Special Issue "Discrete Mathematics and Symmetry"

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

Deadline for manuscript submissions: closed (30 June 2020).

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A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Angel Garrido
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Guest Editor
Department of Fundamental Mathematics, Faculty of Sciences, UNED, Paseo Senda del Rey No. 9, 28040 Madrid, Spain
Interests: Mathematical Analysis; Measure Theory; Fuzzy Measures, in particular symmetry and entropy; Graph Theory; Discrete Mathematics; Automata Theory; Mathematical Education; Heuristics; Automata Theory; Artificial Intelligence
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Special Issue Information

Dear Colleagues,

One of the core concepts essential to understanding natural phenomena and the dynamics of social systems is the concept of “relation”. Furthermore, scientists rely on relational structures with high levels of symmetry because of their optimal behavior and high performance. Human friendships, social and interconnection networks, traffic systems, chemical structures, etc., can be expressed as relational structures. A mathematical model capturing the essence of this situation is a combinatorial object exhibiting a high level of symmetry, and the underlying mathematical discipline is algebraic combinatorics—the most vivid expression of the concept of symmetry in discrete mathematics.

The purpose of this Special Issue of the journal Symmetry is to present some recent developments, as well as possible future directions in algebraic combinatorics. Special emphasis will be given to the concept of symmetry in graphs, finite geometries, and designs.

Contributions pursuing solutions of long standing open problems in algebraic combinatorics, as well as contributions opening up new research topics encompassing symmetry within the boundaries of discrete mathematics but with the possibility of transcending these boundaries, are welcome.

Prof. Dr. Angel Garrido
Guest Editor

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 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

  • algebraic combinatorics
  • discrete mathematics
  • symmetries of graphs
  • symmetry in finite geometries and designs

Published Papers (35 papers)

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Open AccessFeature PaperArticle
Analysis of Recurrent Neural Network and Predictions
Symmetry 2020, 12(4), 615; https://doi.org/10.3390/sym12040615 - 13 Apr 2020
Abstract
This paper analyzes the operation principle and predicted value of the recurrent-neural-network (RNN) structure, which is the most basic and suitable for the change of time in the structure of a neural network for various types of artificial intelligence (AI). In particular, an [...] Read more.
This paper analyzes the operation principle and predicted value of the recurrent-neural-network (RNN) structure, which is the most basic and suitable for the change of time in the structure of a neural network for various types of artificial intelligence (AI). In particular, an RNN in which all connections are symmetric guarantees that it will converge. The operating principle of a RNN is based on linear data combinations and is composed through the synthesis of nonlinear activation functions. Linear combined data are similar to the autoregressive-moving average (ARMA) method of statistical processing. However, distortion due to the nonlinear activation function in RNNs causes the predicted value to be different from the predicted ARMA value. Through this, we know the limit of the predicted value of an RNN and the range of prediction that changes according to the learning data. In addition to mathematical proofs, numerical experiments confirmed our claims. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessFeature PaperArticle
Topological Symmetry Groups of the Heawood Graph
Symmetry 2020, 12(4), 546; https://doi.org/10.3390/sym12040546 - 04 Apr 2020
Abstract
We classify all groups which can occur as the topological symmetry group of some embedding of the Heawood graph in S 3 . [...] Read more.
We classify all groups which can occur as the topological symmetry group of some embedding of the Heawood graph in S 3 . Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Pythagorean Fuzzy Matroids with Application
Symmetry 2020, 12(3), 423; https://doi.org/10.3390/sym12030423 - 05 Mar 2020
Abstract
The Pythagorean fuzzy models deal with graphical and algebraic structures in case of vague information related to membership and non-membership grades. Here, we use Pythagorean fuzzy sets to generalize the concept of vector spaces and discuss their basis and dimensions. We also highlight [...] Read more.
The Pythagorean fuzzy models deal with graphical and algebraic structures in case of vague information related to membership and non-membership grades. Here, we use Pythagorean fuzzy sets to generalize the concept of vector spaces and discuss their basis and dimensions. We also highlight the concept of Pythagorean fuzzy matroids and examine some of their fundamental characteristics like circuits, basis, dimensions, and rank functions. Additionally, we explore the concept of Pythagorean fuzzy matroids in linear algebra, graph theory, and combinatorics. Finally, we demonstrate the use of Pythagorean fuzzy matroids for minimizing the time taken by a salesman in delivering given products. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Some Results on Various Cancellative CA-Groupoids and Variant CA-Groupoids
Symmetry 2020, 12(2), 315; https://doi.org/10.3390/sym12020315 - 22 Feb 2020
Abstract
Cyclic associativity can be regarded as a kind of variation symmetry, and cyclic associative groupoid (CA-groupoid) is a generalization of commutative semigroup. In this paper, the various cancellation properties of CA-groupoids, including cancellation, quasi-cancellation and power cancellation, are studied. The relationships among cancellative [...] Read more.
