#!/usr/bin/env python
# coding: utf-8

# # Distance-regular graph parameter checking in Sage
# 
# [![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.1418409.svg)](https://doi.org/10.5281/zenodo.1418409)
# 
# A Sage package for checking the feasibility of distance-regular graph parameter sets.
# A more detailed description, along with some results, is available in a [manuscript](https://arxiv.org/abs/1803.10797) currently available on arXiv.

# ## Contents
# 
# ### `drg`
# 
# The `drg` folder contains the package source. After you make sure that Sage sees this folder, you can import it as a Python module.

# In[1]:


get_ipython().run_line_magic('display', 'latex')
import drg
p = drg.DRGParameters([80, 63, 12], [1, 12, 60])
p.check_feasible()


# You can also give an intersection array with parameters.

# In[2]:


r = var("r")
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)])
fam.check_feasible()


# In[3]:


fam1 = fam.subs(r == 1)
fam1


# In[4]:


fam2 = fam.subs(r == 2)
fam2


# In[5]:


fam2.check_feasible()


# ### `jupyter`
# 
# A collection of sample Jupyter notebooks giving some nonexistence results.
# 
# * [`Demo.ipynb`](jupyter/Demo.ipynb) - demonstration of the `sage-drg` package
# * [`DRG-104-70-25-1-7-80.ipynb`](jupyter/DRG-104-70-25-1-7-80.ipynb) - proof of nonexistence of a distance-regular graph with intersection array $\{104, 70, 25; 1, 7, 80\}$ with $1470$ vertices
# * [`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
# * [`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
# * [`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
# * [`DRG-d3-2param.ipynb`](jupyter/DRG-d3-2param.ipynb) - proof of nonexistence of a family of distance-regular graphs with intersection arrays $\{(2r+1)(4r+1)(4t-1), 8r(4rt-r+2t), (r+t)(4r+1); 1, (r+t)(4r+1), 4r(2r+1)(4t-1)\}$ ($r, t \ge 1$)
# * [`QPoly-24-20-36_11-1-30_11-24.ipynb`](jupyter/QPoly-24-20-36_11-1-30_11-24.ipynb) - proof of nonexistence of a $Q$-polynomial association scheme with Krein array $\{24, 20, 36/11; 30/11, 24\}$ with $225$ vertices
# * [`QPoly-d3-1param-odd.ipynb`](jupyter/QPoly-d3-1param-odd.ipynb) - proof of nonexistence of a $Q$-polynomial association scheme with Krein array $\{2r^2-1, 2r^2-2, r^2+1; 1, 2, r^2-1\}$ and $r$ odd
# * [`QPoly-d4-LS-odd.ipynb`](jupyter/QPoly-d4-LS-odd.ipynb) - proof of nonexistence of a $Q$-polynomial association scheme with Krein array $\{m, m-1, m(r^2-1)/r^2, m-r^2+1; 1, m/r^2, r^2-1, m\}$ and $m$ odd
# * [`QPoly-d4-tight4design.ipynb`](jupyter/QPoly-d4-tight4design.ipynb) - proof of nonexistence of a $Q$-polynomial association scheme with Krein array $\{r^2-4, r^2-9, 12(s-1)/s, 1; 1, 12/s, r^2-9, r^2-4\}$ ($s \ge 4$)
# * [`QPoly-d5-1param-3mod4.ipynb`](jupyter/QPoly-d5-1param-3mod4.ipynb) - proof of nonexistence of a $Q$-polynomial association scheme with Krein array $\{(r^2+1)/2, (r^2-1)/2, (r^2+1)^2/(2r(r+1)), (r-1)(r^2+1)/(4r), (r^2+1)/(2r); 1, (r-1)(r^2+1)/(2r(r+1)), (r+1)(r^2+1)/(4r), (r-1)(r^2+1)/(2r), (r^2+1)/2\}$ and $r \equiv 3 \pmod{4}$
# 
# A conference presentation is also available as a [RISE](https://github.com/damianavila/RISE) slideshow.
# 
# * [`FPSAC-sage-drg.ipynb`](jupyter/2019-07-04-fpsac/FPSAC-sage-drg.ipynb) - *Computing distance-regular graph and association scheme parameters in SageMath with `sage-drg`* software presentation at FPSAC 2019, Ljubljana, Slovenia

