Figure 2 - uploaded by Tomaz Pisanski
Content may be subject to copyright.
Source publication
Graver and Watkins classified edge-transitive maps on closed surfaces into fourteen types. In this note we study these types for maps in orientable and non-orientable surfaces of small genus, including the Euclidean and hyperbolic plane. We revisit both finite and infinite one-ended edge-transitive maps. For the finite ones we give precise descript...
Contexts in source publication
Context 1
... may be easily shown that these 14 symmetry pre-graphs correspond exactly to the 14 types of edge-transitive maps (see [6,19,16]). The pre-graphs are depicted in Figure 2. Only type 1 consists of flag-transitive maps. ...
Similar publications
Several algorithms based on PageRank algorithm have been proposed to rank the document sentences in the multi-document summarization field and LexRank and T-LexRank algorithms are well known examples. In literature different concepts such as weighted inter-cluster edge, cluster-sensitive graph model and document-sensitive graph model have been prop...
Let \(k\) be a positive integer, \(G\) be a graph on \(V(G)\) containing a path on \(k\) vertices, and \(w\) be a weight function assigning each vertex \(v\in V(G)\) a real weight \(w(v)\). Upper bounds on the weight \(w(P)=\sum_{v\in V(P)}w(v)\) of \(P\) are presented, where \(P\) is chosen among all paths of \(G\) on \(k\) vertices with smallest...
In this paper we take a problem of unsupervised nodes clustering on graphs and show how recent advances in attention models can be applied successfully in a "hard" regime of the problem. We propose an unsupervised algorithm that encodes Bethe Hessian embeddings by optimizing soft modularity loss and argue that our model is competitive to both class...
Citations
... For the definition of a regular covering projection, the reader is referred, for example, to [1]. 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): ...
... 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. ...
... In [20], Širáň, Tucker and Watkins showed that each of the 14 types admits a realization by an oriented map. In [6], Orbanić et al. showed that the 14 types can naturally be described by 14 symmetry type graphs, shown in Figure 11. ...
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.
... Repeated operations were used in other contexts, see for instance [25]. Operations on maps have been studied in connection with symmetry in several papers, see for instance [23,22,4,10]. ...
Prior to crashes, the stock index price time series is characterised by the Log-Periodic Power Law (LPPL) equation of the Johansen-Ledoit-Sornette (JLS) model, which leads to a critical point indicating a change to a new market regime. In this paper, we describe the hierarchical diamond lattice, upon which the JLS model is derived, using the diamond lattice operation Di and derive the recursion for the coefficients of the growth function in a diamond lattice rooted at the main root vertex rm. Further, to verify the adequacy of the JLS model, we analyse the model's residuals and propose its generalization, using the ARMA/GARCH error model. We determine the ARMA/GARCH orders using the extended autocorrelation function (EACF) method and compare the results with those of the Akaike and Bayesian Information Criteria. Using the data for 33 major world stock indices we show, that proposed generalization of the JLS model in general performs better in predicting the market regime changes and has also the ability to recognise false bubble identification, indicated by the JLS model.
... In this section we will describe and analyse their classification, but from a rather more group-theoretic point of view. This is largely motivated by the work ofŠiráň, Tucker and Watkins in [45], where finite symmetric groups are used as automorphism groups in order to construct finite maps of particular types in all 14 classes, together with that of Orbanić, Pellicer, Pisanski and Tucker in [41], where edge-transitive maps of small genus are classified. (In the present paper, 'class' refers to the 14 equivalence classes of edge-transitive maps, whereas 'type' refers to the parameters giving the valencies of the faces and vertices of a map, as in [7].) Figure 4 shows that, except when T = 1, M always has such an automorphism, so that N (T ) is not in class T . ...
For each of the 14 classes of edge-transitive maps described by Graver and Watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map, or of an orientable map without boundary, in that class. Extending earlier results of Siran, Tucker and Watkins, these are used to determine which symmetric groups $S_n$ can arise in this way for each class. Similar results are obtained for all finite simple groups, building on work of Leemans and Liebeck, Nuzhin and others on generating sets for such groups. It is also shown that each edge-transitive class realises finite groups of every sufficiently large nilpotence class or derived length, and also realises uncountably many non-isomorphic infinite groups.
... A map is edge-transitive if the (full) automorphism group is transitive on the set of edges. Edge-transitive † maps were investigated by several authors including Graver and Watkins [42] and Orbanić et al. [84]. In the latter paper the characterisation in terms of 'symmetry diagrams' on surfaces with boundary induced by the action of the full automorphism group is given. ...
... It proves the results obtained by Graver and Watkins. However, the method used in [84] cannot be practically used for construction and classification of edge-transitive maps of higher genera. One have to look for normal subgroups of rather high index if certain finitely-presented groups (N.E.C. groups), which is due to to the complexity of involved algorithms almost intractable. ...
... similarly as in the classification of Archimedean maps. Compared to the method used by Orbanić et al. in [84], we control the genus g of the underlying surface by choosing a proper g-admissible orbifold. Moreover, the list of g-admissible groups of proper signatures is processed independently on the problem. ...
Discrete actions of finite groups on surfaces appears in many situations in numerous branches of mathematics, cryptography, quantum physics, and many other fields of science. In topological graph theory they can be used to derive lists of highly symmetrical maps of fixed genus: regular maps, vertex-transitive maps, Cayley maps, or edge-transitive maps. For example, the classification of actions of cyclic groups is essential for solving enumeration problems of combinatorial objects, i.e. maps, graphs and others. Lists of discrete group actions are used as an experimental material for further research.
