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23 pages, 925 KiB

Open AccessArticle

Bi-Symmetric Polyhedral Cages with Nearly Maximally Connected Faces and Small Holes

by Bernard Piette

Symmetry 2025, 17(6), 940; https://doi.org/10.3390/sym17060940 - 12 Jun 2025

Viewed by 307

Polyhedral cages (p-cages) provide a good description of the geometry of some families of artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces/protein rings, where each face has at least four neighbours and where [...] Read more.

Polyhedral cages (p-cages) provide a good description of the geometry of some families of artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces/protein rings, where each face has at least four neighbours and where the holes are contributed by at most four faces. We start the construction from a planar graph made out of two families of equivalent nodes. We construct the dual of the solid corresponding to that graph, and we tile its faces with regular or nearly regular polygons. We define an energy function describing the amount of irregularity of the p-cages, which we then minimise using a simulated annealing algorithm. We analyse over 600,000 possible geometries but restrict ourselves to p-cages made out of faces with deformations not exceeding 10%. We then present graphically some of the most promising geometries for protein nanocages. Full article

(This article belongs to the Special Issue Chemistry: Symmetry/Asymmetry—Feature Papers and Reviews)

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58 pages, 16345 KiB

Open AccessArticle

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 5313

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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33 pages, 3753 KiB

Open AccessArticle

Matching Polynomials of Symmetric, Semisymmetric, Double Group Graphs, Polyacenes, Wheels, Fans, and Symmetric Solids in Third and Higher Dimensions

by Krishnan Balasubramanian

Symmetry 2025, 17(1), 133; https://doi.org/10.3390/sym17010133 - 17 Jan 2025

Cited by 1 | Viewed by 1689

Abstract

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and [...] Read more.

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and combinatorial complexity of the problem. We also consider a series of recursive graphs possessing symmetries such as D_2h-polyacenes, wheels, and fans. The double group graphs of the Möbius types, which find applications in chemically interesting topologies and stereochemistry, are considered for the matching polynomials. Hence, the present study features a number of vertex- or edge-transitive regular graphs, Archimedean solids, truncated polyhedra, prisms, and 4D and 5D polyhedra. Such polyhedral and Möbius graphs present stereochemically and topologically interesting applications, including in chirality, isomerization reactions, and dynamic stereochemistry. The matching polynomials of these systems are shown to contain interesting combinatorics, including Stirling numbers of both kinds, Lucas polynomials, toroidal tree-rooted map sequences, and Hermite, Laguerre, Chebychev, and other orthogonal polynomials. Full article

(This article belongs to the Collection Feature Papers in Chemistry)

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20 pages, 731 KiB

Open AccessArticle

Bi-Symmetric Polyhedral Cages with Maximally Connected Faces and Small Holes

by Bernard Piette and Árpad Lukács

Symmetry 2025, 17(1), 101; https://doi.org/10.3390/sym17010101 - 10 Jan 2025

Cited by 2 | Viewed by 740

Abstract

Polyhedral cages (p-cages) describe the geometry of some families of artificial protein cages. We identify the p-cages made out of families of equivalent polygonal faces such that the faces of one family have five neighbors and

P_{1}

edges, while those of the [...] Read more.

Polyhedral cages (p-cages) describe the geometry of some families of artificial protein cages. We identify the p-cages made out of families of equivalent polygonal faces such that the faces of one family have five neighbors and

P_{1}

edges, while those of the other family have six neighbors and

P_{2}

edges. We restrict ourselves to polyhedral cages where the holes are adjacent to four faces at most. We characterize all p-cages with a deformation of the faces, compared to regular polygons, not exceeding 10%. Full article

(This article belongs to the Section Mathematics)

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12 pages, 33470 KiB

Open AccessArticle

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 933

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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18 pages, 2622 KiB

Open AccessReview

Automorphism Groups in Polyhedral Graphs

by Modjtaba Ghorbani, Razie Alidehi-Ravandi and Matthias Dehmer

Symmetry 2024, 16(9), 1157; https://doi.org/10.3390/sym16091157 - 5 Sep 2024

Cited by 1 | Viewed by 2048

Abstract

The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, [...] Read more.

The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, as well as the mathematical structure of the automorphism group, which captures all of the symmetries of a graph. The paper also discusses the significance of groups in Abstract Algebra and their relevance to understanding the behavior of mathematical systems. Overall, the findings offer an inclusive understanding of the relationship between symmetry groups and automorphism groups, paving the way for further research in this area. Full article

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22 pages, 693 KiB

Open AccessArticle

Biequivalent Planar Graphs

by Bernard Piette

Axioms 2024, 13(7), 437; https://doi.org/10.3390/axioms13070437 - 28 Jun 2024

Cited by 3 | Viewed by 950

Abstract

We define biequivalent planar graphs, which are a generalisation of the uniform polyhedron graphs, as planar graphs made out of two families of equivalent nodes. Such graphs are required to identify polyhedral cages with geometries suitable for artificial protein cages. We use an [...] Read more.