# ## Citing
# 
# If you use `sage-drg` in your research, please cite both the paper and the software:
# 
# * J. Vidali. Using symbolic computation to prove nonexistence of distance-regular graphs. *Electron. J. Combin.*, 25(4)#P4.21, 2018. [`http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i4p21`](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i4p21).
# 
# * J. Vidali. `jaanos/sage-drg`: `sage-drg` v0.9, 2019. [`https://github.com/jaanos/sage-drg/`](https://github.com/jaanos/sage-drg/), [`doi:10.5281/zenodo.1418409`](https://doi.org/10.5281/zenodo.1418409).
# 
# You may also want to cite other documents containing descriptions of features that were added since the above paper was written:
# 
# * J. Vidali. Computing distance-regular graph and association scheme parameters in SageMath with `sage-drg`. *Sém. Lothar. Combin.* 82B#105, 2019. [`https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2019/105.pdf`](https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2019/105.pdf)
#     + support for general and *Q*-polynomial association schemes
# 
# * A. L. Gavrilyuk, J. Vidali, J. S. Williford. On few-class *Q*-polynomial association schemes: feasible parameters and nonexistence results, 2019. [`arXiv:1908.10081`](http://arxiv.org/abs/1908.10081).
#     + triple intersection number solution finder and forbidden quadruples check
#     + support for quadruple intersection numbers
# 
# ### BibTeX
# 
# The above citations are given here in BibTeX format.
# 
# ```latex
# @article{v18,
#     AUTHOR = {Vidali, Jano\v{s}},
#      TITLE = {Using symbolic computation to prove nonexistence of distance-regular graphs},
#    JOURNAL = {Electron. J. Combin.},
#   FJOURNAL = {Electronic Journal of Combinatorics},
#     VOLUME = {25},
#     NUMBER = {4},
#      PAGES = {P4.21},
#       NOTE = {\url{http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i4p21}},
#       YEAR = {2018},
# }
# 
# @software{v19a,
#    AUTHOR = {Vidali, Jano\v{s}},
#     TITLE = {{\tt jaanos/sage-drg}: {\tt sage-drg} v0.9},
#      NOTE = {\url{https://github.com/jaanos/sage-drg/},
#              \href{https://doi.org/10.5281/zenodo.1418409}{\texttt{doi:10.5281/zenodo.1418409}}},
#      YEAR = {2019},
# }
# 
# @article{v19b,
#     AUTHOR = {Vidali, Jano\v{s}},
#      TITLE = {Computing distance-regular graph and association scheme parameters in SageMath with {\tt sage-drg}},
#    JOURNAL = {S\'{e}m. Lothar. Combin.},
#   FJOURNAL = {S\'{e}minaire Lotharingien de Combinatoire},
#     VOLUME = {82B},
#      PAGES = {#105},
#       NOTE = {\url{https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2019/105.pdf}},
#       YEAR = {2019},
# }
# 
# @article{gvw21,
#     AUTHOR = {Gavrilyuk, Alexander L. and Vidali, Jano\v{s} and Williford, Jason S.},
#      TITLE = {On few-class $Q$-polynomial association schemes: feasible parameters and nonexistence results},
#    JOURNAL = {Ars Math. Contemp.},
#   FJOURNAL = {Ars Mathematica Contemporanea},
#       NOTE = {\href{https://doi.org/10.26493/1855-3974.2101.b76}{\texttt{doi:10.26493/1855-3974.2101.b76}}},
#       YEAR = {2021},
# }
# ```
# ### Other uses
# 
# Additionally, `sage-drg` has been used in the following research:
# 
# * A. Gavrilyuk, S. Suda and J. Vidali. On tight 4-designs in Hamming association schemes, *Combinatorica*, 40(3):345-362, 2020. [`doi:10.1007/s00493-019-4115-z`](https://doi.org/10.1007/s00493-019-4115-z).
# 
# If you would like your research to be listed here, feel free to open an issue or pull request.