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Symmetry
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Symmetry and Graph Theory, 2nd Edition
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Symmetry and Graph Theory, 2nd Edition

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

Deadline for manuscript submissions: 30 September 2025 | Viewed by 5257

Special Issue Editors

Dr. Abel Cabrera Martínez

E-Mail Website
Guest Editor
Departamento de Matemáticas, Universidad de Córdoba, Campus de Rabanales, 14071 Córdoba, Spain
Interests: graph theory; discrete mathematics
Special Issues, Collections and Topics in MDPI journals
Dr. Alejandro Estrada-Moreno

E-Mail Website
Guest Editor
Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
Interests: graph theory; applied mathematics; discrete mathematics; computer science
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We warmly invite you to submit original research papers or reviews to this Special Issue entitled “”Symmetry and Graph Theory”, which aims to focus on recent developments in graph theoretical research, particularly in relation to symmetry. Graph theory has widespread applications across various research domains, making this Special Issue a platform for valuable contributions to this significant field. This Special Issue encompasses a range of topics in graph theory, such as metric dimension, topological indices, coloring, domination, independence, Ramsey numbers, and polynomials in graphs, among others. We welcome your valuable insights regarding any of these areas.

Dr. Abel Cabrera Martínez
Dr. Alejandro Estrada-Moreno
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

  • metric dimension of graphs
  • topological indices
  • graph coloring
  • domination in graphs
  • Ramsey number
  • polynomials in graphs

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Related Special Issue

Published Papers (7 papers)

