Symmetry and Graph Theory

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

Deadline for manuscript submissions: closed (17 June 2024) | Viewed by 13174

Special Issue Editors


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

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 extend a warm invitation to submit original research papers or reviews for this Special Issue on "Symmetry and Graph Theory", focusing on recent developments in graph theoretical research, particularly in relation to symmetry. Graph theory has widespread applications across various research domains, making this Issue a platform for valuable contributions to this significant field. The 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. Your valuable insights in any of these areas are highly encouraged.

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

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue polices can be found here.

Related Special Issue

Published Papers (11 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

10 pages, 1215 KiB  
Article
Exploring Ring Structures: Multiset Dimension Analysis in Compressed Zero-Divisor Graphs
by Nasir Ali, Hafiz Muhammad Afzal Siddiqui, Muhammad Imran Qureshi, Suhad Ali Osman Abdallah, Albandary Almahri, Jihad Asad and Ali Akgül
Symmetry 2024, 16(7), 930; https://doi.org/10.3390/sym16070930 - 20 Jul 2024
Viewed by 610
Abstract
This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring R and the associated compressed zero-divisor graph. An undirected graph consisting of [...] Read more.
This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring R and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set Z(RE)\{[0]}=RE\{[0],[1]}, where RE={[x] :xR} and [x]={yR : ann(x)=ann(y)} is called a compressed zero-divisor graph, denoted by ΓER. An edge is formed between two vertices [x] and [y] of Z(RE) if and only if [x][y]=[xy]=[0], that is, iff xy=0. For a ring R, graph G is said to be realizable as ΓER if G is isomorphic to ΓER. We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
Show Figures

Figure 1

18 pages, 1214 KiB  
Article
Characterizations of Minimal Dominating Sets in γ-Endowed and Symmetric γ-Endowed Graphs with Applications to Structure-Property Modeling
by Sakander Hayat, Raman Sundareswaran, Marayanagaraj Shanmugapriya, Asad Khan, Venkatasubramanian Swaminathan, Mohamed Hussian Jabarullah and Mohammed J. F. Alenazi
Symmetry 2024, 16(6), 663; https://doi.org/10.3390/sym16060663 - 27 May 2024
Viewed by 640
Abstract
Claude Berge (1987) introduced the concept of k-extendable graphs, wherein any independent set of size k is inherently a constituent of a maximum independent set within a graph H=(V,E). Graphs possessing the property of being [...] Read more.
Claude Berge (1987) introduced the concept of k-extendable graphs, wherein any independent set of size k is inherently a constituent of a maximum independent set within a graph H=(V,E). Graphs possessing the property of being 1-extendable are termedas Berge graphs. This introduction gave rise to the notion of well-covered graphs and well-dominated graphs. A graph is categorized as well-covered if each of its maximal independent sets is, in fact, a maximum independent set. Similarly, a graph attains the classification of well-dominated if every minimal dominating set (DS) within it is a minimum dominating set. In alignment with the concept of k-extendable graphs, the framework of (k,γ)-endowed graphs and symmetric (k,γ)-endowed graphs are established. In these graphs, each DS of size k encompasses a minimum DS of the graph. In this article, a study of γ-endowed dominating sets is initiated. Various results providing a deep insight into γ-endowed dominating sets in graphs such as those characterizing the ones possessing a unique minimum DS are proven. We also introduce and study the symmetric γ-endowed graphs and minimality of dominating sets in them. In addition, we give a solution to an open problem in the literature. which seeks to find a domination-based parameter that has a correlation coefficient of ρ>0.9967 with the total π-electronic energy of lower benzenoid hydrocarbons. We show that the upper dominating number Γ(H) studied in this paper delivers a strong prediction potential. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
Show Figures

Figure 1

13 pages, 267 KiB  
Article
Plane Partitions and Divisors
by Cristina Ballantine and Mircea Merca
Symmetry 2024, 16(1), 5; https://doi.org/10.3390/sym16010005 - 19 Dec 2023
Viewed by 1259
Abstract
In this paper, we consider the sum of divisors d of n such that n/d is a power of 2 and derive a new decomposition for the number of plane partitions of n in terms of binomial coefficients as a sum [...] Read more.
In this paper, we consider the sum of divisors d of n such that n/d is a power of 2 and derive a new decomposition for the number of plane partitions of n in terms of binomial coefficients as a sum over partitions of n. In this context, we introduce a new combinatorial interpretation of the number of plane partitions of n. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
Show Figures