Cyclic associativity can be regarded as a kind of variation symmetry, and cyclic associative groupoid (CA-groupoid) is a generalization of commutative semigroup. In this paper, the various cancellation properties of CA-groupoids, including cancellation, quasi-cancellation and power cancellation, are studied. The relationships among cancellative CA-groupoids, quasi-cancellative CA-groupoids and power cancellative CA-groupoids are found out. Moreover, the concept of variant CA-groupoid is proposed firstly, some examples are presented. It is shown that the structure of variant CA-groupoid is very interesting, and the construction methods and decomposition theorem of variant CA-groupoids are established. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph
Symmetry 2020, 12(1), 135; https://doi.org/10.3390/sym12010135 - 09 Jan 2020
Abstract
In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product G + D n for the graph G on six vertices consists [...] Read more.
In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product G + D n for the graph G on six vertices consists of one vertex which is adjacent with three non-consecutive vertices of the 5-cycle. The proofs were based on the idea of establishing minimum values of crossings between two different subgraphs that cross the edges of the graph G exactly once. These minimum symmetrical values are described in the individual symmetric tables. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Bounds for the Generalized Distance Eigenvalues of a Graph
Symmetry 2019, 11(12), 1529; https://doi.org/10.3390/sym11121529 - 17 Dec 2019
Cited by 3
Abstract
Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and [...] Read more.
Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( G ) + ( 1 α ) D ( G ) , where α [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Weak Embeddable Hypernear-Rings
Symmetry 2019, 11(8), 964; https://doi.org/10.3390/sym11080964 - 01 Aug 2019
Cited by 2
Abstract
In this paper we extend one of the main problems of near-rings to the framework of algebraic hypercompositional structures. This problem states that every near-ring is isomorphic with a near-ring of the transformations of a group. First we endow the set of all [...] Read more.
In this paper we extend one of the main problems of near-rings to the framework of algebraic hypercompositional structures. This problem states that every near-ring is isomorphic with a near-ring of the transformations of a group. First we endow the set of all multitransformations of a hypergroup (not necessarily abelian) with a general hypernear-ring structure, called the multitransformation general hypernear-ring associated with a hypergroup. Then we show that any hypernear-ring can be weakly embedded into a multitransformation general hypernear-ring, generalizing the similar classical theorem on near-rings. Several properties of hypernear-rings related with this property are discussed and illustrated also by examples. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
Open AccessArticle
Selective Maintenance Optimization for a Multi-State System Considering Human Reliability
Symmetry 2019, 11(5), 652; https://doi.org/10.3390/sym11050652 - 09 May 2019
Cited by 3
Abstract
In an actual industrial or military operations environment, a multi-state system (MSS) consisting of multi-state components often needs to perform multiple missions in succession. To improve the probability of the system successfully completing the next mission, all the maintenance activities need to be [...] Read more.
In an actual industrial or military operations environment, a multi-state system (MSS) consisting of multi-state components often needs to perform multiple missions in succession. To improve the probability of the system successfully completing the next mission, all the maintenance activities need to be performed during maintenance breaks between any two consecutive missions under limited maintenance resources. In such case, selective maintenance is a widely used maintenance policy. As a typical discrete mathematics problem, selective maintenance has received widespread attention. In this work, a selective maintenance model considering human reliability for multi-component systems is investigated. Each maintenance worker can be in one of multiple discrete working levels due to their human error probability (HEP). The state of components after maintenance is assumed to be random and follow an identified probability distribution. To solve the problem, this paper proposes a human reliability model and a method to determine the state distribution of components after maintenance. The objective of selective maintenance scheduling is to find the maintenance action with the optimal reliability for each component in a maintenance break subject to constraints of time and cost. In place of an enumerative method, a genetic algorithm (GA) is employed to solve the complicated optimization problem taking human reliability into account. The results show the importance of considering human reliability in selective maintenance scheduling for an MSS. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Edge Even Graceful Labeling of Cylinder Grid Graph
Symmetry 2019, 11(4), 584; https://doi.org/10.3390/sym11040584 - 22 Apr 2019
Cited by 2
Abstract
Edge even graceful labeling (e.e.g., l.) of graphs is a modular technique of edge labeling of graphs, introduced in 2017. An e.e.g., l. of simple finite undirected graph G = ( V ( G ) , E ( G ) ) of order [...] Read more.
