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Article

Fine-Tuning of Aspects of Chirality by Co-Authorship Networks

by
Béla Barabás
1,
Ottilia Fülöp
2 and
Gyula Pályi
3,*
1
Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
2
Department of Analysis and Operations Research, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
3
Department of Life Sciences, University of Modena and Reggio Emilia, Via Campi 103, I-41125 Modena, Italy
*
Author to whom correspondence should be addressed.
Symmetry 2025, 17(6), 825; https://doi.org/10.3390/sym17060825 (registering DOI)
Submission received: 29 January 2025 / Revised: 28 April 2025 / Accepted: 22 May 2025 / Published: 26 May 2025
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Abstract

:
In the present article, we illustrate and analyze the co-authorship network of Paul G. Mezey, focusing only on his collaborations on chirality-related papers. We consider scientific works from the Web of Science database as of 10 April 2024. Unlike previous studies on co-authorship networks, this network allows parallel edges, indicating multiple collaborations between the scientists involved. We also present a co-authorship network based on articles citing Mezey’s chirality-related papers (excluding self-citations), examining its main communities detected. Publications on the development of the theoretical and mathematical background of the new ideas on chirality are also considered.

Graphical Abstract

1. Introduction

The mathematical aspects of symmetry, or its lack (asymmetry), are not physical phenomena but products of human abstraction. However, symmetry (or asymmetry) has an especially fundamental role in the scientific description of the structure and/or behavior of atomic/molecular-sized objects—that is, of structural and reaction chemistry [1,2,3]. Symmetry can be defined as a structural feature of objects, making them equal to their specular image. This general statement can be refined by the following considerations:
(a)
Symmetry, or asymmetry, can “extend” over the whole object in question or only to its part or parts. This approach leads to the concept of partial symmetry/asymmetry [4,5].
(b)
Symmetry/asymmetry may describe small or large structural differences. This approach leads to measures of symmetry/asymmetry [6,7].
(c)
In a more specific chemistry-related sense, these efforts led to the development of molecular shape analysis [8,9].
(d)
The physical description of molecular size objects needs a refined mathematical apparatus. The mathematical principles/formulae may have a logical behavior, which is very similar to the above description of symmetry/asymmetry. The symmetry/asymmetry of these mathematical formulae has a very important influence on the “technical” handling (either by deductions or by computer-aided calculations) of these formulae, e.g., Refs. [10,11,12].
Such diverse aspects of complexity can provide a useful basis for considering another level of complexity, which can be modeled by networks representing interactions among scientific representations of chirality, as reflected by actual network systems manifested by co-authorships of scientific publications on relevant subjects. In the last few decades, these options of chirality representations underwent fundamental changes. These developments originated from ingenious combinations of principles of mathematics with new experimental observations of structural [13,14,15], preparative [16,17,18,19,20,21], and biological [22] chemistry. Research collaboration brings together the knowledge, expertise, and experience of the participants, generating new and fruitful ideas. Co-authorship networks have been studied from various perspectives, such as the collaboration network of research communities publishing in a particular journal, or which are from a specific country or institution, or even those who cite a certain important study or quote a particular author [23,24,25,26,27,28].
Doubtless, Paul G. Mezey (University of Saskatchewan, Memorial University of Newfoundland, Canada) is one of the key figures of this scientific advancement, which—on the other hand—had a deep-impact influence on one of the main challenges of theoretical, preparative, and biological chemistry: molecular chirality. In the present paper, we describe an attempt at studying the propagation of the modern ideas on chirality by means of a graph-theory-based, and also statistical analysis of chirality-related publications co-authored by Paul G. Mezey. In addition, we show various keyword networks to visualize and interpret the most influential aspects of the keywords of these papers. After investigating these networks, we construct and analyze the co-authorship network determined by the collaborating authors of the set of scientific documents citing the chirality-related papers co-authored by Paul G. Mezey, according to the Web of Science citation service, excluding self-citation. We study the scientific influence of Paul G. Mezey through these collaboration networks, providing some statistical features of these publications using standard social network analysis techniques.
In the present paper, we consider only scientific works from the Web of Science database [29] as of 10 April 2024, which can be found in Tables S1 and S2 of the Supplementary Materials. Note that Paul G. Mezey has also written several articles not included in the WoS database and many non-chirality-related papers with a higher number of references. For example, his co-authored article with J. Pipek, “A Fast Intrinsic Localization Procedure Applicable for ab Initio and Semiempirical Linear Combination of Atomic Orbital Wave Functions” [30], has received over 1571 citations. In this article, we focus specifically on Mezey’s chirality-related papers, presenting the co-authorship network of these papers and the network of citing papers, to study the impact of Mezey’s work on the academic writing on chirality. All the graphs shown here were made using the Igraph package [31], a library of the R programming language for network analysis.