Figure 1

14 pages, 340 KiB  
Article
Labeling Circulant Graphs: Distance Two Condition
by Laila Loudiki, Ez-Zobair Bidine and Mustapha Kchikech
Symmetry 2023, 15(12), 2160; https://doi.org/10.3390/sym15122160 - 5 Dec 2023
Viewed by 1025
Abstract
Given n6, D={1,2,,n2}, and a generating set SD, the circulant graph Cn(S) has Zn as a vertex set [...] Read more.
Given n6, D={1,2,,n2}, and a generating set SD, the circulant graph Cn(S) has Zn as a vertex set in which two distinct vertices i and j are adjacent if and only if |ij|nS, where |x|n=min(|x|,n|x|) is the circular distance modulo n. In this paper, we determine the L(2,1)-labeling number of Cn(DX), referred to as λ(Cn(DX)), for X={n2}, X={a}, X={a,b}, and in the general case when |X|<n2n4, where a,bD. Furthermore, we demonstrate that for all n6 and any given set S, λ(Cn(S))=n+gcd(n,S¯)2 if and only if gcd(n,S¯)2, and λ(Cn(S))n1 if and only if gcd(n,S¯)=1. Additionally, we establish that when the diameter of Cn(S) equals 2, λ(Cn(S))=n1. This observation motivated us to investigate the properties of S that lead to a diameter of Cn(S) equal to 2. Then, we introduce a highly distinctive family, denoted as An, that generates a large number of generating sets. For each value of n, we acquire a circulant graph Cn(An) with a diameter of 2, λ(Cn(An))=n1, and various additional interesting properties. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
Show Figures

Figure 1

20 pages, 315 KiB  
Article
n-Color Partitions into Distinct Parts as Sums over Partitions
by Mircea Merca and Emil Simion
Symmetry 2023, 15(11), 2067; https://doi.org/10.3390/sym15112067 - 15 Nov 2023
Viewed by 1365
Abstract
The partitions in which the parts of size n can come in n different colors are known as n-color partitions. For r{0,1}, let QLr(n) be the number of n-color [...] Read more.
The partitions in which the parts of size n can come in n different colors are known as n-color partitions. For r{0,1}, let QLr(n) be the number of n-color partitions of n into distinct parts which have a number of parts congruent to r modulo 2. In this paper, we consider specializations of complete and elementary symmetric functions in order to establish two kinds of formulas for QL0(n)±QL1(n) as sums over partitions of n in terms of binomial coefficients. The first kind of formulas only involve partitions in which the parts of size n appear at most n times, while the second kind of formulas involve unrestricted partitions. Similar results are obtained for the first differences of QL0(n)±QL1(n) and the partial sums of QL0(n)±QL1(n). Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
16 pages, 315 KiB  
Article
Statistical Analyses of a Class of Random Cyclooctatetraene Chain Networks with Respect to Several Topological Properties
by Chen Tao, Shengjun Tang and Xianya Geng
Symmetry 2023, 15(11), 1971; https://doi.org/10.3390/sym15111971 - 24 Oct 2023
Viewed by 913
Abstract
In recent years, the research on complex networks has created a boom. The objective of the present paper is to study a random cyclooctatetraene chain whose graph-theoretic mathematical properties arose scientists’ interests. By applying the concept of symmetry and probability theory, we obtain [...] Read more.
In recent years, the research on complex networks has created a boom. The objective of the present paper is to study a random cyclooctatetraene chain whose graph-theoretic mathematical properties arose scientists’ interests. By applying the concept of symmetry and probability theory, we obtain the explicit analytical expressions for the variances of Schultz index, multiplicative degree-Kirchhoff index Gutman index, and additive degree-Kirchhoff index of a random cyclooctatetraene chain with n octagons, which plays a crucial role in the research and application of topological indices. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
Show Figures

Figure 1

14 pages, 306 KiB  
Article
The λ-Fold Spectrum Problem for the Orientations of the Eight-Cycle
by Şafak Durukan-Odabaşı and Uğur Odabaşı
Symmetry 2023, 15(10), 1930; https://doi.org/10.3390/sym15101930 - 18 Oct 2023
Viewed by 1333
Abstract
A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, [...] Read more.
A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is K*λKv*, the λ-fold complete symmetric digraph on v vertices, that is, the digraph with λ directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which K*λKv* has a D-decomposition is called the λ-fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The λ-fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the λ-fold spectrum problem for each of the remaining 21 orientations. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
Show Figures