Edge even graceful labeling (e.e.g., l.) of graphs is a modular technique of edge labeling of graphs, introduced in 2017. An e.e.g., l. of simple finite undirected graph G = ( V ( G ) , E ( G ) ) of order P = | ( V ( G ) | and size q = | E ( G ) | is a bijection f : E ( G ) { 2 , 4 , , 2 q } , such that when each vertex v V ( G ) is assigned the modular sum of the labels (images of f ) of the edges incident to v , the resulting vertex labels are distinct mod 2 r , where r = max ( p , q ) . In this work, the family of cylinder grid graphs are studied. Explicit formulas of e.e.g., l. for all of the cases of each member of this family have been proven. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups
Symmetry 2019, 11(4), 553; https://doi.org/10.3390/sym11040553 - 17 Apr 2019
Cited by 1
Abstract
In this paper, some basic properties and structure characterizations of AG-groups are further studied. First, some examples of infinite AG-groups are given, and weak commutative, alternative and quasi-cancellative AG-groups are discussed. Second, two new concepts of involution AG-group and generalized involution AG-group are [...] Read more.
In this paper, some basic properties and structure characterizations of AG-groups are further studied. First, some examples of infinite AG-groups are given, and weak commutative, alternative and quasi-cancellative AG-groups are discussed. Second, two new concepts of involution AG-group and generalized involution AG-group are proposed, the relationships among (generalized) involution AG-groups, commutative groups and AG-groups are investigated, and the structure theorems of (generalized) involution AG-groups are proved. Third, the notion of filter of an AG-group is introduced, the congruence relation is constructed from arbitrary filter, and the corresponding quotient structure and homomorphism theorems are established. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Isoperimetric Numbers of Randomly Perturbed Intersection Graphs
Symmetry 2019, 11(4), 452; https://doi.org/10.3390/sym11040452 - 01 Apr 2019
Cited by 1
Abstract
Social networks describe social interactions between people, which are often modeled by intersection graphs. In this paper, we propose an intersection graph model that is induced by adding a sparse random bipartite graph to a given bipartite graph. Under some mild conditions, we [...] Read more.
Social networks describe social interactions between people, which are often modeled by intersection graphs. In this paper, we propose an intersection graph model that is induced by adding a sparse random bipartite graph to a given bipartite graph. Under some mild conditions, we show that the vertex–isoperimetric number and the edge–isoperimetric number of the randomly perturbed intersection graph on n vertices are Ω ( 1 / ln n ) asymptomatically almost surely. Numerical simulations for small graphs extracted from two real-world social networks, namely, the board interlocking network and the scientific collaboration network, were performed. It was revealed that the effect of increasing isoperimetric numbers (i.e., expansion properties) on randomly perturbed intersection graphs is presumably independent of the order of the network. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Generalized Permanental Polynomials of Graphs
Symmetry 2019, 11(2), 242; https://doi.org/10.3390/sym11020242 - 16 Feb 2019
Abstract
The search for complete graph invariants is an important problem in graph theory and computer science. Two networks with a different structure can be distinguished from each other by complete graph invariants. In order to find a complete graph invariant, we introduce the [...] Read more.
The search for complete graph invariants is an important problem in graph theory and computer science. Two networks with a different structure can be distinguished from each other by complete graph invariants. In order to find a complete graph invariant, we introduce the generalized permanental polynomials of graphs. Let G be a graph with adjacency matrix A ( G ) and degree matrix D ( G ) . The generalized permanental polynomial of G is defined by P G ( x , μ ) = per ( x I ( A ( G ) μ D ( G ) ) ) . In this paper, we compute the generalized permanental polynomials for all graphs on at most 10 vertices, and we count the numbers of such graphs for which there is another graph with the same generalized permanental polynomial. The present data show that the generalized permanental polynomial is quite efficient for distinguishing graphs. Furthermore, we can write P G ( x , μ ) in the coefficient form i = 0 n c μ i ( G ) x n i and obtain the combinatorial expressions for the first five coefficients c μ i ( G ) ( i = 0 , 1 , , 4 ) of P G ( x , μ ) . Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Dynamics Models of Synchronized Piecewise Linear Discrete Chaotic Systems of High Order
Symmetry 2019, 11(2), 236; https://doi.org/10.3390/sym11020236 - 15 Feb 2019
Cited by 14
Abstract
This paper deals with the methods for investigating the nonlinear dynamics of discrete chaotic systems (DCS) applied to piecewise linear systems of the third order. The paper proposes an approach to the analysis of the systems under research and their improvement. Thus, effective [...] Read more.