2. Basic Graph Theoretical Concepts

Before presenting the above-mentioned networks, here, we summarize the main graph theoretical definitions used in this paper (here, the terms “network” and “graph” are used interchangeably). We fix the notations used hereafter in the present work following the notations and definitions given in our previous works [24,25,26,27].
A network or a (finite undirected) graph denoted by G = V , E is represented by a set of nodes (or vertices) V and a set of edges  E joining some pairs of nodes. Social network analysis (SNA) is the mapping of relationships between nodes (e.g., people, groups, countries, keywords, or other connected entities) of the studied network. We say that two nodes x and y are adjacent or connected if they are joined by at least one edge x y . A loop is an edge that joins a node to itself. Two or more edges that are incident to the same two nodes are called parallel edges.
A finite undirected graph (or network) is called simple if it does not contain loops or parallel edges. Otherwise, it is called a multigraph.
For simplicity, let us consider a finite undirected graph G = V , E that can contain parallel edges but no loops. Let us suppose that G has n nodes and denote its node set by V = { 1 ,   2 ,   . . . , n } . A subset of nodes of a graph is called a clique if every two nodes of the subset are adjacent. A clique with the largest possible number of nodes is called maximum clique of the graph.
The adjacency matrix A of the graph G is a symmetric square matrix with elements A i j = 0 , if there is no edge joining nodes i and j of the graph, and A i j = p , if there are exactly p edges joining nodes i and j .
The degree (or degree centrality) of a node i (denoted by deg i ) is the number of edges incident to i , where loops are counted twice. Thus, in a multigraph without loops, we have deg i = ∑ j = 1 n A i j , where n denotes the number of nodes in the graph and A i j are the elements of the adjacency matrix.
The average degree of a network with n nodes and e edges provides information about the number of edges compared to the number of nodes. Each edge is incident to two nodes and counts in the degree of both nodes; thus, the average degree of an undirected network is 2 e / n . Nodes with more connections tend to be more influential; thus, the degree is often a very effective measure of the “influence” of the nodes in any network.
Let λ denote the largest positive eigenvalue of the adjacency matrix A and x denote the corresponding eigenvector, with the property that its maximum coordinate is equal to 1. The eigenvector centrality  x i of a node i is defined as the i -th coordinate of the eigenvector x . The correctness of the definition of eigenvector centrality is based on an extension of the Perron–Frobenius theorem to matrices with non-negative elements, which guarantees the existence of the above eigenvector with non-negative coordinates [31,32]. Eigenvector centrality acknowledges that not all neighbors of node i are equivalent, and connection to nodes that are themselves influential results in an even more affecting node.
The density of node set S ⊆ V of a graph G = V , E is the number of edges that connect nodes from S divided by S · ( S − 1 ) 2 . In the case of graphs with parallel edges, the density can be higher than 1; in the case of simple graphs, it belongs to the interval [ 0 , 1 ] .
Unlike in our previous papers [24,25,26,27], in this paper, we present networks that are finite undirected multigraphs, providing insights into the links between nodes. For example, when considering the degree or the eigenvector centrality of a node, or when calculating the average degree of a network, it matters if two nodes are connected by a single edge or a certain number of parallel edges; thus, we do not introduce edge-weighted simple graphs instead of multigraphs. All the multigraphs in this paper do not contain loops.
Communities or clusters of a network are densely connected subgraphs of the network. Nodes with a central position in their communities play an important role within the mentioned communities. All the communities of the graphs in this paper were found using the leading eigenvector algorithm of the Igraph package, based on the leading eigenvector method developed by M. E. J. Newman [32,33]. This method uses eigenvalues and eigenvectors of a matrix called the modularity matrix, calculates the largest positive eigenvalue of the modularity matrix and its eigenvectors, and separates nodes into communities based on the sign of the corresponding elements in the eigenvectors [31,32,33].

