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PreprintCC BY-NC-ND 4.0arXiv

Four class association scheme double covers of strongly regular graphs

  • Patrick Cesarz
  • Jason Williford
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Abstract

In this paper we give necessary conditions for 4-class association schemes that are generated by double covers of strongly regular graphs. These conditions are applied to open cases for diameter 4 antipodal distance-regular graphs. Using these conditions we are able to show the nonexistence of four cases in the table of Brouwer, Cohen and Neumeier: {20,18,3,1;1,3,18,20}, {22,21,3,1;1,3,21,22}, { 54,50,5,1;,1,5,50,54 }, { 170,162,9,1;,1,9,162,170 }. More generally, we show there is no distance-regular graph with intersection array {k,b1,b2,1;1,b2,b1,k } where b2 ≠ 1 and ( k b1 )/4(1+k+( k b1 )/2b2) is odd. Tables are also given for more general 4-class association schemes generated by double covers of strongly regular graphs.

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Cite

BibTeXRIS
BibTeX
@misc{cesarz2026four,
  title = {{Four class association scheme double covers of strongly regular graphs}},
  author = {Patrick Cesarz and Jason Williford},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05420},
  eprint = {2610.05420},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05420},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Four class association scheme double covers of strongly regular graphs
AU  - Patrick Cesarz
AU  - Jason Williford
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05420
UR  - https://arxiv.org/abs/2610.05420
AB  - In this paper we give necessary conditions for 4-class association schemes that are generated by double covers of strongly regular graphs. These conditions are applied to open cases for diameter 4 antipodal distance-regular graphs. Using these conditions we are able to show the nonexistence of four cases in the table of Brouwer, Cohen and Neumeier: {20,18,3,1;1,3,18,20}, {22,21,3,1;1,3,21,22}, { 54,50,5,1;,1,5,50,54 }, { 170,162,9,1;,1,9,162,170 }. More generally, we show there is no distance-regular graph with intersection array {k,b_1,b_2,1;1,b_2,b_1,k } where b_2 ≠ 1 and ( k b_1 )/4(1+k+( k b_1 )/2b_2) is odd. Tables are also given for more general 4-class association schemes generated by double covers of strongly regular graphs.
ER  -