Page 274 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 274
CONFIGURATIONS (MS-81)
Configurations from strong deficient difference sets
Marién Abreu, marien.abreu@unibas.it
Università degli Studi della Basilicata, Italy
Coauthors: Martin Funk, Domenico Labbate, Vedran Krcˇadinac
In [1] we have studied combinatorial configurations with the associated point and line graphs
being strongly regular, which we call strongly regular configurations. In the talk “Strongly
regular configurations” of this minisymposium, Vedran Krcˇadinac will present existing known
families of strongly regular configurations; constructions of several other families; necessary
existence conditions and a table of feasible parameters with at most 200 points.
Let G be a group of order v. A subset D ⊂ G of size k is a deficient difference set the left
differences d−1 1d2 are all distinct. Considering the elements of G as points and the development
devD = {gD|g ∈ G} as lines a symmetric (vk) configuration is obtained and it has G as its
automorphism group acting regularly on the points and lines. Let ∆(D) = {d−1 1d2|d1, d2 ∈
D, d1 = d2} be the set of left differences of D. For a group element x ∈ G \ {1}, denote by
n(x) = |∆(D) ∩ x∆(D)|. If n(x) = λ for every x ∈ ∆(D), and n(x) = µ for every x ∈/ ∆(D),
D is said to be a strong deficient difference set (SDDS) for (vk; λ, µ).
Here, we present one of the new families of strongly regular configurations constructed in
[1], with parameters different from semipartial geometries and arising from strong deficient dif-
ference sets, as well as two examples arising from Hall’s plane and its dual. Moreover, from the
exhaustive search performed in groups of order v ≤ 200 further four examples corresponding
to strong deficient difference sets, but not in the previous families, are obtained.
References
[1] M. Abreu, M. Funk, V. Krcˇadinac, D. Labbate, Strongly regular configurations, preprint,
2021. https://arxiv.org/abs/2104.04880
Splittability of cubic bicirculants and their related configurations
Nino Bašic´, nino.basic@famnit.upr.si
University of Primorska, Slovenia, and IMFM, Slovenia
Coauthors: Matjaž Krnc, Tomaž Pisanski
Recently, it was shown that there exist infinitely many splittable and also infinitely many unsplit-
table cyclic (n3) configurations. This was achieved by studying splittability of trivalent cyclic
Haar graphs. We extend this study to include cubic bicirculant and their related configurations.
Connected (nk) configurations exist for almost all n
Leah Berman, lwberman@alaska.edu
University of Alaska Fairbanks, United States
Coauthors: Gábor Gévay, Tomaž Pisanski
A geometric (nk) configuration is a collection of points and straight lines, typically in the Eu-
clidean plane, so that each line passes through k of the points and each of the points lies on k of
272
Configurations from strong deficient difference sets
Marién Abreu, marien.abreu@unibas.it
Università degli Studi della Basilicata, Italy
Coauthors: Martin Funk, Domenico Labbate, Vedran Krcˇadinac
In [1] we have studied combinatorial configurations with the associated point and line graphs
being strongly regular, which we call strongly regular configurations. In the talk “Strongly
regular configurations” of this minisymposium, Vedran Krcˇadinac will present existing known
families of strongly regular configurations; constructions of several other families; necessary
existence conditions and a table of feasible parameters with at most 200 points.
Let G be a group of order v. A subset D ⊂ G of size k is a deficient difference set the left
differences d−1 1d2 are all distinct. Considering the elements of G as points and the development
devD = {gD|g ∈ G} as lines a symmetric (vk) configuration is obtained and it has G as its
automorphism group acting regularly on the points and lines. Let ∆(D) = {d−1 1d2|d1, d2 ∈
D, d1 = d2} be the set of left differences of D. For a group element x ∈ G \ {1}, denote by
n(x) = |∆(D) ∩ x∆(D)|. If n(x) = λ for every x ∈ ∆(D), and n(x) = µ for every x ∈/ ∆(D),
D is said to be a strong deficient difference set (SDDS) for (vk; λ, µ).
Here, we present one of the new families of strongly regular configurations constructed in
[1], with parameters different from semipartial geometries and arising from strong deficient dif-
ference sets, as well as two examples arising from Hall’s plane and its dual. Moreover, from the
exhaustive search performed in groups of order v ≤ 200 further four examples corresponding
to strong deficient difference sets, but not in the previous families, are obtained.
References
[1] M. Abreu, M. Funk, V. Krcˇadinac, D. Labbate, Strongly regular configurations, preprint,
2021. https://arxiv.org/abs/2104.04880
Splittability of cubic bicirculants and their related configurations
Nino Bašic´, nino.basic@famnit.upr.si
University of Primorska, Slovenia, and IMFM, Slovenia
Coauthors: Matjaž Krnc, Tomaž Pisanski
Recently, it was shown that there exist infinitely many splittable and also infinitely many unsplit-
table cyclic (n3) configurations. This was achieved by studying splittability of trivalent cyclic
Haar graphs. We extend this study to include cubic bicirculant and their related configurations.
Connected (nk) configurations exist for almost all n
Leah Berman, lwberman@alaska.edu
University of Alaska Fairbanks, United States
Coauthors: Gábor Gévay, Tomaž Pisanski
A geometric (nk) configuration is a collection of points and straight lines, typically in the Eu-
clidean plane, so that each line passes through k of the points and each of the points lies on k of
272