3. The Co-Authorship Network of Paul G. Mezey’s Chirality-Related Papers

Let us call any scientific document (article, book, book chapter) from the Web of Science database [29] an M-paper if it is related to chirality and is authored or co-authored by Paul G. Mezey. The total number of M-papers is 163, of which 80 are single-author papers and 83 are co-authored papers. The node set of the co-authorship network constructed with the Igraph package is formed using all 56 scientists who have at least one M-paper.
Paul G. Mezey is also added to the node set of this network; thus, the network has 57 nodes. Two nodes are linked by p edges in this network if and only if the corresponding authors have exactly p different joint M-papers. (The absence of an edge between two scientists in this network does not necessarily imply that they have no chirality-related joint works; it just means that they do not have any joint M-papers.)
In the first step of our analysis, we mapped the co-authorship network of the M-papers showing Paul G. Mezey’s serious scientific collaboration on the theory of chirality. This network has 270 edges; its average degree is approximately 9.39, with a density of approximately 0.17. The network is shown in Figure 1.
The co-authorship network of M-papers is colored according to communities that were detected by the leading eigenvector algorithm, which found six communities. The largest has 23 nodes (authors), and it is shown in orange in Figure 1. The second largest community has 20 nodes, which is shown in blue, and the third largest community, colored yellow, has 5 nodes.
The graph has one maximum clique that contains eight nodes: P. G. Mezey, P. D. Walker, P. L. Warburton, Z. Zimpel, D. G. Irvine, D. G. Dixon, X. D. Huang, and B. M. Greenberg. This clique can be seen in Figure 2.
The co-authorship network of M-papers indicates a cooperation with a high number of scientists interested in chirality problems. Here, we mention that G. A. Arteca has 25 co-authored chirality-related scientific works with P. G. Mezey, and P. D. Walker has 14. These three authors have the highest eigenvector centralities: P. G. Mezey (1.00), G. A. Arteca (0.73), and P. D. Walker (0.48).
Other frequent co-authors of P. G. Mezey are S. Arimoto, (with six joint M-papers), P. L. Warburton (with six), and Z. Zimpel (with six).
The papers cited under [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71] are products of numerous collaborations, as mentioned earlier. These contain the step-by-step construction of an entirely new way of viewing the structure of molecules, in terms of the quantified shape and quantified similarity of the structure of molecules. These new viewpoints are built based on the classical concepts of conformation [34,35] and configuration [36]. The shape concept was defined [37,38,39,40,41] and described (in more detail) in terms of electron density [42,43,44,45,46], electrostatic potential [47,48], or molecular charge densities [49]. This kind of discussion enabled us to consider the fine details of the molecular structure as “roughness” of cross sections of molecular surfaces [50], as well as to define the concept (and degree) of molecular similarity [35,51,52]. The molecular shape could also be discussed in terms of “shape groups” [35,51]. The shape-based discussion of molecules enabled us to follow the stereochemical course of chemical reactions [53,54,55,56,57]. All these studies culminated in the definition of measures of chirality [58,59] and the development of the holographic electron density theorem [10], which enables an entirely new approach to problems of chirality.
