jupyter/DRG-104-70-25-1-7-80.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Nonexistence of a distance-regular graph with intersection array $\\{104, 70, 25; 1, 7, 80\\}$\n",
    "\n",
    "We will show that a distance-regular graph with intersection array $\\{104, 70, 25; 1, 7, 80\\}$ 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 $1470$ vertices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}1470</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}1470\n",
       "\\end{math}"
      ],
      "text/plain": [
       "1470"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = drg.DRGParameters([104, 70, 25], [1, 7, 80])\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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\n",
      "text/plain": [
       "Graphics object consisting of 8 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 29 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 38 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\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 be formally self-dual for the natural ordering of eigenspaces and thus also $Q$-polynomial, so we have $q^3_{11} = q^1_{13} = q^1_{31} = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{aligned}0: &\\ \\left(\\begin{array}{rrrr}\n",
       "1 & 0 & 0 & 0 \\\\\n",
       "0 & 104 & 0 & 0 \\\\\n",
       "0 & 0 & 1040 & 0 \\\\\n",
       "0 & 0 & 0 & 325\n",
       "\\end{array}\\right) \\\\\n",
       "1: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "1 & 33 & 70 & 0 \\\\\n",
       "0 & 70 & 720 & 250 \\\\\n",
       "0 & 0 & 250 & 75\n",
       "\\end{array}\\right) \\\\\n",
       "2: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 1 & 0 \\\\\n",
       "0 & 7 & 72 & 25 \\\\\n",
       "1 & 72 & 742 & 225 \\\\\n",
       "0 & 25 & 225 & 75\n",
       "\\end{array}\\right) \\\\\n",
       "3: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 80 & 24 \\\\\n",
       "0 & 80 & 720 & 240 \\\\\n",
       "1 & 24 & 240 & 60\n",
       "\\end{array}\\right) \\\\\\end{aligned}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{aligned}0: &\\ \\left(\\begin{array}{rrrr}\n",
       "1 & 0 & 0 & 0 \\\\\n",
       "0 & 104 & 0 & 0 \\\\\n",
       "0 & 0 & 1040 & 0 \\\\\n",
       "0 & 0 & 0 & 325\n",
       "\\end{array}\\right) \\\\\n",
       "1: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "1 & 33 & 70 & 0 \\\\\n",
       "0 & 70 & 720 & 250 \\\\\n",
       "0 & 0 & 250 & 75\n",
       "\\end{array}\\right) \\\\\n",
       "2: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 1 & 0 \\\\\n",
       "0 & 7 & 72 & 25 \\\\\n",
       "1 & 72 & 742 & 225 \\\\\n",
       "0 & 25 & 225 & 75\n",
       "\\end{array}\\right) \\\\\n",
       "3: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 1 \\\\\n",
       "0 & 0 & 80 & 24 \\\\\n",
       "0 & 80 & 720 & 240 \\\\\n",
       "1 & 24 & 240 & 60\n",
       "\\end{array}\\right) \\\\\\end{aligned}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "0: [   1    0    0    0]\n",
       "   [   0  104    0    0]\n",
       "   [   0    0 1040    0]\n",
       "   [   0    0    0  325]\n",
       "\n",
       "1: [  0   1   0   0]\n",
       "   [  1  33  70   0]\n",
       "   [  0  70 720 250]\n",
       "   [  0   0 250  75]\n",
       "\n",
       "2: [  0   0   1   0]\n",
       "   [  0   7  72  25]\n",
       "   [  1  72 742 225]\n",
       "   [  0  25 225  75]\n",
       "\n",
       "3: [  0   0   0   1]\n",
       "   [  0   0  80  24]\n",
       "   [  0  80 720 240]\n",
       "   [  1  24 240  60]"
      ]
     },