This paper deals with the methods for investigating the nonlinear dynamics of discrete chaotic systems (DCS) applied to piecewise linear systems of the third order. The paper proposes an approach to the analysis of the systems under research and their improvement. Thus, effective and mathematically sound methods for the analysis of nonlinear motions in the models under consideration are proposed. It makes it possible to obtain simple calculated relations for determining the basic dynamic characteristics of systems. Based on these methods, the authors developed algorithms for calculating the dynamic characteristics of discrete systems, i.e. areas of the existence of steady-state motion, areas of stability, capture band, and parameters of transients. By virtue of the developed methods and algorithms, the dynamic modes of several models of discrete phase synchronization systems can be analyzed. They are as follows: Pulsed and digital different orders, dual-ring systems of various types, including combined ones, and systems with cyclic interruption of auto-tuning. The efficiency of various devices for information processing, generation and stabilization could be increased by using the mentioned discrete synchronization systems on the grounds of the results of the analysis. We are now developing original software for analyzing the dynamic characteristics of various classes of discrete phase synchronization systems, based on the developed methods and algorithms. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five
Symmetry 2019, 11(2), 123; https://doi.org/10.3390/sym11020123 - 22 Jan 2019
Cited by 3
Abstract
The main aim of the paper is to give the crossing number of the join product G + D n for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the [...] Read more.
The main aim of the paper is to give the crossing number of the join product G + D n for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the three-cycle, and D n consists of n isolated vertices. In the proofs, the idea of the new representation of the minimum numbers of crossings between two different subgraphs that do not cross the edges of the graph G by the graph of configurations G D in the considered drawing D of G + D n will be used. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph D n and with the path P n on n vertices for three other graphs. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Edge Even Graceful Labeling of Polar Grid Graphs
Symmetry 2019, 11(1), 38; https://doi.org/10.3390/sym11010038 - 02 Jan 2019
Cited by 4
Abstract
Edge Even Graceful Labelingwas first defined byElsonbaty and Daoud in 2017. An edge even graceful labeling of a simple graph G with p vertices and q edges is a bijection f from the edges of the graph to the set { 2 , [...] Read more.
Edge Even Graceful Labelingwas first defined byElsonbaty and Daoud in 2017. An edge even graceful labeling of a simple graph G with p vertices and q edges is a bijection f from the edges of the graph to the set { 2 , 4 , , 2 q } such that, when each vertex is assigned the sum of all edges incident to it mod 2 r where r = max { p , q } , the resulting vertex labels are distinct. In this paper we proved necessary and sufficient conditions for the polar grid graph to be edge even graceful graph. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessFeature PaperArticle
Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices of Generalized Bridge Molecular Graphs
Symmetry 2018, 10(12), 751; https://doi.org/10.3390/sym10120751 - 14 Dec 2018
Cited by 13
Abstract
Topological indices are graph invariants computed by the distance or degree of vertices of the molecular graph. In chemical graph theory, topological indices have been successfully used in describing the structures and predicting certain physicochemical properties of chemical compounds. In this paper, we [...] Read more.
Topological indices are graph invariants computed by the distance or degree of vertices of the molecular graph. In chemical graph theory, topological indices have been successfully used in describing the structures and predicting certain physicochemical properties of chemical compounds. In this paper, we propose a definition of generalized bridge molecular graphs that can model more kinds of long chain polymerization products than the bridge molecular graphs, and provide some results of the edge versions of atom-bond connectivity ( A B C e ) and geometric arithmetic ( G A e ) indices for some generalized bridge molecular graphs, which have regular, periodic and symmetrical structures. The results of this paper offer promising prospects in the applications for chemical and material engineering, especially in chemical industry research. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Sufficient Conditions for Triangular Norms Preserving ⊗-Convexity
Symmetry 2018, 10(12), 729; https://doi.org/10.3390/sym10120729 - 07 Dec 2018
Abstract
The convexity in triangular norm (for short, ⊗−convexity) is a generalization of Zadeh’s quasiconvexity. The aggregation of two ⊗−convex sets is under the aggregation operator ⊗ is also ⊗−convex, but the aggregation operator ⊗ is not unique. To solve it in complexity, in [...] Read more.