The problem of classification of discrete actions of groups on orientable surfaces of genus g ≥ 2 is nowadays challenge. The classification of groups acting on the sphere is a classical part of crystallography. In case of torus the situation is in principle known, though there are infinitely many group actions. Thanks to Riemann-Hurwitz formula we know that for higher genera there are just finitely many finite groups acting on a surface of a given genus. Published lists of actions go up to genus five (Broughton; Bogopolskij; Kuribayashi and Kimura). For small genera, the classification can be done with help of computer algebra systems. For example, using Magma we derived the list of actions of discrete groups on surfaces of genus 2 ≤ g ≤ 21.
This text is intended as a brief introduction to the problematic and a summary of results achieved by the author (in collaboration with other researchers); the extensive use of discrete group actions is emphasised. We also deal with algorithmic and implementation details, extend and comment results contained in attached reprints of author’s papers.
... In [20], Orbanić, Pellicer, Pisanski and Tucker (2011), show the 14 symmetry type graphs of edge-transitive maps. Later, in [4] and [5], the complete list of possible symmetry type graphs with at most 6 vertices is determined. ...
A map, as a 2-cell embedding of a graph on a closed surface, is called a k-orbit map if the group of automorphisms (or symmetries) of the map partitions its set of flags into k orbits. Orbani, Pellicer and Weiss studied the effects of operations as medial and trun-cation on k-orbit maps. In this paper we study the possible symmetry types of maps that result from other maps after applying the chamfering operation and we give the number of possible flag-orbits that has the chamfering map of a k-orbit map, even if we repeat this operation t times.
... Dress and Brinkmann give an application to mathematical chemistry in [7]. Orbanić, Pellicer, Pisanski and Tucker (2011), show the 14 symmetry type graphs of edge-transitive maps, [16]. Edge-transitive maps were studied in [13] by Siran, Tucker and Watkins. ...
There are operations that transform a map M (an embedding of a graph on a
surface) into another map in the same surface, modifying its structure and
consequently its set of flags F(M). For instance, by truncating all the
vertices of a map M, each flag in F(M) is divided into three flags of the
truncated map. Orbanic, Pellicer and Weiss studied the truncation of k-orbit
maps for k < 4. They introduced the notion of T-compatible maps in order to
give a necessary condition for a truncation of a k-orbit map to be either k-,
3k/2- or 3k-orbit map. Using a similar notion, by introducing an appropriate
partition on the set of flags of the maps, we extend the results on truncation
of k-orbit maps for k < 8 and k=9.
... Thus, for a reflexible maniplex, the connection group and the symmetry group are isomorphic. For more on symmetry in maps see [15,16]. ...
This paper introduces the idea of a maniplex, a common generalization of map and of polytope. The paper then discusses operators, orientability, symmetry and the action of the symmetry group.
The automorphism group of a map on a surface acts naturally on its flags (triples of incident vertices, edges, and faces). We will study the action of the automorphism group of a map on its edges. A map is semi- equivelar if all of its vertices have the same type of face-cycles. A semi-equivelar toroidal map refers to a semi- equivelar map embedded on a torus. If a map has k edge orbits under its automorphism group, it is referred to as a k-edge orbital or k-orbital. Specifically, it is referred to as an edge-transitive map if k = 1. If any two edges have the same edge-symbol, a map is said to be edge-homogeneous. Every edge-homogeneous toroidal map has an edge-transitive cover, as proved in [A. Orbanić, D. Pellicer, T. Pisanski and T. W. Tucker, Edge-transitive maps of low genus, Ars Math. Contemp. 4 (2011), no. 2, 385–402]. In this article, we show the existence and give a classification of k-edge covers of semi-equivelar toroidal maps.
Based on the procedure given in [15] we describe an algorithm, implemented in a computer program, for complete enumeration of combinatorial equivalence classes of fundamental polygons for any fixed plane discontinuous group given by its signature. This is a solution of a long standing problem, we call it Poincaré-Delone problem to honour of Henry Poincaré and Boris Nikolaevich Delone (Delaunay). We give examples and computations together with some complete lists of combinatorially different polygons which serve as fundamental domains for the groups with the Macbeath signatures, e.g. \((2,+,[\ ];\{\,\})\),\((3,-,[\ ];\{\,\})\) and \((3,+,[\ ];\{\,\})\).
Let ${\cal M}$ be a map with the underlying graph $\Gamma$. The automorphism group $Aut({\cal M})$ induces a natural action on the set of all vertex-edge-face incident triples, called {\em flags} of ${\cal M}$. The map ${\cal M}$ is said to be a {\em $k$-orbit} map if $Aut({\cal M})$ has $k$ orbits on the set of all flags of ${\cal M}$. It is known that there are seven different classes of $2$-orbit maps, with only four of them corresponding to arc-transitive maps, that is maps for which $Aut{\cal M}$ acts arc-transitively on the underlying graph $\Gamma$. The Petrie dual operator links these four classes in two pairs, one of which corresponds to the chiral maps and their Petrie duals. In this paper we focus on the other pair of classes of $2$-orbit arc-transitive maps. We investigate the connection of these maps to consistent cycles of the underlying graph with special emphasis on such maps of smallest possible valence, namely $4$. We then give a complete classification of such maps whose underlying graphs are arc-transitive Rose Window graphs.