Some publications by the Mezey group dealt with the shape-based description of some specific achiral molecules [60,61,62,63,64,65], which also supported research efforts centered on achiral aromatic molecules, for example, for analyzing fragments of drug-precursor molecules [66] or predicting the toxicity of polycyclic aromatic hydrocarbons [65]. Beyond the practically useful results served, from the viewpoint of the present review, these studies also assisted in the “sharpening” of the mathematical apparatus and the refining of the physical background behind the new kinds of modeling efforts.
Other important publications coming from the cliques described above dealt with biomolecules, which have a chiral structure (e.g., various proteins [58,67,68,69], bovine insulin [70], taxol and HIV-1 protease [71]). Other models analyzed by the new mathematical/physical approaches included carboxypeptidase-A inhibitors and chloroplast ribosomal proteins [67], liver alcohol dehydrogenase inhibitors [48], and histamine H-2 receptor agonists; nicotine agonists were analyzed with special attention to shape vs. activity and shape vs. similarity relations [51]. It should be mentioned that the descriptions of chain molecules (proteins) by the molecular shape concept [68,69,70] contributed to the roots of the theory used for “protein structure prediction” by Demis Hassabis and John M. Jumper, which became highly recognized recently [72].
It is worth mentioning that in the two papers where the Arteca/Mezey and Walker/Mezey cliques “merge” [46,49], both communications deal with the description of molecular shape in terms of electronic charge densities. Thus, it can be supposed that this kind of approach was thought to be a very important element of the new type of description of molecules.
The degree of a node in this network shows the number of all co-authors of M-papers considered with multiplicity. Some of the highest degree nodes in this network are P. G. Mezey (145), G. A. Arteca (42), P. D. Walker (34), P. L. Warburton (23), Z. Zimpel (22), D. G. Dixon (17), and S. Arimoto (15). There are only six nodes with degree 1.
Figure 3 shows the co-authorship network of M-papers obtained by omitting the main node, Paul G. Mezey, and the edges incident to this node. This network contains 56 nodes (9 of them isolated) and 124 edges.
Here, we mention only the three largest communities: the largest one contains 14 nodes, and it is shown in purple in Figure 3. The second-largest community has 10 nodes, which are shown in orange, and the third-largest community, colored yellow, has 6 nodes. In this co-authorship network, the two nodes connected by the most parallel edges (four edges) are S. Arimoto–K. Fukui, followed by three pairs of nodes connected with three parallel edges, respectively, S. Arimoto–K. F. Taylor, D. G. Dixon–Z. Zimpel, and K. Fukui–K. F. Taylor. The joint effort leading to the idea of the congruity of certain linear operators to chemical network systems (molecules) [39] leads to an exceptionally deep view of the application of mathematics in chemistry.
If we omit the parallel edges of the co-authorship network in Figure 1 (considering a simple graph for better visibility), then the 57 nodes of the simple graph version of the co-authorship network of M-papers are connected by 149 edges, resulting an average degree of approximately 5.23 and a density of approximately 0.09, as shown in Figure 4. Here, the highest degree nodes are almost the same: P. G. Mezey (56), G. A. Arteca (15), P. D. Walker (14), P. L. Warburton (14), Z. Zimpel (12), D. G. Dixon (10), and B. M. Greenberg (9). There are only nine nodes with degree 1.
In this graph, the three highest eigenvector centralities belong to P. G. Mezey (1.00), P. D. Walker (0.44) and P. L. Warburton (0.42).