The convexity in triangular norm (for short, ⊗−convexity) is a generalization of Zadeh’s quasiconvexity. The aggregation of two ⊗−convex sets is under the aggregation operator ⊗ is also ⊗−convex, but the aggregation operator ⊗ is not unique. To solve it in complexity, in the present paper, we give some sufficient conditions for aggregation operators preserve ⊗−convexity. In particular, when aggregation operators are triangular norms, we have that several results such as arbitrary triangular norm preserve D convexity and a convexity on bounded lattices, M preserves H convexity in the real unite interval [ 0 , 1 ] . Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessFeature PaperArticle
A Novel Edge Detection Method Based on the Regularized Laplacian Operation
Symmetry 2018, 10(12), 697; https://doi.org/10.3390/sym10120697 - 03 Dec 2018
Abstract
In this paper, an edge detection method based on the regularized Laplacian operation is given. The Laplacian operation has been used extensively as a second-order edge detector due to its variable separability and rotation symmetry. Since the image data might contain some noises [...] Read more.
In this paper, an edge detection method based on the regularized Laplacian operation is given. The Laplacian operation has been used extensively as a second-order edge detector due to its variable separability and rotation symmetry. Since the image data might contain some noises inevitably, regularization methods should be introduced to overcome the instability of Laplacian operation. By rewriting the Laplacian operation as an integral equation of the first kind, a regularization based on partial differential equation (PDE) can be used to compute the Laplacian operation approximately. We first propose a novel edge detection algorithm based on the regularized Laplacian operation. Considering the importance of the regularization parameter, an unsupervised choice strategy of the regularization parameter is introduced subsequently. Finally, the validity of the proposed edge detection algorithm is shown by some comparison experiments. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
The Complexity of Some Classes of Pyramid Graphs Created from a Gear Graph
Symmetry 2018, 10(12), 689; https://doi.org/10.3390/sym10120689 - 02 Dec 2018
Cited by 4
Abstract
The methods of measuring the complexity (spanning trees) in a finite graph, a problem related to various areas of mathematics and physics, have been inspected by many mathematicians and physicists. In this work, we defined some classes of pyramid graphs created by a [...] Read more.
The methods of measuring the complexity (spanning trees) in a finite graph, a problem related to various areas of mathematics and physics, have been inspected by many mathematicians and physicists. In this work, we defined some classes of pyramid graphs created by a gear graph then we developed the Kirchhoff’s matrix tree theorem method to produce explicit formulas for the complexity of these graphs, using linear algebra, matrix analysis techniques, and employing knowledge of Chebyshev polynomials. Finally, we gave some numerical results for the number of spanning trees of the studied graphs. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Novel Three-Way Decisions Models with Multi-Granulation Rough Intuitionistic Fuzzy Sets
Symmetry 2018, 10(11), 662; https://doi.org/10.3390/sym10110662 - 21 Nov 2018
Cited by 5
Abstract
The existing construction methods of granularity importance degree only consider the direct influence of single granularity on decision-making; however, they ignore the joint impact from other granularities when carrying out granularity selection. In this regard, we have the following improvements. First of all, [...] Read more.
The existing construction methods of granularity importance degree only consider the direct influence of single granularity on decision-making; however, they ignore the joint impact from other granularities when carrying out granularity selection. In this regard, we have the following improvements. First of all, we define a more reasonable granularity importance degree calculating method among multiple granularities to deal with the above problem and give a granularity reduction algorithm based on this method. Besides, this paper combines the reduction sets of optimistic and pessimistic multi-granulation rough sets with intuitionistic fuzzy sets, respectively, and their related properties are shown synchronously. Based on this, to further reduce the redundant objects in each granularity of reduction sets, four novel kinds of three-way decisions models with multi-granulation rough intuitionistic fuzzy sets are developed. Moreover, a series of concrete examples can demonstrate that these joint models not only can remove the redundant objects inside each granularity of the reduction sets, but also can generate much suitable granularity selection results using the designed comprehensive score function and comprehensive accuracy function of granularities. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Maximum Detour–Harary Index for Some Graph Classes
Symmetry 2018, 10(11), 608; https://doi.org/10.3390/sym10110608 - 07 Nov 2018
Abstract
The definition of a Detour–Harary index is ω H ( G ) = 1 2 u , v V ( G ) 1 l ( u , v | G ) , where G is a simple and connected graph, and [...] Read more.