We define biequivalent planar graphs, which are a generalisation of the uniform polyhedron graphs, as planar graphs made out of two families of equivalent nodes. Such graphs are required to identify polyhedral cages with geometries suitable for artificial protein cages. We use an algebraic method, which is followed by an algorithmic method, to determine all such graphs with up to 300 nodes each with valencies ranging between three and six. We also present a graphic representation of every graph found. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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13 pages, 251 KiB

Open AccessArticle

Efficiency of Various Tiling Strategies for the Zuker Algorithm Optimization

by Piotr Blaszynski, Marek Palkowski, Wlodzimierz Bielecki and Maciej Poliwoda

Mathematics 2024, 12(5), 728; https://doi.org/10.3390/math12050728 - 29 Feb 2024

Viewed by 1428

Abstract

This paper focuses on optimizing the Zuker RNA folding algorithm, a bioinformatics task with non-serial polyadic dynamic programming and non-uniform loop dependencies. The intricate dependence pattern is represented using affine formulas, enabling the automatic application of tiling strategies via the polyhedral method. Three [...] Read more.

This paper focuses on optimizing the Zuker RNA folding algorithm, a bioinformatics task with non-serial polyadic dynamic programming and non-uniform loop dependencies. The intricate dependence pattern is represented using affine formulas, enabling the automatic application of tiling strategies via the polyhedral method. Three source-to-source compilers—PLUTO, TRACO, and DAPT—are employed, utilizing techniques such as affine transformations, the transitive closure of dependence relation graphs, and space–time tiling to generate cache-efficient codes, respectively. A dedicated transpose code technique for non-serial polyadic dynamic programming codes is also examined. The study evaluates the performance of these optimized codes for speed-up and scalability on multi-core machines and explores energy efficiency using RAPL. The paper provides insights into related approaches and outlines future research directions within the context of bioinformatics algorithm optimization. Full article

(This article belongs to the Special Issue Numerical Algorithms: Computer Aspects and Related Topics)

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19 pages, 3759 KiB

Open AccessArticle

Effect of Calcium and Fullerene Symmetry Spatial Minimization on Angiogenesis

by Manuel Rivas and Manuel Reina

Symmetry 2024, 16(1), 55; https://doi.org/10.3390/sym16010055 - 31 Dec 2023

Viewed by 1463

Abstract

The topological partition theory states that icosahedral group affine extensions (fullerenes symmetry) are the most effective way to energetically optimize the surface covering. In recent decades, potential applications of fullerene symmetry have emerged in the major fields of biology, like enzyme inhibition and [...] Read more.

The topological partition theory states that icosahedral group affine extensions (fullerenes symmetry) are the most effective way to energetically optimize the surface covering. In recent decades, potential applications of fullerene symmetry have emerged in the major fields of biology, like enzyme inhibition and antiviral therapy. This research suggests a novel perspective to interpret the underlying spatial organization of cell populations in tissues from the polyhedral graph theory. We adopted this theoretical framework to study HUVEC cell in vitro angiogenesis assays on Matrigel. This work underscores the importance of extracellular

{C a}^{2 +}

gradients, both from conditioned BJ and pretreated HUVEC cells, in angiogenesis fullerene-rule spatial minimization. Full article

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Graphical abstract

29 pages, 756 KiB

Open AccessArticle

Near-Miss Bi-Homogenous Symmetric Polyhedral Cages

by Bernard Piette and Árpad Lukács

Symmetry 2023, 15(9), 1804; https://doi.org/10.3390/sym15091804 - 21 Sep 2023

Cited by 4 | Viewed by 1686

Abstract

Following the discovery of an artificial protein cage with a paradoxical geometry, we extend the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages, for which all the faces are equivalent, and define bi-homogeneous symmetric polyhedral cages made of two different types of [...] Read more.

Following the discovery of an artificial protein cage with a paradoxical geometry, we extend the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages, for which all the faces are equivalent, and define bi-homogeneous symmetric polyhedral cages made of two different types of faces, where all the faces of a given type are equivalent. We parametrise the possible connectivity configurations for such cages, analytically derive p-cages that are regular, and numerically compute near-symmetric p-cages made of polygons with 6 to 18 edges and with deformation not exceeding 10%. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Nature-Inspired, Bio-Based Materials)

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28 pages, 825 KiB

Open AccessEditor’s ChoiceArticle

Near-Miss Symmetric Polyhedral Cages

by Bernard M. A. G. Piette and Árpad Lukács

Symmetry 2023, 15(3), 717; https://doi.org/10.3390/sym15030717 - 13 Mar 2023

Cited by 6 | Viewed by 3033

Abstract

Following the experimental discovery of several nearly symmetric protein cages, we define the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages made out of P-gons. We use group theory to parameterize the possible configurations and we minimize the irregularity of the P-gons [...] Read more.