4. Keyword Network Analysis of the M-Papers

The author keywords of the M-papers provide a concise overview of the M-paper’s main ideas and content. Papers with many keywords in common tend to cover common topics and issues. Instead of presenting the frequently used two-dimensional density map of the keywords, i.e., a set of keywords without linking, where a more intense color implies a larger number of keywords and higher connectivity in the neighborhood, here, we want to look more closely at the keywords and at the connections between them. For more visibility, we provide two types of keyword-related networks for the M-papers.
The node set of the first type of keyword network of the M-papers, presented in Figure 5, contains all the M-papers in which the authors provided keywords, i.e., 58 papers from the total amount of 163. Books, book chapters, and other M-papers without author keywords were not included in the node set of this type of network. For the sake of simplicity, each M-paper was given a number, namely, the number in the table downloaded from the Web of Science database [29]. Thus, the node set in this network consists of only those 58 numbers between 1 and 163, which indicate the M-papers with author keywords in the above-mentioned table. Two nodes are linked by p edges in this network if and only if the corresponding M-papers have exactly p keywords in common.
The more keywords that a paper has in common with the others in this network, the higher the corresponding node degree. The highest degree node in this network is node 111 (i.e., article [73] in the network in Figure 5), with degree 21, closely followed by nodes 123 and 149 (i.e., articles [74,75]), both on rational drug design, both with degree 19. These papers are equally valid for chiral and/or achiral drug research.
The two nodes connected by the most parallel edges (10 edges) are 122 and 127. These nodes represent papers on molecular modeling [76] and on the significance of the “fragmentation principles” in the computer-based treatment of molecules [77], both by P. G. Mezey.
The maximum clique contains 11 nodes, which is shown in Figure 6. Five of these eleven papers were single-authored by P. G. Mezey; the remaining six papers were written by P. G. Mezey in collaboration with at least one of the following scientists: G. A. Arteca, A. Frolov, E. Jako, G. A. Heal, P. D. Walker, M. Ramek, and Q. S. Du. Not surprisingly, here, the most parallel edges also connect the papers with the same authors, that is, the papers corresponding to nodes 43 and 51 (papers [78,79]) and publications corresponding to nodes 123 and 149 (Refs. [74,75]). The content of these articles is pairwise-similar: the publications corresponding to nodes 43 and 51 deal with the possibilities of computer-aided drug design, while those corresponding to nodes 123 and 149 describe details of the shape theory of molecules.
The second type of keyword network (network of co-occurrence of the keywords), presented in Figure 7, shows the number of connections of the keywords: here, the nodes are the 313 keywords of all the M-papers. Two nodes are linked by p edges in this network if and only if the corresponding two keywords co-occur as keywords in exactly p different M-papers. For more visibility, the higher the degree of a node (keyword), the larger its size and lettering, and only the keywords of at least degree 9 are listed by name in the network.
In the second type of keyword network, the following pairs of nodes (keywords) are connected by the maximum number of parallel edges: (holographic electron density theorem, fragment similarity), (fragment similarity, molecular similarity), (molecular similarity, holographic electron density theorem), and (macromolecular quantum chemistry, molecular modeling). Each pair of nodes is connected by three parallel edges; thus, no two keywords in this vast network from Figure 7 occur together in at least four different M-papers.
The most frequent nodes in the second type of keyword network are presented in Table 1.
It should be pointed out that almost all keywords listed in Table 1 can be brought into contact with molecular chirality, or, in other words, these concepts serve the new kinds of geometric/mathematical description of chiral molecules.
The maximum clique in this network contains 10 nodes (keywords), as shown in Figure 8.
The largest community contains 49 nodes (keywords), and it is shown in Figure 9. For more visibility, only the keywords of at least degree 9 are listed by name in this community.

