Configurations from a Graphical Viewpoint

Pisanski, Tomaž; Servatius, Brigitte

doi:10.1007/978-0-8176-8364-1

Cited by 56 publications

(62 citation statements)

References 5 publications

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“…We see that all defective lines are of the same form, namely {α, α, α}. A configuration C 4 whose Veldkamp space reproduces the above-described partitions of points and lines of PG(3, 2) is, as demonstrated below, the famous Desargues (10 3 )-configuration, D, which is one of the most prominent point-line incidence structures (see, for example, [13]). Up to isomorphism, there exist altogether 10 (10 3 )-configurations.…”

Section: Sedenions and The Desargues (10 3 )-Configurationmentioning

confidence: 66%

From Cayley-Dickson Algebras to Combinatorial Grassmannians

2015

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Abstract:Given a 2 N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6, we first observe that the multiplication table of its imaginary units e a , 1 ≤ a ≤ 2 N − 1, is encoded in the properties of the projective space PG(N − 1, 2) if these imaginary units are regarded as points and distinguished triads of them {e a , e b , e c }, 1 ≤ a < b < c ≤ 2 N − 1 and e a e b = ±e c , as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b = c. Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG (N − 1, 2), the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration C N ; in particular, C 3 (octonions) is isomorphic to the Pasch (6 2 , 4 3 )-configuration, C 4 (sedenions) is the famous Desargues (10 3 )-configuration, C 5 (32-nions) coincides with the Cayley-Salmon (15 4 , 20 3 )-configuration found in the well-known Pascal mystic hexagram and C 6 (64-nions) is identical with a particular (21 5 , 35 3 )-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic C N leads to a conjecture that C N is isomorphic to a combinatorial Grassmannian of type G 2 (N + 1).

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Section: Sedenions and The Desargues (10 3 )-Configurationmentioning

confidence: 66%

From Cayley-Dickson Algebras to Combinatorial Grassmannians

2015

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“…The square of the Levi graph L(C) of a configuration C is called the Grünbaum graph of C in [19] and [20]. In [8], it is called the independence graph.…”

Section: Splittable and Unsplittable Configurations (And Graphs)mentioning

confidence: 99%

“…The graph can be written as LCF [5, −5] n . (For the LCF notation see [19].) This means that the edges determined by symbols 0 and 1 form a Hamiltonian cycle while the edges arising from the symbol 3 form chords of length 5.…”

Section: Splittable and Unsplittable Cyclic (N ) Configurationsmentioning

confidence: 99%

Splittable and unsplittable graphs and configurations

Bašić¹,

Grošelj²,

Grünbaum³

et al. 2018

Ars Math. Contemp.

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“…Symmetry type graphs have been used previously in the analysis of maps; see, for instance, [5][6][7][8][9] . We extend these results of maps to oriented maps in the obvious way by introducing a useful tool that we call an oriented symmetry type graph (see Figure 3):…”

Section: Definitionmentioning

confidence: 99%

“…Operations on maps with the property that the new map resides in the same surface as the original map have been the subject of active investigation (see [6,7,[10][11][12][13]). If the underlying surface is orientable, we may choose one orientation that induces an orientation of the corresponding map, making it an oriented map.…”

Section: Operations On Oriented Mapsmentioning

confidence: 99%

Operations on Oriented Maps

2017

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A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves may be considered as maps embedded in the same surface as the original graph. The flag graph is the underlying graph of the dual of the barycentric subdivision of the original map. Certain operations on maps can be defined by appropriate operations on flag graphs. Orientable surfaces may be given consistent orientations, and oriented maps can be described by a generating pair consisting of a permutation and an involution on the set of arcs (or darts) defining a partially directed arc graph. In this paper we describe how certain operations on maps can be described directly on oriented maps via arc graphs.Keywords: map; oriented map; truncation; dual; medial; snub; flag graph; arc graph MSC: 52C20, 05C10, 51M20 Maps and Oriented MapsA map M on a closed surface is a two-cell embedding of a finite connected graph G = (V, E) (see, e.g., [1] or [2]). Equivalently, every map can be viewed as a set of three fixed-point-free involutions r 0 , r 1 and r 2 acting on a set of flags Ω with the property that r 0 r 2 is also a fixed-point-free involution, in which case we denote the map as M = (Ω, r 0 , r 1 , r 2 ); see, for instance, [3] (p. 415). With this second point of view, each map can be described completely using a three-edge colored cubic graph, called the flag graph, whose vertices are elements of Ω, where ω 1 and ω 2 are connected by an edge colored i in the flag graph if and only if r i (ω 1 ) = ω 2 . Generally, only connected flag graphs are considered. The graph G is called the skeleton or the underlying graph of a map M.In some cases, one may relax the conditions and allow fixed points in some of the involutions r 0 , r 1 and r 2 . Such a structure describes a map in a surface with a boundary and with its flag graph containing semi-edges. We call such a map and respective flag graph degenerate.If the surface in which the map resides is orientable, then M is said to be an orientable map. There is a well-known test involving flag graphs to determine whether a given map is orientable: Proposition 1. A map is orientable if and only if its flag graph is bipartite.In a bipartite flag graph, the flags (i.e., vertices) of the map come in two color classes; the vertices in a single color class of flags are called arcs. Restricting attention to one set of arcs corresponds to choosing an orientation of the orientable map, and we call the restricted graph an oriented map; each orientable map gives rise to two oppositely oriented oriented maps. We note that in a map there are four flags per edge of the skeleton, while in an oriented map there are two arcs per edge. That is,

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Configurations from a Graphical Viewpoint

Cited by 56 publications

References 5 publications

From Cayley-Dickson Algebras to Combinatorial Grassmannians

From Cayley-Dickson Algebras to Combinatorial Grassmannians

Splittable and unsplittable graphs and configurations

Operations on Oriented Maps

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