,
    "editable": true
   },
   "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": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "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": {
    "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 $2$. Note that we have $p^2_{22} = 5$, so such triples must exist. The parameter $a$ will denote the number of vertices adjacent to 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 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      a      0 -a + 5      0      0]\n",
       "   [     0      0      0      0      0      0]\n",
       "   [     0 -a + 5      0 a + 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*a + 92           0 -20*a + 150           0]\n",
       "   [          0           0           0           0           0           0]\n",
       "   [          0           0 -20*a + 150           0  20*a + 200           0]\n",
       "   [          0           0           0           0           0           0]\n",
       "\n",
       "3: [          0           0           0           0           0           0]\n",
       "   [          0      -a + 5           0      a + 45           0           0]\n",
       "   [          0           0           0           0           0           0]\n",
       "   [          0      a + 45           0 11*a + 1055           0 -12*a + 160]\n",
       "   [          0           0           0           0           0           0]\n",
       "   [          0           0           0 -12*a + 160           0   12*a + 15]\n",
       "\n",
       "4: [          0           0           0           0           0           0]\n",
       "   [          0           0           0           0           0           0]\n",
       "   [          0           0 -20*a + 150           0  20*a + 200           0]\n",
       "   [          0           0           0           0           0           0]\n",
       "   [          0           0  20*a + 200           0 -20*a + 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*a + 160           0   12*a + 15]\n",
       "   [          0           0           0           0           0           0]\n",
       "   [          0           0           0   12*a + 15           0  -12*a + 20]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S222 = p.tripleEquations(2, 2, 2, params = {\"a\": (1, 1, 1)})\n",
    "S222"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The number of vertices at distance $5$ from all of $u, v, w$ is a nonnegative integer, implying that the parameter $a$ is either $0$ or $1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a\n",
      "-12*a + 20\n"
     ]
    }
   ],
   "source": [
    "print(S222[1, 1, 1])\n",
    "print(S222[5, 5, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "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": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5\n",
      "243\n"
     ]
    }
   ],
   "source": [
    "print(p.p[2, 1, 1])\n",
    "print(p.p[2, 2, 2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "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**."
   ]
  }
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