The definition of a Detour–Harary index is ω H ( G ) = 1 2 u , v V ( G ) 1 l ( u , v | G ) , where G is a simple and connected graph, and l ( u , v | G ) is equal to the length of the longest path between vertices u and v. In this paper, we obtained the maximum Detour–Harary index about unicyclic graphs, bicyclic graphs, and cacti, respectively. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessFeature PaperArticle
Q-Filters of Quantum B-Algebras and Basic Implication Algebras
Symmetry 2018, 10(11), 573; https://doi.org/10.3390/sym10110573 - 01 Nov 2018
Cited by 19
Abstract
The concept of quantum B-algebra was introduced by Rump and Yang, that is, unified algebraic semantics for various noncommutative fuzzy logics, quantum logics, and implication logics. In this paper, a new notion of q-filter in quantum B-algebra is proposed, and quotient structures are [...] Read more.
The concept of quantum B-algebra was introduced by Rump and Yang, that is, unified algebraic semantics for various noncommutative fuzzy logics, quantum logics, and implication logics. In this paper, a new notion of q-filter in quantum B-algebra is proposed, and quotient structures are constructed by q-filters (in contrast, although the notion of filter in quantum B-algebra has been defined before this paper, but corresponding quotient structures cannot be constructed according to the usual methods). Moreover, a new, more general, implication algebra is proposed, which is called basic implication algebra and can be regarded as a unified frame of general fuzzy logics, including nonassociative fuzzy logics (in contrast, quantum B-algebra is not applied to nonassociative fuzzy logics). The filter theory of basic implication algebras is also established. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
Open AccessArticle
Fuzzy Normed Rings
Symmetry 2018, 10(10), 515; https://doi.org/10.3390/sym10100515 - 16 Oct 2018
Cited by 2
Abstract
In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy [...] Read more.
In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal, and fuzzy normed maximal ideal of a normed ring, respectively. We show some algebraic properties of normed ring theory on fuzzy sets, prove theorems, and give relevant examples. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
Open AccessArticle
Invariant Graph Partition Comparison Measures
Symmetry 2018, 10(10), 504; https://doi.org/10.3390/sym10100504 - 15 Oct 2018
Abstract
Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a [...] Read more.
Symmetric graphs have non-trivial automorphism groups. This article starts with the proof that all partition comparison measures we have found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms. By the construction of a pseudometric space of equivalence classes of permutations and with Hausdorff’s and von Neumann’s methods of constructing invariant measures on the space of equivalence classes, we design three different families of invariant measures, and we present two types of invariance proofs. Last, but not least, we provide algorithms for computing invariant partition comparison measures as pseudometrics on the partition space. When combining an invariant partition comparison measure with its classical counterpart, the decomposition of the measure into a structural difference and a difference contributed by the group automorphism is derived. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
A Multi-Granularity 2-Tuple QFD Method and Application to Emergency Routes Evaluation
Symmetry 2018, 10(10), 484; https://doi.org/10.3390/sym10100484 - 11 Oct 2018
Cited by 3
Abstract
Quality function deployment (QFD) is an effective approach to satisfy the customer requirements (CRs). Furthermore, accurately prioritizing the engineering characteristics (ECs) as the core of QFD is considered as a group decision making (GDM) problem. In order to availably [...] Read more.
Quality function deployment (QFD) is an effective approach to satisfy the customer requirements (CRs). Furthermore, accurately prioritizing the engineering characteristics (ECs) as the core of QFD is considered as a group decision making (GDM) problem. In order to availably deal with various preferences and the vague information of different experts on a QFD team, multi-granularity 2-tuple linguistic representation is applied to elucidate the relationship and correlation between CRs and ECs without loss of information. In addition, the importance of CRs is determined using the best worst method (BWM), which is more applicable and has good consistency. Furthermore, we propose considering the relationship matrix and correlation matrix method to prioritize ECs. Finally, an example about evaluating emergency routes of metro station is proposed to illustrate the validity of the proposed methodology. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal
Symmetry 2018, 10(10), 465; https://doi.org/10.3390/sym10100465 - 08 Oct 2018
Cited by 1
Abstract
For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. Pseudo-BCI algebra is a class of non-classical logic algebras, which is closely related to various non-commutative fuzzy logic systems. The aim of this paper is focus on the [...] Read more.
