Distance-regular graph parameter checking in Sage

A SageMath package for checking the feasibility of distance-regular graph parameter sets. A more detailed description, along with some results, is available in a paper published in the Electronic Journal of Combinatorics.

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.

sage: import drg
sage: p = drg.DRGParameters([80, 63, 12], [1, 12, 60])
sage: p.check_feasible()

You can also give an intersection array with parameters.

sage: r = var("r")
sage: 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)])
sage: fam.check_feasible()
sage: fam1 = fam.subs(r == 1)
sage: fam1
Parameters of a distance-regular graph with intersection array {6, 4, 2; 1, 2, 3}
sage: fam2 = fam.subs(r == 2)
sage: fam2
Parameters of a distance-regular graph with intersection array {40, 33, 8; 1, 8, 30}
sage: fam2.check_feasible()
...
InfeasibleError: nonexistence by JurišićVidali12

jupyter

A collection of sample Jupyter notebooks giving some nonexistence results. Also includes conference and seminar presentations.

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.

• J. Vidali. jaanos/sage-drg: sage-drg v0.9, 2019. https://github.com/jaanos/sage-drg/, doi: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

  • 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, Ars Math. Contemp., 20(1):103-127, 2021. doi:10.26493/1855-3974.2101.b76.

  • triple intersection number solution finder and forbidden quadruples check
  • support for quadruple intersection numbers

BibTeX

The above citations are given here in BibTeX format.

@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},
    VOLUME = {20},
    NUMBER = {1},
     PAGES = {103--127},
      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.

If you would like your research to be listed here, feel free to open an issue or pull request.