Special Issue "Recent Advances in Discrete and Fractional Mathematics"

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

Deadline for manuscript submissions: closed (5 July 2020).

Special Issue Editors

Prof. Dr. Jose M. Rodriguez
Website
Guest Editor
Department of Mathematics, Carlos III University of Madrid-Leganés Campus, Avenida de la Universidad 30, CP-28911, Leganés, Madrid, Spain
Interests: discrete mathematics; fractional calculus; topological indices; polynomials in graphs; geometric function theory; geometry; approximation theory
Special Issues and Collections in MDPI journals
Prof. Dr. José M. Sigarreta
Website
Guest Editor
Faculty of Mathematics. Autonomous University of Guerrero-Acapulco Campus, Calle Carlos E. Adame 54, Garita, CP-39650, Acapulco, Guerrero, Mexico
Interests: discrete mathematics; alliances in graphs; conformable and non-conformable calculus; geometry; topological indices
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

Although Discrete and Fractional Mathematics has always played an important role in Mathematics, in recent years, this role has significantly increased in several branches of these fields, including, but not limited to:

Gromov hyperbolic graphs, domination theory, differential of graphs, polynomials in graphs, alliances in graphs, complex systems, topological indices, discrete geometry, fractional differential equations, fractional integral operators, and discrete and fractional inequalities.

The aim of this Special Issue is to attract leading researchers in these areas in order to include new high-quality results on these topics involving their symmetry properties, both from a theoretical and an applied point of view.

Prof. Dr. Jose M. Rodriguez
Prof. Dr.  José M. Sigarreta
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 papers will be 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 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

  • Discrete Mathematics
  • Graph Theory
  • Hyperbolic Graphs
  • Domination in Graphs
  • Differential of Graphs
  • Polynomials in Graphs
  • Alliances in Graphs
  • Complex Systems
  • Topological Indices
  • Discrete Geometry
  • Fractional Calculus
  • Fractional Differential Equations
  • Fractional Integral Operators
  • Conformable and Non-Conformable Calculus
  • Discrete and Fractional Inequalities

Published Papers (14 papers)