For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. Pseudo-BCI algebra is a class of non-classical logic algebras, which is closely related to various non-commutative fuzzy logic systems. The aim of this paper is focus on the structure of a special class of pseudo-BCI algebras in which every element is quasi-maximal (call it QM-pseudo-BCI algebras in this paper). First, the new notions of quasi-maximal element and quasi-left unit element in pseudo-BCK algebras and pseudo-BCI algebras are proposed and some properties are discussed. Second, the following structure theorem of QM-pseudo-BCI algebra is proved: every QM-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an anti-group pseudo-BCI algebra. Third, the new notion of weak associative pseudo-BCI algebra (WA-pseudo-BCI algebra) is introduced and the following result is proved: every WA-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an Abel group. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons
Symmetry 2018, 10(10), 447; https://doi.org/10.3390/sym10100447 - 29 Sep 2018
Abstract
Let P be a planar point set with no three points collinear, k points of P be a k-hole of P if the k points are the vertices of a convex polygon without points of P. This article proves 13 is the [...] Read more.
Let P be a planar point set with no three points collinear, k points of P be a k-hole of P if the k points are the vertices of a convex polygon without points of P. This article proves 13 is the smallest integer such that any planar points set containing at least 13 points with no three points collinear, contains a 3-hole, a 4-hole and a 5-hole which are pairwise disjoint. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Multi-Granulation Graded Rough Intuitionistic Fuzzy Sets Models Based on Dominance Relation
Symmetry 2018, 10(10), 446; https://doi.org/10.3390/sym10100446 - 28 Sep 2018
Cited by 2
Abstract
From the perspective of the degrees of classification error, we proposed graded rough intuitionistic fuzzy sets as the extension of classic rough intuitionistic fuzzy sets. Firstly, combining dominance relation of graded rough sets with dominance relation in intuitionistic fuzzy ordered information systems, we [...] Read more.
From the perspective of the degrees of classification error, we proposed graded rough intuitionistic fuzzy sets as the extension of classic rough intuitionistic fuzzy sets. Firstly, combining dominance relation of graded rough sets with dominance relation in intuitionistic fuzzy ordered information systems, we designed type-I dominance relation and type-II dominance relation. Type-I dominance relation reduces the errors caused by single theory and improves the precision of ordering. Type-II dominance relation decreases the limitation of ordering by single theory. After that, we proposed graded rough intuitionistic fuzzy sets based on type-I dominance relation and type-II dominance relation. Furthermore, from the viewpoint of multi-granulation, we further established multi-granulation graded rough intuitionistic fuzzy sets models based on type-I dominance relation and type-II dominance relation. Meanwhile, some properties of these models were discussed. Finally, the validity of these models was verified by an algorithm and some relative examples. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Four Operators of Rough Sets Generalized to Matroids and a Matroidal Method for Attribute Reduction
Symmetry 2018, 10(9), 418; https://doi.org/10.3390/sym10090418 - 19 Sep 2018
Cited by 5
Abstract
Rough sets provide a useful tool for data preprocessing during data mining. However, many algorithms related to some problems in rough sets, such as attribute reduction, are greedy ones. Matroids propose a good platform for greedy algorithms. Therefore, it is important to study [...] Read more.
Rough sets provide a useful tool for data preprocessing during data mining. However, many algorithms related to some problems in rough sets, such as attribute reduction, are greedy ones. Matroids propose a good platform for greedy algorithms. Therefore, it is important to study the combination between rough sets and matroids. In this paper, we investigate rough sets and matroids through their operators, and provide a matroidal method for attribute reduction in information systems. Firstly, we generalize four operators of rough sets to four operators of matroids through the interior, closure, exterior and boundary axioms, respectively. Thus, there are four matroids induced by these four operators of rough sets. Then, we find that these four matroids are the same one, which implies the relationship about operators between rough sets and matroids. Secondly, a relationship about operations between matroids and rough sets is presented according to the induced matroid. Finally, the girth function of matroids is used to compute attribute reduction in information systems. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
Open AccessArticle
Some Results on Multigranulation Neutrosophic Rough Sets on a Single Domain
Symmetry 2018, 10(9), 417; https://doi.org/10.3390/sym10090417 - 19 Sep 2018
Cited by 1
Abstract
As a generalization of single value neutrosophic rough sets, the concept of multi-granulation neutrosophic rough sets was proposed by Bo et al., and some basic properties of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators were studied. However, they did not do a [...] Read more.
As a generalization of single value neutrosophic rough sets, the concept of multi-granulation neutrosophic rough sets was proposed by Bo et al., and some basic properties of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators were studied. However, they did not do a comprehensive study on the algebraic structure of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators. In the present paper, we will provide the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators. In particular, in the one-dimensional case, for special neutrosophic relations, the completely lattice isomorphic relationship between upper neutrosophic rough approximation operators and lower neutrosophic rough approximation operators is proved. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Fixed Points Results in Algebras of Split Quaternion and Octonion
Symmetry 2018, 10(9), 405; https://doi.org/10.3390/sym10090405 - 17 Sep 2018
Cited by 1
Abstract
Fixed points of functions have applications in game theory, mathematics, physics, economics and computer science. The purpose of this article is to compute fixed points of a general quadratic polynomial in finite algebras of split quaternion and octonion over prime fields Z p [...] Read more.