Figure 1

8 pages, 231 KiB  
Article
Inexact Iterates of Nonexpansive Mappings with Summable Errors in Metric Spaces with Graphs
by Simeon Reich and Alexander J. Zaslavski
Symmetry 2023, 15(10), 1927; https://doi.org/10.3390/sym15101927 - 17 Oct 2023
Viewed by 1017
Abstract
In our joint work with Dan Butnariu (2006) we established the stability of the convergence of iterates of a nonexpansive mapping on a complete metric space in the presence of summable computational errors. In a recent paper of ours, we extended this result [...] Read more.
In our joint work with Dan Butnariu (2006) we established the stability of the convergence of iterates of a nonexpansive mapping on a complete metric space in the presence of summable computational errors. In a recent paper of ours, we extended this result to inexact iterates of nonexpansive mappings on complete metric spaces with graphs under a certain assumption on the iterates. In the present paper we obtain an analogous result by removing that assumption on the iterates and replacing it with an additional assumption on the graph. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
17 pages, 743 KiB  
Article
On Laplacian Eigenvalues of Wheel Graphs
by Manal Alotaibi, Ahmad Alghamdi and Hanan Alolaiyan
Symmetry 2023, 15(9), 1737; https://doi.org/10.3390/sym15091737 - 11 Sep 2023
Cited by 1 | Viewed by 1363
Abstract
Consider G to be a simple graph with n vertices and m edges, and L(G) to be a Laplacian matrix with Laplacian eigenvalues of μ1,μ2,,μn=zero. [...] Read more.
Consider G to be a simple graph with n vertices and m edges, and L(G) to be a Laplacian matrix with Laplacian eigenvalues of μ1,μ2,,μn=zero. Write Sk(G)=i=1kμi as the sum of the k-largest Laplacian eigenvalues of G, where k{1,2,,n}. The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer’s conjecture states that Sk(G)m+k+12, where k=1,2,,n. This paper proves Brouwer’s conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph Wn+1, which provides a better approximation for this upper bound using Brouwer’s conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of Wn+1, n3. We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
Show Figures

Figure 1

15 pages, 780 KiB  
Article
Total Perfect Roman Domination
by Ahlam Almulhim
Symmetry 2023, 15(9), 1676; https://doi.org/10.3390/sym15091676 - 31 Aug 2023
Cited by 2 | Viewed by 1036
Abstract
A total perfect Roman dominating function (TPRDF) on a graph G=(V,E) is a function f from V to {0,1,2} satisfying (i) every vertex v with f(v)=0 [...] Read more.
A total perfect Roman dominating function (TPRDF) on a graph G=(V,E) is a function f from V to {0,1,2} satisfying (i) every vertex v with f(v)=0 is a neighbor of exactly one vertex u with f(u)=2; in addition, (ii) the subgraph of G that is induced by the vertices with nonzero weight has no isolated vertex. The weight of a TPRDF f is vVf(v). The total perfect Roman domination number of G, denoted by γtRp(G), is the minimum weight of a TPRDF on G. In this paper, we initiated the study of total perfect Roman domination. We characterized graphs with the largest-possible γtRp(G). We proved that total perfect Roman domination is NP-complete for chordal graphs, bipartite graphs, and for planar bipartite graphs. Finally, we related γtRp(G) to perfect domination γp(G) by proving γtRp(G)3γp(G) for every graph G, and we characterized trees T of order n3 for which γtRp(T)=3γp(T). This notion can be utilized to develop a defensive strategy with some properties. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
Show Figures

Figure 1

12 pages, 444 KiB  
Article
Convex Polygon Triangulation Based on Symmetry
by Predrag V. Krtolica, Predrag S. Stanimirović, Sead H. Mašović, Islam A. Elshaarawy and Alena Stupina
Symmetry 2023, 15(8), 1526; https://doi.org/10.3390/sym15081526 - 2 Aug 2023
Viewed by 1285
Abstract
A polygon with n nodes can be divided into two subpolygons by an internal diagonal through node n. Splitting the polygon along diagonal δi,n and diagonal δni,n, [...] Read more.
A polygon with n nodes can be divided into two subpolygons by an internal diagonal through node n. Splitting the polygon along diagonal δi,n and diagonal δni,n, i{2,,n/2} results in mirror images. Obviously, there are n/21 pairs of these reflectively symmetrical images. The influence of the observed symmetry on polygon triangulation is studied. The central result of this research is the construction of an efficient algorithm used for generating convex polygon triangulations in minimal time and without generating repeat triangulations. The proposed algorithm uses the diagonal values of the Catalan triangle to avoid duplicate triangulations with negligible computational costs and provides significant speedups compared to known methods. Full article
(This article belongs to the Special Issue Symmetry and Graph Theory)
Show Figures

Figure 1

Back to TopTop