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   "source": [
    "# Distance-regular graph parameter checking in Sage\n",
    "\n",
    "[![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.1418410.svg)](https://doi.org/10.5281/zenodo.1418410)\n",
    "\n",
    "A Sage package for checking the feasibility of distance-regular graph parameter sets.\n",
    "A more detailed description, along with some results, is available in a [manuscript](https://arxiv.org/abs/1803.10797) currently available on arXiv."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Contents\n",
    "\n",
    "### `drg`\n",
    "\n",
    "The `drg` folder contains the package source. After you make sure that Sage sees this folder, you can import it as a Python module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import drg\n",
    "p = drg.DRGParameters([80, 63, 12], [1, 12, 60])\n",
    "p.check_feasible()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also give an intersection array with parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = var(\"r\")\n",
    "fam = drg.DRGParameters([2*r^2*(2*r + 1), (2*r - 1)*(2*r^2 + r + 1), 2*r^2], [1, 2*r^2 , r*(4*r^2 - 1)])\n",
    "fam.check_feasible()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Parameters of a distance-regular graph with intersection array {6, 4, 2; 1, 2, 3}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fam1 = fam.subs(r == 1)\n",
    "fam1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Parameters of a distance-regular graph with intersection array {40, 33, 8; 1, 8, 30}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fam2 = fam.subs(r == 2)\n",
    "fam2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "ename": "InfeasibleError",
     "evalue": "nonexistence by JurišićVidali12",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mInfeasibleError\u001b[0m                           Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-5-58e3e45d7a79>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mfam2\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcheck_feasible\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/home/janos/repos/git/sage-drg/drg/drg.pyc\u001b[0m in \u001b[0;36mcheck_feasible\u001b[0;34m(self, checked, skip, derived)\u001b[0m\n\u001b[1;32m    674\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcheck\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mchecks\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    675\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mname\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mskip\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 676\u001b[0;31m                 \u001b[0mcheck\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    677\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mderived\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    678\u001b[0m             \u001b[0;32mreturn\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/home/janos/repos/git/sage-drg/drg/drg.pyc\u001b[0m in \u001b[0;36mcheck_family\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    635\u001b[0m                            in zip(self.b[:-1] + self.c[1:], b + c)], vars)\n\u001b[1;32m    636\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0many\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcheckConditions\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcond\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msol\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0msol\u001b[0m \u001b[0;32min\u001b[0m \u001b[0msols\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 637\u001b[0;31m                 \u001b[0;32mraise\u001b[0m \u001b[0mInfeasibleError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrefs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mref\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    638\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    639\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mcheck_feasible\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mchecked\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mskip\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mderived\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mInfeasibleError\u001b[0m: nonexistence by JurišićVidali12"
     ]
    }
   ],
   "source": [
    "fam2.check_feasible()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### `jupyter`\n",
    "\n",
    "A collection of sample Jupyter notebooks giving some nonexistence results.\n",
    "\n",
    "* [Demo.ipynb](jupyter/Demo.ipynb) - demonstration of the `sage-drg` package\n",
    "* [DRG-135-128-16-1-16-120.ipynb](jupyter/DRG-135-128-16-1-16-120.ipynb) - proof of nonexistence of a distance-regular graph with intersection array $\\{135, 128, 16; 1, 16, 120\\}$ with $1360$ vertices\n",
    "* [DRG-234-165-12-1-30-198.ipynb](jupyter/DRG-234-165-12-1-30-198.ipynb) - proof of nonexistence of a distance-regular graph with intersection array $\\{234, 165, 12; 1, 30, 198\\}$ with $1600$ vertices\n",
    "* [DRG-55-54-50-35-10-bipartite.ipynb](jupyter/DRG-55-54-50-35-10-bipartite.ipynb) - proof of nonexistence of a bipartite distance-regular graph with intersection array $\\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\\}$ with $3500$ vertices\n",
    "* [DRG-d3-2param.ipynb](jupyter/DRG-d3-2param.ipynb) - proof of nonexistence of a family of distance-regular graphs with intersection arrays $\\{(2r+1)(4r+1)(4t-1), 8r(4rt-r+2t), (r+t)(4r+1); 1, (r+t)(4r+1), 4r(2r+1)(4t-1)\\}$ ($r, t \\ge 1$)\n",
    "* [QPoly-d4-tight4design.ipynb](jupyter/QPoly-d4-tight4design.ipynb) - proof of nonexistence of a $Q$-polynomial association scheme with Krein array $\\{r^2-4, r^2-9, 10, 1; 1, 2, r^2-9, r^2-4\\}$ ($r \\ne 9$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Citing\n",
    "\n",
    "If you use `sage-drg` in your research, please cite both the manuscript and the repository:\n",
    "\n",
    "* J. Vidali. Using symbolic computation to prove nonexistence of distance-regular graphs, 2018. [`arXiv:1803.10797`](http://arxiv.org/abs/1803.10797).\n",
    "\n",
    "* J. Vidali. `jaanos/sage-drg`: `sage-drg` v0.8, 2018. [`https://github.com/jaanos/sage-drg/`](https://github.com/jaanos/sage-drg/), [`doi:10.5281/zenodo.1418410`](http://dx.doi.org/10.5281/zenodo.1418410).\n",
    "\n",
    "```latex\n",
    "@other {v18a,\n",
    "    AUTHOR = {Vidali, Jano\\v{s}},\n",
    "     TITLE = {Using symbolic computation to prove nonexistence of distance-regular graphs},\n",
    "      NOTE = {\\href{http://arxiv.org/abs/1803.10797}{\\texttt{arXiv:1803.10797}}},\n",
    "      YEAR = {2018},\n",
    "}\n",
    "\n",
    "@software{v18b,\n",
    "   AUTHOR = {Vidali, Jano\\v{s}},\n",
    "    TITLE = {{\\tt jaanos/sage-drg}: {\\tt sage-drg} v0.8},\n",
    "     NOTE = {\\url{https://github.com/jaanos/sage-drg/},\n",
    "             \\href{http://dx.doi.org/10.5281/zenodo.1418410}{\\texttt{10.5281/zenodo.1418410}}},\n",
    "     YEAR = {2018},\n",
    "}\n",
    "```\n",
    "\n",
    "Additionally, `sage-drg` has been used in the following research:\n",
    "\n",
    "* A. Gavrilyuk, S. Suda and J. Vidali. On tight 4-designs in Hamming association schemes, 2018. [`arXiv:1809.07553`](http://arxiv.org/abs/1809.07553).\n",
    "\n",
    "If you would like your research to be listed here, feel free to open an issue or pull request."
   ]
  }
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