Fixed points of functions have applications in game theory, mathematics, physics, economics and computer science. The purpose of this article is to compute fixed points of a general quadratic polynomial in finite algebras of split quaternion and octonion over prime fields Z p. Some characterizations of fixed points in terms of the coefficients of these polynomials are also given. Particularly, cardinalities of these fixed points have been determined depending upon the characteristics of the underlying field. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
Open AccessArticle
Optimizing the High-Level Maintenance Planning Problem of the Electric Multiple Unit Train Using a Modified Particle Swarm Optimization Algorithm
Symmetry 2018, 10(8), 349; https://doi.org/10.3390/sym10080349 - 19 Aug 2018
Cited by 2
Abstract
Electric multiple unit (EMU) trains’ high-level maintenance planning is a discrete problem in mathematics. The high-level maintenance process of the EMU trains consumes plenty of time. When the process is undertaken during peak periods of the passenger flow, the transportation demand may not [...] Read more.
Electric multiple unit (EMU) trains’ high-level maintenance planning is a discrete problem in mathematics. The high-level maintenance process of the EMU trains consumes plenty of time. When the process is undertaken during peak periods of the passenger flow, the transportation demand may not be fully satisfied due to the insufficient supply of trains. In contrast, if the process is undergone in advance, extra costs will be incurred. Based on the practical requirements of high-level maintenance, a 0–1 programming model is proposed. To simplify the description of the model, candidate sets of delivery dates, i.e., time windows, are generated according to the historical data and maintenance regulations. The constraints of the model include maintenance regulations, the passenger transportation demand, and capacities of workshop. The objective function is to minimize the mileage losses of all EMU trains. Moreover, a modified particle swarm algorithm is developed for solving the problem. Finally, a real-world case study of Shanghai Railway is conducted to demonstrate the proposed method. Computational results indicate that the (approximate) optimal solution can be obtained successfully by our method and the proposed method significantly reduces the solution time to 500 s. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation
Symmetry 2018, 10(8), 341; https://doi.org/10.3390/sym10080341 - 15 Aug 2018
Cited by 26
Abstract
In this work, we study the completely integrable sixth-order nonlinear Ramani equation. By applying the Lie symmetry analysis technique, the Lie point symmetries and the optimal system of one-dimensional sub-algebras of the equation are derived. The optimal system is further used to derive [...] Read more.
In this work, we study the completely integrable sixth-order nonlinear Ramani equation. By applying the Lie symmetry analysis technique, the Lie point symmetries and the optimal system of one-dimensional sub-algebras of the equation are derived. The optimal system is further used to derive the symmetry reductions and exact solutions. In conjunction with the Riccati Bernoulli sub-ODE (RBSO), we construct the travelling wave solutions of the equation by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction. We show that the equation is nonlinearly self-adjoint and construct the conservation laws (CL) associated with the Lie symmetries by invoking the conservation theorem due to Ibragimov. Some figures are shown to show the physical interpretations of the acquired results. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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Open AccessArticle
Binary Icosahedral Group and 600-Cell
Symmetry 2018, 10(8), 326; https://doi.org/10.3390/sym10080326 - 07 Aug 2018
Abstract
In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, [...] Read more.
In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of H4. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
Open AccessArticle
Enumeration of Strongly Regular Graphs on up to 50 Vertices Having S3 as an Automorphism Group
Symmetry 2018, 10(6), 212; https://doi.org/10.3390/sym10060212 - 11 Jun 2018
Abstract
One of the main problems in the theory of strongly regular graphs (SRGs) is constructing and classifying SRGs with given parameters. Strongly regular graphs with parameters (37,18,8,9), (41,20,9, [...] Read more.
One of the main problems in the theory of strongly regular graphs (SRGs) is constructing and classifying SRGs with given parameters. Strongly regular graphs with parameters (37,18,8,9), (41,20,9,10), (45,22,10,11), (49,24,11,12), (49,18,7,6) and (50,21,8,9) are the only strongly regular graphs on up to 50 vertices that still have to be classified. In this paper, we give the enumeration of SRGs with these parameters having S3 as an automorphism group. The construction of SRGs in this paper is a step in the classification of SRGs on up to 50 vertices. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry) Printed Edition available
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