5. The Network of Citing Papers

The co-authorship network based on the citations of the M-papers (in this paper, this network is referred to as the network of citing papers) is a social network determined by a multigraph with a node set formed by all scientists who have at least one work that cites at least one M-paper (according to the citation database of the Web of Science from 10 April 2024) without self-citations. Two nodes of this graph are connected by p edges if the corresponding two authors have exactly p common papers that cite at least one M-paper. According to the Web of Science citation service, there are 933 articles that cite at least one M-paper. Among them, 111 are single-authored articles; we omitted these because isolated nodes are unnecessary in this graph. The network of citing papers has 1623 nodes (authors of the above-mentioned 822 multi-authored citing papers) and 4786 edges (considered with multiplicity). Its degree is approximately 5.9, as shown in Figure 10, which is proof of the profound influence of P. G Mezey’s work on the development of the theory of chirality.
The maximum clique of this graph contains 19 nodes. It is a simple graph because of a joint paper by 19 authors, including V. W. Z. Yu, C. Campos, W. Dawson, A. García, V. Havu, B. Hourahine, W. P. Huhn, M. Jacquelin, W. L. Jia, M. Keçeli, R. Laasner, Y. Z. Li, L. Lin, J. F. Lu, J. Moussa, J. E. Roman, A. Vázquez-Mayagoitia, C. Yang, and V. Blum, which was published in 2020 in Computer Physics Communications under the title ELSI—An open infrastructure for electronic structure solvers [80]. A part of the theoretical background of this paper is Mezey’s publication on quantum similarity measures and Löwdin’s transformation [59].
The degree of a node in the network of citing papers shows the number of all co-authors of all scientific documents that cite at least one M-paper, accounting for multiplicity. Here, again, only studies in the Web of Science citation database are considered. Some of the highest degree nodes are Y. Marrero-Ponce (93), F. Torrens (78), R. Carbo-Dorca (75), X. F. Xu (73), and G.C. Wang (72), the last two scientists having the highest eigenvector centralities in the network (1.00 and 0.99, respectively).

6. The Largest Communities in the Network of Citing Papers

In the network of citing papers, 284 communities were detected by the leading eigenvalue algorithm. Figure 11 and Figure 12 show the two largest communities of the network of citing papers. Both contain 50 scientists, and they are centered around Y. Marrero-Ponce and A. Genoni, respectively.
The nodes of the highest degree in the community in Figure 11, namely, Y. Marrero-Ponce (93) and F. Torrens (78), have the highest eigenvector centrality scores in this community (1.00 and 0.95, respectively). Note that they are also the two highest degree nodes in the network of citing papers with unchanged degrees, which means that all of their neighbors from the network of citing papers are detected in this community.
The nodes of the highest degree in the community in Figure 12 are A. Genoni (41) and M. Sironi (26), with eigenvector centralities 1.00 and 0.74, respectively.
The maximum clique of the community of Y. Marrero-Ponce is presented in Figure 13. It contains 15 nodes due to a joint paper (from 2020) by the following 15 authors: O. M. Rivera-Borroto, Y. Marrero-Ponce, A. Meneses-Marcel, J. A. Escario, A. G. Barrio, V. J. Arán, M. A. M. Alho, D. M. Pereira, J. J. Nogal, F. Torrens, F. Ibarra-Velarde, Y. V. Montenegro, A. Huesca-Guillén, N. Rivera, and C. Vogel [81]. The 15 nodes in this clique are connected by 137 edges. This publication deals with the computer-aided research for new medicines against trichomoniasis using the “Lego” approach of molecule building elaborated by P. D. Walker and P. G. Mezey [69].
An article published in the Journal of Chemical Physics in 2023 [82] provides the maximum clique of the community of A. Genoni, as presented in Figure 14. This clique contains the authors E. Hupf, F. Kleemiss, T. Borrmann, R. Pal, J. M. Krzeszczakowska, M. Woinska, D. Jayatilaka, A. Genoni, and S. Grabowsky, linked by 37 edges. These authors deal with the effects of experimentally obtained electron correlation and polarization; as one of their basic concepts, they use the “Lego” approach of molecule building suggested by Walker and Mezey [69].

