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Symmetry
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Symmetry in Combinatorial Structures
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Symmetry in Combinatorial Structures

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 April 2025) | Viewed by 9558

Special Issue Editors

Prof. Dr. Jürgen Bokowski

E-Mail Website
Guest Editor
Department Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany
Interests: computational geometry; combinatorial geometry; discrete mathematics
Prof. Dr. Isabel Hubard

E-Mail Website
Guest Editor
Institute of Mathematics, National Autonomous University of Mexico (IM UNAM), 04510 Mexico City, Mexico
Interests: abstract polytopes; symmetries of discrete objects; polyhedra

Special Issue Information

Dear Colleagues,

The Special Issue 'Symmetry in Combinatorial Structures' should be related to geometric objects, for which their abstract properties are given with combinatorial automorphisms, and their existence (perhaps with geometric symmetries) has to be decided. Here are some examples of possible topics.

Regular maps (a source of symmetry groups and a natural generalization of Platonic solids) and their questionable topological or polyhedral embeddings in 3-space.

Symmetric abstract point-line configurations in the sense of Branco Grünbaum and their embeddings.

Possible additional candidates of regular maps for regular Leonardo polyhedra.

Non-convex 2-spheres with properties like Archimedean bodies similar to the union of the 15 truncated cubes depicted above (refer to the graphical abstract).

Please click on the following video link for more details: https://youtu.be/AailDBdIPVY.

We invite contributions (original research and review articles) covering a broad range of topics around combinatorial and geometric symmetries.

Prof. Dr. Jürgen Bokowski
Prof. Dr. Isabel Hubard
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

 

Keywords

  • combinatorial and geometrical automorphisms of surfaces, sphere systems and manifolds
  • graph theory
  • embeddings
  • surfaces and manifolds
  • discrete geometry
  • symmetry groups
  • geometric polyhedra
  • regular maps

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Published Papers (5 papers)