     "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 local eigenvalues nonexistence check, as it turns out that their multiplicity cannot be integral. Instead, we will use triple intersection numbers to prove nonexistence."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "p.check_feasible(skip=[\"localEigenvalues\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $w, x, y, z$ be vertices such that $x$ is adjacent to $y$ and $z$, $y$ is at distance $2$ from $w$ and $z$, and $w$ is at distance $3$ from $x$. Note that we have $p^1_{12} = 70$ and $p^1_{32} = 250$, so such vertices must exist. We first compute the triple intersection numbers with respect to $x, y, z$. The parameter $\\alpha$ will denote the number of vertices adjacent to $x, y, z$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{aligned}0: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0\n",
       "\\end{array}\\right) \\\\\n",
       "1: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 1 & 0 \\\\\n",
       "0 & \\alpha & -\\alpha + 33 & 0 \\\\\n",
       "1 & -\\alpha + 33 & \\alpha + 36 & 0 \\\\\n",
       "0 & 0 & 0 & 0\n",
       "\\end{array}\\right) \\\\\n",
       "2: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & -\\alpha + 6 & \\alpha + 39 & 25 \\\\\n",
       "0 & \\alpha + 39 & 24 \\, \\alpha + 306 & -25 \\, \\alpha + 375 \\\\\n",
       "0 & 25 & -25 \\, \\alpha + 375 & 25 \\, \\alpha - 150\n",
       "\\end{array}\\right) \\\\\n",
       "3: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & -25 \\, \\alpha + 400 & 25 \\, \\alpha - 150 \\\\\n",
       "0 & 0 & 25 \\, \\alpha - 150 & -25 \\, \\alpha + 225\n",
       "\\end{array}\\right) \\\\\\end{aligned}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{aligned}0: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0\n",
       "\\end{array}\\right) \\\\\n",
       "1: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 1 & 0 \\\\\n",
       "0 & \\alpha & -\\alpha + 33 & 0 \\\\\n",
       "1 & -\\alpha + 33 & \\alpha + 36 & 0 \\\\\n",
       "0 & 0 & 0 & 0\n",
       "\\end{array}\\right) \\\\\n",
       "2: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & -\\alpha + 6 & \\alpha + 39 & 25 \\\\\n",
       "0 & \\alpha + 39 & 24 \\, \\alpha + 306 & -25 \\, \\alpha + 375 \\\\\n",
       "0 & 25 & -25 \\, \\alpha + 375 & 25 \\, \\alpha - 150\n",
       "\\end{array}\\right) \\\\\n",
       "3: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & -25 \\, \\alpha + 400 & 25 \\, \\alpha - 150 \\\\\n",
       "0 & 0 & 25 \\, \\alpha - 150 & -25 \\, \\alpha + 225\n",
       "\\end{array}\\right) \\\\\\end{aligned}\n",
       "\\end{math}"
      ],
      "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           0           1           0]\n",
       "   [          0       alpha -alpha + 33           0]\n",
       "   [          1 -alpha + 33  alpha + 36           0]\n",
       "   [          0           0           0           0]\n",
       "\n",
       "2: [              0               0               0               0]\n",
       "   [              0      -alpha + 6      alpha + 39              25]\n",
       "   [              0      alpha + 39  24*alpha + 306 -25*alpha + 375]\n",
       "   [              0              25 -25*alpha + 375  25*alpha - 150]\n",
       "\n",
       "3: [              0               0               0               0]\n",
       "   [              0               0               0               0]\n",
       "   [              0               0 -25*alpha + 400  25*alpha - 150]\n",
       "   [              0               0  25*alpha - 150 -25*alpha + 225]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.tripleEquations(1, 1, 2, params={\"alpha\": (1, 1, 1)})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From $[2\\ 1\\ 1] \\ge 0$ and $[3\\ 2\\ 3] \\ge 0$, it follows that there is a single solution with $\\alpha = 6$ and therefore $[3\\ 2\\ 3] = 0$, implying that $w$ cannot be at distance $3$ from $z$ for any choice of $w, x, y, z$ as above.\n",
    "\n",