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Research

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Open AccessArticle
Closed Knight’s Tours on (m,n,r)-Ringboards
Symmetry 2020, 12(8), 1217; https://doi.org/10.3390/sym12081217 - 25 Jul 2020
Abstract
A (legal) knight’s move is the result of moving the knight two squares horizontally or vertically on the board and then turning and moving one square in the perpendicular direction. A closed knight’s tour is a knight’s move that visits every square on [...] Read more.
A (legal) knight’s move is the result of moving the knight two squares horizontally or vertically on the board and then turning and moving one square in the perpendicular direction. A closed knight’s tour is a knight’s move that visits every square on a given chessboard exactly once and returns to its start square. A closed knight’s tour and its variations are studied widely over the rectangular chessboard or a three-dimensional rectangular box. For m,n>2r, an (m,n,r)-ringboard or (m,n,r)-annulus-board is defined to be an m×n chessboard with the middle part missing and the rim contains r rows and r columns. In this paper, we obtain that a (m,n,r)-ringboard with m,n3 and m,n>2r has a closed knight’s tour if and only if (a) m=n=3 and r=1 or (b) m,n7 and r3. If a closed knight’s tour on an (m,n,r)-ringboard exists, then it has symmetries along two diagonals. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessArticle
Trees with Minimum Weighted Szeged Index Are of a Large Diameter
Symmetry 2020, 12(5), 793; https://doi.org/10.3390/sym12050793 - 09 May 2020
Abstract
The weighted Szeged index ( w S z ) has gained considerable attention recently because of its unusual mathematical properties. Searching for a tree (or trees) that minimizes the w S z is still going on. Several structural details of a minimal tree [...] Read more.
The weighted Szeged index ( w S z ) has gained considerable attention recently because of its unusual mathematical properties. Searching for a tree (or trees) that minimizes the w S z is still going on. Several structural details of a minimal tree were described. Here, it is shown a surprising property that these trees have maximum degree at most 16, and as a consequence, we promptly conclude that these trees are of large diameter. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessArticle
The Differential on Graph Operator Q(G)
Symmetry 2020, 12(5), 751; https://doi.org/10.3390/sym12050751 - 06 May 2020
Abstract
If G = ( V ( G ) , E ( G ) ) is a simple connected graph with the vertex set V ( G ) and the edge set E ( G ) , S is a subset of V ( [...] Read more.
If G = ( V ( G ) , E ( G ) ) is a simple connected graph with the vertex set V ( G ) and the edge set E ( G ) , S is a subset of V ( G ) , and let B ( S ) be the set of neighbors of S in V ( G ) S . Then, the differential of S ( S ) is defined as | B ( S ) | | S | . The differential of G, denoted by ( G ) , is the maximum value of ( S ) for all subsets S V ( G ) . The graph operator Q ( G ) is defined as the graph that results by subdividing every edge of G once and joining pairs of these new vertices iff their corresponding edges are incident in G. In this paper, we study the relations between ( G ) and ( Q ( G ) ) . Besides, we exhibit some results relating the differential ( G ) and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessArticle
Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag–Leffler Memories Involving the Truncated M-Derivative
Symmetry 2020, 12(4), 626; https://doi.org/10.3390/sym12040626 - 15 Apr 2020
Cited by 4
Abstract
In this research, novel M-truncated fractional derivatives with three orders have been proposed. These operators involve truncated Mittag–Leffler function to generalize the Khalil conformable derivative as well as the M-derivative. The new operators proposed are the convolution of truncated M-derivative [...] Read more.
In this research, novel M-truncated fractional derivatives with three orders have been proposed. These operators involve truncated Mittag–Leffler function to generalize the Khalil conformable derivative as well as the M-derivative. The new operators proposed are the convolution of truncated M-derivative with a power law, exponential decay and the complete Mittag–Leffler function. Numerical schemes based on Lagrange interpolation to predict chaotic behaviors of Rucklidge, Shimizu–Morioka and a hybrid strange attractors were considered. Additionally, numerical analysis based on 0–1 test and sensitive dependence on initial conditions were carried out to verify and show the existence of chaos in the chaotic attractor. These results showed that these novel operators involving three orders, two for the truncated M-derivative and one for the fractional term, depict complex chaotic behaviors. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessArticle
On the Metric Dimension of Arithmetic Graph of a Composite Number
Symmetry 2020, 12(4), 607; https://doi.org/10.3390/sym12040607 - 11 Apr 2020
Abstract
This paper is devoted to the study of the arithmetic graph of a composite number m, denoted by A m . It has been observed that there exist different composite numbers for which the arithmetic graphs are isomorphic. It is proved that [...] Read more.
This paper is devoted to the study of the arithmetic graph of a composite number m, denoted by A m . It has been observed that there exist different composite numbers for which the arithmetic graphs are isomorphic. It is proved that the maximum distance between any two vertices of A m is two or three. Conditions under which the vertices have the same degrees and neighborhoods have also been identified. Symmetric behavior of the vertices lead to the study of the metric dimension of A m which gives minimum cardinality of vertices to distinguish all vertices in the graph. We give exact formulae for the metric dimension of A m , when m has exactly two distinct prime divisors. Moreover, we give bounds on the metric dimension of A m , when m has at least three distinct prime divisors. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessArticle
On the Generalized Hermite–Hadamard Inequalities via the Tempered Fractional Integrals
Symmetry 2020, 12(4), 595; https://doi.org/10.3390/sym12040595 - 08 Apr 2020
Cited by 12
Abstract
Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if [...] Read more.
Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of λ -incomplete gamma function. Using the new notation, we established a few inequalities of the Hermite–Hadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, Riemann–Liouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
Open AccessArticle
Modeling Alcohol Concentration in Blood via a Fractional Context
Symmetry 2020, 12(3), 459; https://doi.org/10.3390/sym12030459 - 13 Mar 2020
Cited by 2
Abstract
