| 1. |
- Ball, Simeon, et al.
(författare)
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An alternative way to generalise the pentagon
- 2013
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Ingår i: Journal of combinatorial designs (Print). - : John Wiley & Sons. - 1063-8539 .- 1520-6610. ; 21:4, s. 163-179
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Tidskriftsartikel (refereegranskat)abstract
- We introduce the concept of a pentagonal geometry as a generalization of the pentagon and the Desargues configuration, in the same vein that the generalized polygons share the fundamental properties of ordinary polygons. In short, a pentagonal geometry is a regular partial linear space in which for all points x, the points not collinear with the point x, form a line. We compute bounds on their parameters, give some constructions, obtain some nonexistence results for seemingly feasible parameters and suggest a cryptographic application related to identifying codes of partial linear spaces.
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| 2. |
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| 3. |
- Bras-Amorós, Maria, et al.
(författare)
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On the existence of combinatorial configurations
- 2010
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Ingår i: 3rd International Workshop on Optimal Networks Topologies, 2010. - 9788476535653 ; , s. 145-167
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)
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| 4. |
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| 5. |
- Bras-Amorós, Maria, et al.
(författare)
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The semigroup of combinatorial configurations
- 2012
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Ingår i: Semigroup Forum. - : Springer. - 0037-1912 .- 1432-2137. ; 84:1, s. 91-96
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Tidskriftsartikel (refereegranskat)abstract
- We elaborate on the existence and construction of the so-called combinatorial configurations. The main result is that for fixed degrees the existence of such configurations is given by a numerical semigroup. The proof is constructive giving a method to obtain combinatorial configurations with parameters large enough.
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| 6. |
- Conder, Marston, et al.
(författare)
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New methods for finding minimum genus embeddings of graphs on orientable and non-orientable surfaces
- 2019
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Ingår i: Ars Mathematica Contemporanea. - : Drustvo Matematikov, Fizikov in Astronomov,Society of Mathematicians, Physicists and Astronomers. - 1855-3966 .- 1855-3974. ; 17:1, s. 1-35
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Tidskriftsartikel (refereegranskat)abstract
- The question of how to find the smallest genus of all embeddings of a given finite connected graph on an orientable (or non-orientable) surface has a long and interesting history. In this paper we introduce four new approaches to help answer this question, in both the orientable and non-orientable cases. One approach involves taking orbits of subgroups of the automorphism group on cycles of particular lengths in the graph as candidates for subsets of the faces of an embedding. Another uses properties of an auxiliary graph defined in terms of compatibility of these cycles. We also present two methods that make use of integer linear programming, to help determine bounds for the minimum genus, and to find minimum genus embeddings. This work was motivated by the problem of finding the minimum genus of the Hoffman-Singleton graph, and succeeded not only in solving that problem but also in answering several other open questions.
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| 7. |
- Forbes, Anthony D., et al.
(författare)
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Existence results for pentagonal geometries
- 2022
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Ingår i: The Australasian Journal of Combinatorics. - : University of Queensland Press. - 1034-4942. ; 82:1, s. 95-114
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Tidskriftsartikel (refereegranskat)abstract
- New results on pentagonal geometries PENT(k, r) with block sizes k = 3 or k = 4 are given. In particular we completely determine the existence spectra for PENT(3, r) systems with the maximum number of opposite line pairs as well as those without any opposite line pairs. A wide-ranging result about PENT(3, r) with any number of opposite line pairs is proved. We also determine the existence spectrum of PENT(4, r) systems with eleven possible exceptions.
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| 8. |
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| 9. |
- Izquierdo, Milagros, 1963-, et al.
(författare)
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Isometric Point-Circle Configurations on Surfaces from Uniform Maps
- 2016
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer-Verlag New York. - 2194-1009. ; 159, s. 201-212, s. 201-212
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Tidskriftsartikel (refereegranskat)abstract
- We embed neighborhood geometries of graphs on surfaces as point-circle configurations. We give examples coming from regular maps on surfaces with a maximum number of automorphisms for their genus, and survey geometric realization of pentagonal geometries coming from Moore graphs. An infinite family of point-circle v4'>v4v4 configurations on p-gonal surfaces with two p-gonal morphisms is given. The image of these configurations on the sphere under the two p-gonal morphisms is also described.
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| 10. |
- Leemans, Dimitri, et al.
(författare)
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Incidence geometries with trialities coming from maps with Wilson trialities
- 2023
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Ingår i: Innovations in Incidence Geometry. - : Mathematical Sciences Publishers. - 2640-7337 .- 2640-7345. ; 20:2-3, s. 325-340
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Tidskriftsartikel (refereegranskat)abstract
- Triality is a classical notion in geometry that arose in the context of the Lie groups of type D4. Another notion of triality, Wilson triality, appears in the context of reflexible maps. We build a bridge between these two notions, showing how to construct an incidence geometry with a triality from a map that admits a Wilson triality. We also extend a result by Jones and Poulton, showing that for every prime power q, the group L2 (q3) has maps that admit Wilson trialities but no dualities.
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