Following the experimental discovery of several nearly symmetric protein cages, we define the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages made out of P-gons. We use group theory to parameterize the possible configurations and we minimize the irregularity of the P-gons numerically to construct all such polyhedral cages for

P = 6

to

P = 20

with deformation of up to 10%. Full article

(This article belongs to the Section Mathematics)

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22 pages, 1621 KiB

Open AccessFeature PaperArticle

Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group

by Modjtaba Ghorbani, Mardjan Hakimi-Nezhaad, Matthias Dehmer and Xueliang Li

Symmetry 2020, 12(9), 1411; https://doi.org/10.3390/sym12091411 - 25 Aug 2020

Cited by 1 | Viewed by 2582

Abstract

The Graovac–Pisanski (GP) index of a graph is a modified version of the Wiener index based on the distance between each vertex x and its image

α (x)

, where

α

is an automorphism of graph. The aim of this paper [...] Read more.

The Graovac–Pisanski (GP) index of a graph is a modified version of the Wiener index based on the distance between each vertex x and its image

α (x)

, where

α

is an automorphism of graph. The aim of this paper is to compute the automorphism group of some classes of cubic polyhedral graphs and then we determine their Wiener index. In addition, we investigate the GP-index of these classes of graphs. Full article

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30 pages, 2291 KiB

Open AccessArticle

A Survey on Symmetry Group of Polyhedral Graphs

by Modjtaba Ghorbani, Matthias Dehmer, Shaghayegh Rahmani and Mina Rajabi-Parsa

Symmetry 2020, 12(3), 370; https://doi.org/10.3390/sym12030370 - 2 Mar 2020

Cited by 6 | Viewed by 3481

Abstract

Every three-connected simple planar graph is a polyhedral graph and a cubic polyhedral graph with pentagonal and hexagonal faces is called as a classical fullerene. The aim of this paper is to survey some results about the symmetry group of cubic polyhedral graphs. [...] Read more.

Every three-connected simple planar graph is a polyhedral graph and a cubic polyhedral graph with pentagonal and hexagonal faces is called as a classical fullerene. The aim of this paper is to survey some results about the symmetry group of cubic polyhedral graphs. We show that the order of symmetry group of such graphs divides 240. Full article

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15 pages, 871 KiB

Open AccessArticle

Symmetry and Correspondence of Algorithmic Complexity over Geometric, Spatial and Topological Representations

by Hector Zenil, Narsis A. Kiani and Jesper Tegnér

Entropy 2018, 20(7), 534; https://doi.org/10.3390/e20070534 - 18 Jul 2018

Cited by 2 | Viewed by 5477

Abstract

We introduce a definition of algorithmic symmetry in the context of geometric and spatial complexity able to capture mathematical aspects of different objects using as a case study polyominoes and polyhedral graphs. We review, study and apply a method for approximating the algorithmic [...] Read more.

We introduce a definition of algorithmic symmetry in the context of geometric and spatial complexity able to capture mathematical aspects of different objects using as a case study polyominoes and polyhedral graphs. We review, study and apply a method for approximating the algorithmic complexity (also known as Kolmogorov–Chaitin complexity) of graphs and networks based on the concept of Algorithmic Probability (AP). AP is a concept (and method) capable of recursively enumerate all properties of computable (causal) nature beyond statistical regularities. We explore the connections of algorithmic complexity—both theoretical and numerical—with geometric properties mainly symmetry and topology from an (algorithmic) information-theoretic perspective. We show that approximations to algorithmic complexity by lossless compression and an Algorithmic Probability-based method can characterize spatial, geometric, symmetric and topological properties of mathematical objects and graphs. Full article

(This article belongs to the Special Issue Entropic Aspects Arising from Geometric Descriptions of Physical Phenomena)

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21 pages, 291 KiB

Open AccessArticle

Ricci Curvature on Polyhedral Surfaces via Optimal Transportation

by Benoît Loisel and Pascal Romon

Axioms 2014, 3(1), 119-139; https://doi.org/10.3390/axioms3010119 - 6 Mar 2014

Cited by 29 | Viewed by 5716

Abstract

The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse [...] Read more.

The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu and Yau and Jost and Liu have used and extended this notion for graphs, giving estimates for the curvature and, hence, the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific, but crucial case of polyhedral surfaces. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

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