7. Conclusions

In the present paper, we presented the co-authorship network of the researchers who have a chirality-related joint paper with Paul G. Mezey. We used the “relation to chirality” in a broader sense, where our analysis also included publications that describe the mathematical and/or physical background of those papers that deal with chiral phenomena and chiral molecules in a closer sense. After using various keyword network analyses, we presented some statistical characteristics of the related publications. Furthermore, by using standard social network analysis techniques, we constructed and analyzed the co-authorship network of all scientific papers that cited these chirality-related papers (excluding self-citation) according to the Web of Science citation service. The study of co-authorship networks and the largest communities can be useful in designing new scientific collaborations, and, consequently, it has a growing importance in all fields of science.
In chemistry, chirality is also dependent on the deformability of molecules. Deformation can cause even formally “achiral” molecules to adopt chiral structures, which, on the other hand, can influence chemical reactions (especially in the sense of “chiral induction”). The mathematical description of these fine structural changes and their effects was hardly possible by the mathematical apparatus used for the description of molecular structure until the last decades of the 20th century.
The new concepts for the description of molecular structures by Mezey and his coworkers enabled the conceptual and quantitative description of “temporarily chiral” or “more–less chiral” molecules in terms of molecular shape, similarity, and symmetry deficiency or by the holographic description of molecular structures (reviews: Refs. [10,11,76,77,83]).
These developments, which were also reached by a number of other scientists, enabled a very sensitive “tuning” of the description of chiral molecules (for an overview, refer to Ref. [84]).
The research described in the present paper is intended to show the interactions of ideas contributing to this evolution by means of graphs and network-theoretical and statistical analyses of the relevant publications.
Beyond the above-outlined developments in the description of the structure of molecules, the related concept of “symmorphy” [85] should be mentioned. In symmorphy, the focus is on morphology as being shifted from the condition of the preservation all of the metric properties of the object, that is, all the point-to-point distances of the given object, as it happens in all symmetry operations, to the preservation of the morphology of the object, that is, to the appearance of the actual shape of the object, while allowing all of those homeomorphic transformations of the entire space that transform points of the object to points of the object, as long as the complete set of points of the object is transformed into the complete set of objects, continuously preserving the morphology of the object. Such homeomorphic transformations, specific for the object, also form an algebraic group, the symmorphy group of the object [85]. Of course, such a symmorphy group, that is, the family of these specific homeomorphisms, is specific for the given object since after applying any one of them to the object, the morphology of the object will appear unchanged, even if the points of the object may get shifted in some continuous way within the object, but no longer in a distance-preserving way, as happens in symmetry operations. If in appearance, the morphology of the object is unchanged, then the given homeomorphism is an element of the given symmorphy group, although the space around the object may get distorted in any drastic, but still continuous way. Such symmorphy groups provide a broader characterization of the shape of objects, including molecules, than symmetry groups. Of course, the symmetry operations of a symmetry group for a given object also belong to the symmorphy transformations of the object, and the relevant symmetry group is a subgroup of the relevant symmorphy group.

Supplementary Materials

The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/sym17060825/s1: Table S1: Table of M-papers; Table S2: Table of citing papers.

Author Contributions

Conceptualization, G.P. and O.F.; methodology, O.F. and B.B.; software, B.B.; validation, B.B.; formal analysis, G.P.; writing—original draft, O.F. and G.P. All authors have read and agreed to the published version of the manuscript.

Funding

The research by Ottilia Fülöp reported in this paper is part of project no. BME-NVA-02, implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021 funding scheme.

