Page 304 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 304
GRAPHS AND GROUPS, GEOMETRIES AND GAP - G2G2 (MS-7)
Corollary 2. For every integer m ≥ 3, the m-closure of a solvable permutation group is
solvable.
The work of the speaker is supported by the Mathematical Center in Akademgorodok, the
agreement with Ministry of Science and High Education of the Russian Federation number
075-15-2019-1613.
References
[1] E. A. O’Brien, I. Ponomarenko, A. V. Vasil’ev, E. Vdovin, The 3-closure of a solvable
permutation group is solvable, subm. to J. Algebra (2021), see also arXiv:2012.14166.
[2] C. E. Praeger and J. Saxl, Closures of finite primitive permutation groups, Bull. London
Math. Soc., 24, 251–258 (1992).
[3] Á. Seress, The minimal base size of primitive solvable permutation groups, J. London
Math. Soc., 53, 243–255 (1996).
[4] S. Skresanov, Counterexamples to two conjectures in the Kourovka notebook, Algebra
Logic 58, no. 3, 249–253 (2019).
[5] H. Wielandt, Permutation groups through invariant relations and invariant functions, The
Ohio State University (1969).
Computing distance-regular graph and association scheme parameters in
SageMath with sage-drg
Janoš Vidali, janos.vidali@fmf.uni-lj.si
University of Ljubljana, Slovenia
The sage-drg package [8] for the SageMath computer algebra system has been originally
developed for computation of parameters of distance-regular graphs, and its functionality has
later been extended to handle general association schemes. The package has been used to obtain
nonexistence results for both distance-regular graphs and Q-polynomial association schemes,
mostly using the triple intersection numbers technique, see for example [3, 4, 7].
Recently, checks for two new feasibility conditions have been implemented. The first
technique, developed by Kodalen and Martin [5], relies on Schönberg’s theorem on positive
semidefinite functions in Sm−1 and its application on the minimal idempotents of an association
scheme. The implementation of the relevant checks in sage-drg allows us to replicate their
nonexistence results for several feasible parameter sets for Q-polynomial association schemes.
The second technique derives from Terwilliger’s work on P - and Q-polynomial associa-
tion schemes [6] and has most recently been used by Gavrilyuk and Koolen [1, 2] to obtain
some nonexistence and uniqueness results for Q-polynomial distance-regular graphs. The im-
plementation of the relevant procedures in sage-drg allows us to generalize their approach
and derive nonexistence for many feasible parameter sets of classical distance-regular graphs.
References
[1] A. L. Gavrilyuk and J. H. Koolen. The Terwilliger polynomial of a Q-polynomial
distance-regular graph and its application to pseudo-partition graphs. Linear Algebra
Appl., 466(1):117–140, 2015. doi:10.1016/j.laa.2014.09.048.
302
Corollary 2. For every integer m ≥ 3, the m-closure of a solvable permutation group is
solvable.
The work of the speaker is supported by the Mathematical Center in Akademgorodok, the
agreement with Ministry of Science and High Education of the Russian Federation number
075-15-2019-1613.
References
[1] E. A. O’Brien, I. Ponomarenko, A. V. Vasil’ev, E. Vdovin, The 3-closure of a solvable
permutation group is solvable, subm. to J. Algebra (2021), see also arXiv:2012.14166.
[2] C. E. Praeger and J. Saxl, Closures of finite primitive permutation groups, Bull. London
Math. Soc., 24, 251–258 (1992).
[3] Á. Seress, The minimal base size of primitive solvable permutation groups, J. London
Math. Soc., 53, 243–255 (1996).
[4] S. Skresanov, Counterexamples to two conjectures in the Kourovka notebook, Algebra
Logic 58, no. 3, 249–253 (2019).
[5] H. Wielandt, Permutation groups through invariant relations and invariant functions, The
Ohio State University (1969).
Computing distance-regular graph and association scheme parameters in
SageMath with sage-drg
Janoš Vidali, janos.vidali@fmf.uni-lj.si
University of Ljubljana, Slovenia
The sage-drg package [8] for the SageMath computer algebra system has been originally
developed for computation of parameters of distance-regular graphs, and its functionality has
later been extended to handle general association schemes. The package has been used to obtain
nonexistence results for both distance-regular graphs and Q-polynomial association schemes,
mostly using the triple intersection numbers technique, see for example [3, 4, 7].
Recently, checks for two new feasibility conditions have been implemented. The first
technique, developed by Kodalen and Martin [5], relies on Schönberg’s theorem on positive
semidefinite functions in Sm−1 and its application on the minimal idempotents of an association
scheme. The implementation of the relevant checks in sage-drg allows us to replicate their
nonexistence results for several feasible parameter sets for Q-polynomial association schemes.
The second technique derives from Terwilliger’s work on P - and Q-polynomial associa-
tion schemes [6] and has most recently been used by Gavrilyuk and Koolen [1, 2] to obtain
some nonexistence and uniqueness results for Q-polynomial distance-regular graphs. The im-
plementation of the relevant procedures in sage-drg allows us to generalize their approach
and derive nonexistence for many feasible parameter sets of classical distance-regular graphs.
References
[1] A. L. Gavrilyuk and J. H. Koolen. The Terwilliger polynomial of a Q-polynomial
distance-regular graph and its application to pseudo-partition graphs. Linear Algebra
Appl., 466(1):117–140, 2015. doi:10.1016/j.laa.2014.09.048.
302