    "We now compute the triple intersection numbers with respect to $w, x, y$. The parameter $\\beta$ will denote the number of vertices at distances $(3, 1, 2)$ from $w, x, y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{aligned}0: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0\n",
       "\\end{array}\\right) \\\\\n",
       "1: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 7 & 5 \\, \\beta - 27 & -5 \\, \\beta + 100 \\\\\n",
       "0 & 0 & -5 \\, \\beta + 99 & 5 \\, \\beta - 75\n",
       "\\end{array}\\right) \\\\\n",
       "2: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "1 & \\beta + 9 & -\\beta + 70 & 0 \\\\\n",
       "0 & -\\beta + 63 & -24 \\, \\beta + 882 & 25 \\, \\beta - 225 \\\\\n",
       "0 & 0 & 25 \\, \\beta - 210 & -25 \\, \\beta + 450\n",
       "\\end{array}\\right) \\\\\n",
       "3: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "0 & -\\beta + 24 & \\beta & 0 \\\\\n",
       "0 & \\beta & 19 \\, \\beta - 135 & -20 \\, \\beta + 375 \\\\\n",
       "0 & 0 & -20 \\, \\beta + 360 & 20 \\, \\beta - 300\n",
       "\\end{array}\\right) \\\\\\end{aligned}</script></html>"
      ],
      "text/latex": [
       "\\begin{math}\n",
       "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{aligned}0: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0\n",
       "\\end{array}\\right) \\\\\n",
       "1: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "0 & 7 & 5 \\, \\beta - 27 & -5 \\, \\beta + 100 \\\\\n",
       "0 & 0 & -5 \\, \\beta + 99 & 5 \\, \\beta - 75\n",
       "\\end{array}\\right) \\\\\n",
       "2: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 0 & 0 & 0 \\\\\n",
       "1 & \\beta + 9 & -\\beta + 70 & 0 \\\\\n",
       "0 & -\\beta + 63 & -24 \\, \\beta + 882 & 25 \\, \\beta - 225 \\\\\n",
       "0 & 0 & 25 \\, \\beta - 210 & -25 \\, \\beta + 450\n",
       "\\end{array}\\right) \\\\\n",
       "3: &\\ \\left(\\begin{array}{rrrr}\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "0 & -\\beta + 24 & \\beta & 0 \\\\\n",
       "0 & \\beta & 19 \\, \\beta - 135 & -20 \\, \\beta + 375 \\\\\n",
       "0 & 0 & -20 \\, \\beta + 360 & 20 \\, \\beta - 300\n",
       "\\end{array}\\right) \\\\\\end{aligned}\n",
       "\\end{math}"
      ],
      "text/plain": [
       "0: [0 0 0 0]\n",
       "   [0 0 0 0]\n",
       "   [0 0 0 0]\n",
       "   [0 0 1 0]\n",
       "\n",
       "1: [            0             0             0             0]\n",
       "   [            0             0             0             0]\n",
       "   [            0             7   5*beta - 27 -5*beta + 100]\n",
       "   [            0             0  -5*beta + 99   5*beta - 75]\n",
       "\n",
       "2: [             0              0              0              0]\n",
       "   [             1       beta + 9     -beta + 70              0]\n",
       "   [             0     -beta + 63 -24*beta + 882  25*beta - 225]\n",
       "   [             0              0  25*beta - 210 -25*beta + 450]\n",
       "\n",
       "3: [             0              1              0              0]\n",
       "   [             0     -beta + 24           beta              0]\n",
       "   [             0           beta  19*beta - 135 -20*beta + 375]\n",
       "   [             0              0 -20*beta + 360  20*beta - 300]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.tripleEquations(3, 2, 1, params={\"beta\": (3, 1, 2)})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From $[3\\ 3\\ 2] \\ge 0$ and $[1\\ 3\\ 3] \\ge 0$, it follows that $15 \\le \\beta \\le 18$. This would imply the existence of a vertex $z$ as above that is at distance $3$ from $w$ - a contradiction! We thus conclude that a distance-regular graph with intersection array $\\{104, 70, 25; 1, 7, 80\\}$ **does not exist**."
   ]
  }
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