Data Availability Statement

The original contributions presented in this study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Co-authorship network of Paul G. Mezey’s chirality-related papers (M-papers) constructed with the Igraph package, version 0.10.8, 2023.
Figure 1. Co-authorship network of Paul G. Mezey’s chirality-related papers (M-papers) constructed with the Igraph package, version 0.10.8, 2023.
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Figure 2. The maximum clique of the co-authorship network of the M-papers.
Figure 2. The maximum clique of the co-authorship network of the M-papers.
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Figure 3. Co-authorship network of M-papers after omitting the node Paul G. Mezey and the edges incident to this main node. Communities were detected by the leading eigenvector algorithm.
Figure 3. Co-authorship network of M-papers after omitting the node Paul G. Mezey and the edges incident to this main node. Communities were detected by the leading eigenvector algorithm.
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Figure 4. Simple graph version of the co-authorship network of M-papers constructed with the Igraph package, version 0.10.8, 2023. The graph is colored according to communities that were detected by the leading eigenvector algorithm.
Figure 4. Simple graph version of the co-authorship network of M-papers constructed with the Igraph package, version 0.10.8, 2023. The graph is colored according to communities that were detected by the leading eigenvector algorithm.
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Figure 5. The first type of keyword network of the M-papers: two papers are linked by p edges if and only if they have exactly p keywords in common.
Figure 5. The first type of keyword network of the M-papers: two papers are linked by p edges if and only if they have exactly p keywords in common.
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Figure 6. Maximum clique of the first type of keyword network.
Figure 6. Maximum clique of the first type of keyword network.
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Figure 7. The second type of keyword network: two keywords are linked by p edges if and only if they co-occur as keywords in exactly p different M-papers.
Figure 7. The second type of keyword network: two keywords are linked by p edges if and only if they co-occur as keywords in exactly p different M-papers.
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Figure 8. Maximum clique of the second type of keyword network.
Figure 8. Maximum clique of the second type of keyword network.
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Figure 9. The largest community of the network of co-occurrence of the keywords detected by the leading eigenvector algorithm.
Figure 9. The largest community of the network of co-occurrence of the keywords detected by the leading eigenvector algorithm.
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Figure 10. The network of citing papers constructed with the Igraph package, version 0.10.8, 2023. The graph is colored according to communities that were detected by the leading eigenvector algorithm.
Figure 10. The network of citing papers constructed with the Igraph package, version 0.10.8, 2023. The graph is colored according to communities that were detected by the leading eigenvector algorithm.
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Figure 11. Community of Y. Marrero-Ponce containing 405 edges.
Figure 11. Community of Y. Marrero-Ponce containing 405 edges.
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Figure 12. Community of A. Genoni containing 199 edges.
Figure 12. Community of A. Genoni containing 199 edges.
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Figure 13. Maximum clique of the community of Y. Marrero-Ponce.
Figure 13. Maximum clique of the community of Y. Marrero-Ponce.
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Figure 14. Maximum clique of the community of A. Genoni.
Figure 14. Maximum clique of the community of A. Genoni.
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Table 1. The most frequent keywords, together with their frequency.
Table 1. The most frequent keywords, together with their frequency.
Author KeywordsFrequency
molecular shape11
holographic electron density theorem8
molecular modeling7
electron density6
macromolecular quantum chemistry5
QSAR (quantitative structure-activity relations)4
shape analysis4
chirality3
fragment similarity3
molecular similarity3
symmetry3
symmetry deficiency3
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Barabás, B.; Fülöp, O.; Pályi, G. Fine-Tuning of Aspects of Chirality by Co-Authorship Networks. Symmetry 2025, 17, 825. https://doi.org/10.3390/sym17060825

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Barabás B, Fülöp O, Pályi G. Fine-Tuning of Aspects of Chirality by Co-Authorship Networks. Symmetry. 2025; 17(6):825. https://doi.org/10.3390/sym17060825

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Barabás, Béla, Ottilia Fülöp, and Gyula Pályi. 2025. "Fine-Tuning of Aspects of Chirality by Co-Authorship Networks" Symmetry 17, no. 6: 825. https://doi.org/10.3390/sym17060825

APA Style

Barabás, B., Fülöp, O., & Pályi, G. (2025). Fine-Tuning of Aspects of Chirality by Co-Authorship Networks. Symmetry, 17(6), 825. https://doi.org/10.3390/sym17060825

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