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Research

16 pages, 321 KiB  
Article
Formulas for the Number of Weak Homomorphisms from Paths to Rectangular Grid Graphs
by Penying Rochanakul, Hatairat Yingtaweesittikul and Sayan Panma
Symmetry 2025, 17(4), 497; https://doi.org/10.3390/sym17040497 - 26 Mar 2025
Viewed by 98
Abstract
A weak homomorphism from graph G to graph H is a mapping f:V(G)→V(H), where either f(x)=f(y) or [...] Read more.
A weak homomorphism from graph G to graph H is a mapping f:V(G)→V(H), where either f(x)=f(y) or {f(x),f(y)}∈E(H) hold for all {x,y}∈E(G). A rectangular grid graph is formed by taking the Cartesian product of two paths. Counting weak homomorphisms is a fundamental problem in graph theory. In this paper, we present formulas for calculating the number of weak homomorphisms from paths to rectangular grid graphs. This count directly corresponds to the number of partial walks with length m within the rectangular grid graph, offering a combinatorial solution to this enumeration problem. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)
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16 pages, 1972 KiB  
Article
Edge-Irregular Reflexive Strength of Non-Planar Graphs
by Suleman Khan, Muhammad Waseem Akram, Umar Ishtiaq, Mubariz Garayev and Ioan-Lucian Popa
Symmetry 2025, 17(3), 386; https://doi.org/10.3390/sym17030386 - 4 Mar 2025
Viewed by 379
Abstract
Symmetry in non-planar graphs is a fundamental concept that enhances understanding, simplifies analyses, and has practical implications in diverse fields such as science, engineering, and mathematics. A total κ-labeling for a graph Gνˇ is composed of two labeling: one is [...] Read more.
Symmetry in non-planar graphs is a fundamental concept that enhances understanding, simplifies analyses, and has practical implications in diverse fields such as science, engineering, and mathematics. A total κ-labeling for a graph Gνˇ is composed of two labeling: one is an edge labeling Υe:E(Gνˇ)→{1,2,3,…,κe} and the other is a vertex labeling Υv:V(Gνˇ)→{0,2,4,…,2κv}, where κ=max{κe,2κv}. The weight of an edge under reflexive labeling is defined as wt(pq)=Υv(p)+Υe(pq)+Υv(q)∀e=pq. The total κ−labeling is said to be an edge-irregular reflexive κ−labeling, if for every two edges eδ and ej, the weights are distinct. The lowest value of κ for which the graph Gνˇ has an irregular reflexive edge κ−labeling is called the reflexive edge strength of Gνˇ, denoted as res(Gνˇ). res(Gνˇ) captures irregularity while preserving reflexivity by quantifying edge-weight variability in reflexive graphs. In this article, we are interested in determining the tight lower bound for non-planar prisms, cross prisms, cross-particle modified graphs, and cross-particle glowing graphs under reflexive labeling. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)
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19 pages, 955 KiB  
Article
Resolving the Open Problem by Proving a Conjecture on the Inverse Mostar Index for c-Cyclic Graphs
by Liju Alex and Kinkar Chandra Das
Symmetry 2025, 17(2), 291; https://doi.org/10.3390/sym17020291 - 14 Feb 2025
Viewed by 374
Abstract
Inverse topological index problems involve determining whether a graph exists with a given integer as its topological index. One such index, the Mostar index, Mo(G), is defined as [...] Read more.
Inverse topological index problems involve determining whether a graph exists with a given integer as its topological index. One such index, the Mostar index, Mo(G), is defined as Mo(G)=∑uv∈E(G)|nu(e|G)−nv(e|G)|, where nu(e|G) and nv(e|G) represent the number of vertices closer to vertex u than v and closer to v than u, respectively, for an edge e=uv. The inverse Mostar index problem has gained significant attention recently. In their work, Alizadeh et al. [Solving the Mostar index inverse problem, J. Math. Chem. 62 (5) (2024) 1079–1093] proposed the following open problem: “Which nonnegative integers can be realized as Mostar indices of c-cyclic graphs, for a given positive integer c?”. Subsequently, one of the present authors [On the inverse Mostar index problem for molecular graphs, Trans. Comb. 14 (1) (2024) 65–77] conjectured that, except for finitely many positive integers, all other positive integers can be realized as the Mostar index of a c-cyclic graph, where c≥3. In this paper, we address the inverse Mostar index problem for c-cyclic graphs. Specifically, we construct infinitely many families of symmetric c-cyclic structures, thereby demonstrating a solution to the inverse Mostar index problem using an infinite family of such symmetric structures. By providing a comprehensive proof of the conjecture, we fully resolve this longstanding open problem. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)
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35 pages, 3375 KiB  
Article
Optimization in Symmetric Trees, Unicyclic Graphs, and Bicyclic Graphs with Help of Mappings Using Second Form of Generalized Power-Sum Connectivity Index
by Muhammad Yasin Khan, Gohar Ali and Ioan-Lucian Popa
Symmetry 2025, 17(1), 122; https://doi.org/10.3390/sym17010122 - 15 Jan 2025
Viewed by 638
Abstract
The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. [...] Read more.
The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. The generalized power-sum connectivity index (GPSCI) for the graph is ΩYα(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)+dΩ(ϱ)dΩ(ϱ))α, while the second form of the GPSCI is defined as Yβ(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)×dΩ(ϱ)dΩ(ϱ))β. In this paper, we investigate Yβ in the family of trees, unicyclic graphs, and bicyclic graphs. We determine optimal graphs in the desired families for Yβ using certain mappings. For graphs with maximal values, two mappings are used, namely A and B, while for graphs with minimal values, mapping C and mapping D are considered. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)
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12 pages, 275 KiB  
Article
Graceful Local Antimagic Labeling of Graphs: A Pattern Analysis Using Python
by Luqman Alam, Andrea Semaničová-Feňovčíková and Ioan-Lucian Popa
Symmetry 2025, 17(1), 108; https://doi.org/10.3390/sym17010108 - 12 Jan 2025
Viewed by 652
Abstract
Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results [...] Read more.
Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results and patterns within specific graph classes. The study expands on the existing literature by offering computational evidence, proposing algorithms for the verification of labelings, and exploring the relationship between the local antimagic labeling and the chromatic number. Our results increase the understanding of graph labeling and offer insights into its computational aspects. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)
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25 pages, 2062 KiB  
Article
On Local Fractional Topological Indices and Entropies for Hyper-Chordal Ring Networks Using Local Fractional Metric Dimension
by Shahzad Ali, Shahzaib Ashraf, Shahbaz Ali, Abdullah Afzal and Amal S. Alali
Symmetry 2025, 17(1), 5; https://doi.org/10.3390/sym17010005 - 24 Dec 2024
Viewed by 690
Abstract
An algebraic graph is defined in terms of graph theory as a graph with related algebraic structures or characteristics. If the vertex set of a graph G is a group, a ring, or a field, then G is called an algebraic structure graph. [...] Read more.
An algebraic graph is defined in terms of graph theory as a graph with related algebraic structures or characteristics. If the vertex set of a graph G is a group, a ring, or a field, then G is called an algebraic structure graph. This work uses an algebraic structure graph based on the modular ring Zn, known as a hyper-chordal ring network. The lower and upper bounds of the local fractional metric dimension are computed for certain families of hyper-chordal ring networks. Utilizing the cardinalities of local fractional resolving sets, local fractional resolving (LFR)M-polynomials are computed for hyper-chordal ring networks. Further, new topological indices based on (LFR)M-polynomials are established for the proposed networks. The local fraction entropies are developed by modifying the first three kinds of Zagreb entropies, which are calculated for the chosen hyper-chordal ring networks. Furthermore, numerical and graphical comparisons are discussed to observe the order between newly computed topological indices. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)
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20 pages, 415 KiB  
Article
Efficient Graph Algorithms in Securing Communication Networks
by Syed Ahtsham Ul Haq Bokhary, Athar Kharal, Fathia M. Al Samman, Mhassen. E. E. Dalam and Ameni Gargouri
Symmetry 2024, 16(10), 1269; https://doi.org/10.3390/sym16101269 - 26 Sep 2024
Cited by 1 | Viewed by 1383
Abstract
This paper presents three novel encryption and decryption schemes based on graph theory that aim to improve security and error resistance in communication networks. The novelty of this work lies in the application of complete bipartite graphs in two of the schemes and [...] Read more.
This paper presents three novel encryption and decryption schemes based on graph theory that aim to improve security and error resistance in communication networks. The novelty of this work lies in the application of complete bipartite graphs in two of the schemes and the Cartesian product of graphs in the third, representing a unique approach to cryptographic algorithm development. Unlike traditional cryptographic methods, these graph-based schemes use structural properties of graphs to achieve robust encryption, providing greater resistance to attacks and corruption. Each scheme is illustrated with detailed examples that show how the algorithms can be successfully implemented. The algorithms are written in standard mathematical notation, making them adaptable for machine implementation and scalable for real-world use. The schemes are also rigorously analyzed and compared in terms of their temporal and spatial complexities, using Big O notation. This comprehensive evaluation focuses on their effectiveness, providing valuable insights into their potential for secure communication in modern networks. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)
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