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Research

56 pages, 16345 KiB  
Article
Polyhedral Embeddings of Triangular Regular Maps of Genus g, 2 ⩽ g ⩽ 14, and Neighborly Spatial Polyhedra
by Jürgen Bokowski and Kevin H.
Symmetry 2025, 17(4), 622; https://doi.org/10.3390/sym17040622 - 19 Apr 2025
Viewed by 4745
Abstract
This article provides a survey of polyhedral embeddings of triangular regular maps of genus g, 2⩽g⩽14, and of neighborly spatial polyhedra. An old conjecture of Grünbaum from 1967, although disproved in 2000, lies behind this investigation. We [...] Read more.
This article provides a survey of polyhedral embeddings of triangular regular maps of genus g, 2⩽g⩽14, and of neighborly spatial polyhedra. An old conjecture of Grünbaum from 1967, although disproved in 2000, lies behind this investigation. We discuss all duals of these polyhedra as well, whereby we accept, e.g., the Szilassi torus with its non-convex faces to be a dual of the Möbius torus. A numerical optimization approach by the second author for finding such embeddings was first applied to finding (unsuccessfully) a dual polyhedron of one of the 59 closed oriented surfaces with the complete graph of 12 vertices as their edge graph. The same method has been successfully applied for finding polyhedral embeddings of triangular regular maps of genus g, 2⩽g⩽14. The effectiveness of the new method has led to ten additional new polyhedral embeddings of triangular regular maps and their duals. There do exist symmetrical polyhedral embeddings of all triangular regular maps with genus g, 2⩽g⩽14, except in a single undecided case of genus 13. Among these results, there are three new Leonardo polyhedra, each with 156 vertices, 546 edges, and 364 triangular faces, based on the Hurwitz triplet of genus 14 with Conder notation R14.1, R14.2, and R14.3. Full article
(This article belongs to the Special Issue Symmetry in Combinatorial Structures)
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12 pages, 33470 KiB  
Article
On Symmetrical Equivelar Polyhedra of Type {3, 7} and Embeddings of Regular Maps
by Jürgen Bokowski
Symmetry 2024, 16(10), 1273; https://doi.org/10.3390/sym16101273 - 27 Sep 2024
Cited by 1 | Viewed by 882
Abstract
A regular map is an abstract generalization of a Platonic solid. It describes a group, a topological cell decomposition of a 2-manifold of type {p, q} with only p-gons, such that q of them meet at each vertex [...] Read more.
A regular map is an abstract generalization of a Platonic solid. It describes a group, a topological cell decomposition of a 2-manifold of type {p, q} with only p-gons, such that q of them meet at each vertex in a circular manner, and we have maximal combinatorial symmetry, expressed by the flag transitivity of the symmetry group. On the one hand, we have articles on topological surface embeddings of regular maps by F. Razafindrazaka and K. Polthier, C. Séquin, and J. J. van Wijk.On the other hand, we have articles with polyhedral embeddings of regular maps by J. Bokowski and M. Cuntz, A. Boole Stott, U. Brehm, H. S. M. Coxeter, B. Grünbaum, E. Schulte, and J. M. Wills. The main concern of this partial survey article is to emphasize that all these articles should be seen as contributing to the common body of knowledge in the area of regular map embeddings. This article additionally provides a method for finding symmetrical equivelar polyhedral embeddings of type {3, 7} based on symmetrical graph embeddings on convex surfaces. Full article
(This article belongs to the Special Issue Symmetry in Combinatorial Structures)
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20 pages, 12496 KiB  
Article
Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs
by M. A. Reyes, C. Dalfó and M. A. Fiol
Symmetry 2024, 16(9), 1135; https://doi.org/10.3390/sym16091135 - 2 Sep 2024
Viewed by 1091
Abstract
The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), [...] Read more.
The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. In the case of mixed graphs, we can have edges (without direction) and arcs (with direction). The chordal ring and chordal ring mixed graphs are bipartite and 3-regular. They consist of a number r (for r≥1) of (undirected or directed) cycles with some edges (the chords) joining them. In particular, for CMR, when r=1, that is, with only one undirected cycle, we obtain the known families of chordal ring graphs. Here, we used plane tessellations to represent our chordal multi-ring graphs. This allowed us to obtain their maximum number of vertices for every given diameter. Additionally, we computationally obtained their minimum diameter for any value of the number of vertices. Moreover, when seen as a lift graph (also called voltage graph) of a base graph on Abelian groups, we obtained closed formulas for the spectrum, that is, the eigenvalue multi-set of its adjacency matrix. Full article
(This article belongs to the Special Issue Symmetry in Combinatorial Structures)
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18 pages, 340 KiB  
Article
The p-Frobenius Number for the Triple of the Generalized Star Numbers
by Ruze Yin, Jiaxin Mu and Takao Komatsu
Symmetry 2024, 16(8), 1090; https://doi.org/10.3390/sym16081090 - 22 Aug 2024
Viewed by 1212
Abstract
In this paper, we give closed-form expressions of the p-Frobenius number for the triple of the generalized star numbers an(n−1)+1 for an integer a≥4. When a=6, it is [...] Read more.
In this paper, we give closed-form expressions of the p-Frobenius number for the triple of the generalized star numbers an(n−1)+1 for an integer a≥4. When a=6, it is reduced to the famous star number. For the set of given positive integers {a1,a2,…,ak}, the p-Frobenius number is the largest integer N whose number of non-negative integer representations N=a1x1+a2x2+⋯+akxk is at most p. When p=0, the 0-Frobenius number is the classical Frobenius number, which is the central topic of the famous linear Diophantine problem of Frobenius. Full article
(This article belongs to the Special Issue Symmetry in Combinatorial Structures)
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13 pages, 322 KiB  
Article
The Symmetry Group of the Grand Antiprism
by Barry Monson
Symmetry 2024, 16(8), 1071; https://doi.org/10.3390/sym16081071 - 19 Aug 2024
Viewed by 846
Abstract
The grand antiprism A is an outlier among the uniform 4-polytopes, since it is not obtainable from Wythoff’s construction. Its symmetry group G(A) has been incorrectly described as [[10,2+,10]] or even [...] Read more.
The grand antiprism A is an outlier among the uniform 4-polytopes, since it is not obtainable from Wythoff’s construction. Its symmetry group G(A) has been incorrectly described as [[10,2+,10]] or even as an ‘ionic diminished Coxeter group’. In fact, G(A) is another group of order 400, namely the group ±[D10×D10]·2, in the notation of Conway and Smith. We explain all this and so correct a persistent error in the literature. This fresh look at the beautiful geometry of the polytope A is also a fine opportunity to introduce the reader to the elegance of Wythoff’s construction and to the less familiar use of quaternions to classify the finite 4-dimensional isometry groups. Full article
(This article belongs to the Special Issue Symmetry in Combinatorial Structures)
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