We use a conformable fractional derivative G T α through two kernels T ( t , α ) = e ( α 1 ) t and T ( t , α ) = t 1 α in order to model the [...] Read more.
We use a conformable fractional derivative G T α through two kernels T ( t , α ) = e ( α 1 ) t and T ( t , α ) = t 1 α in order to model the alcohol concentration in blood; we also work with the conformable Gaussian differential equation (CGDE) of this model, to evaluate how the curve associated with such a system adjusts to the data corresponding to the blood alcohol concentration. As a practical application, using the symmetry of the solution associated with the CGDE, we show the advantage of our conformable approaches with respect to the usual ordinary derivative. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessArticle
Entropy Generation in a Mass-Spring-Damper System Using a Conformable Model
Symmetry 2020, 12(3), 395; https://doi.org/10.3390/sym12030395 - 04 Mar 2020
Cited by 1
Abstract
This article studies the entropy generation of a mass-spring-damper mechanical system, under the conformable fractional operator definition. We perform several simulations by varying the fractional order γ and the damping ratio ζ , including the usual dynamic response when γ = 1.0 and [...] Read more.
This article studies the entropy generation of a mass-spring-damper mechanical system, under the conformable fractional operator definition. We perform several simulations by varying the fractional order γ and the damping ratio ζ , including the usual dynamic response when γ = 1.0 and the typical damping cases. We analyze the entropy production for this system and its strong dependency on both γ and ζ parameters. Therefore, we determine their optimal values to obtain the highest efficiency of the MSD response, as well as other impressive features. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessArticle
On the Inverse Degree Polynomial
Symmetry 2019, 11(12), 1490; https://doi.org/10.3390/sym11121490 - 07 Dec 2019
Abstract
Using the symmetry property of the inverse degree index, in this paper, we obtain several mathematical relations of the inverse degree polynomial, and we show that some properties of graphs, such as the cardinality of the set of vertices and edges, or the [...] Read more.
Using the symmetry property of the inverse degree index, in this paper, we obtain several mathematical relations of the inverse degree polynomial, and we show that some properties of graphs, such as the cardinality of the set of vertices and edges, or the cyclomatic number, can be deduced from their inverse degree polynomials. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
Open AccessArticle
Improved Image Splicing Forgery Detection by Combination of Conformable Focus Measures and Focus Measure Operators Applied on Obtained Redundant Discrete Wavelet Transform Coefficients
Symmetry 2019, 11(11), 1392; https://doi.org/10.3390/sym11111392 - 10 Nov 2019
Abstract
The image is the best information carrier in the current digital era and the easiest to manipulate. Image manipulation causes the integrity of this information carrier to be ambiguous. The image splicing technique is commonly used to manipulate images by fusing different regions [...] Read more.
The image is the best information carrier in the current digital era and the easiest to manipulate. Image manipulation causes the integrity of this information carrier to be ambiguous. The image splicing technique is commonly used to manipulate images by fusing different regions in one image. Over the last decade, it has been confirmed that various structures in science and engineering can be demonstrated more precisely by fractional calculus using integrals or derivative operators. Many fractional-order-based techniques have been used in the image-processing field. Recently, a new specific fractional calculus, called conformable calculus, was delivered. Herein, we employ the combination of conformable focus measures (CFMs), and focus measure operators (FMOs) in obtaining redundant discrete wavelet transform (RDWT) coefficients for improving the image splicing forgery detection. The process of image splicing disorders the content of tampered image and causes abnormality in the image features. The spliced region’s boundaries are usually blurring to avoid detection. To make use of the blurred information, both CFMs and FMOs are used to calculate the degree of blurring of the tampered region’s boundaries for image splicing detection. The two public image datasets IFS-TC and CASIA TIDE V2 are used for evaluation of the proposed method. The obtained results of the proposed method achieved accuracy rate 98.30% for Cb channel on IFS-TC image dataset and 98.60% of the Cb channel on CASIA TIDE V2 with 24-D feature vector. The proposed method exhibited superior results compared with other image splicing detection methods. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessArticle
A Multi-Stage Homotopy Perturbation Method for the Fractional Lotka-Volterra Model
Symmetry 2019, 11(11), 1330; https://doi.org/10.3390/sym11111330 - 24 Oct 2019
Abstract
In this work, we propose an efficient multi-stage homotopy perturbation method to find an analytic solution to the fractional Lotka-Volterra model. We obtain its order of accuracy, and we study the stability of the system. Moreover, we present several examples to show of [...] Read more.
In this work, we propose an efficient multi-stage homotopy perturbation method to find an analytic solution to the fractional Lotka-Volterra model. We obtain its order of accuracy, and we study the stability of the system. Moreover, we present several examples to show of the effectiveness of this method, and we conclude that the value of the derivative order plays an important role in the trajectories velocity. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessFeature PaperArticle
A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework
Symmetry 2019, 11(10), 1322; https://doi.org/10.3390/sym11101322 - 22 Oct 2019
Cited by 2
Abstract
The new general theory dedicated to the stability for LTI MIMO, in particular nonsquare, fractional-order systems described by the Grünwald–Letnikov discrete-time state–space domain is presented in this paper. Such systems under inverse model control, principally MV/perfect control, represent a real research challenge due [...] Read more.
The new general theory dedicated to the stability for LTI MIMO, in particular nonsquare, fractional-order systems described by the Grünwald–Letnikov discrete-time state–space domain is presented in this paper. Such systems under inverse model control, principally MV/perfect control, represent a real research challenge due to an infinite number of solutions to the underlying inverse problem for nonsquare matrices. Therefore, the paper presents a new algorithm for fractional-order perfect control with corresponding stability formula involving recently given H- and σ -inverse of nonsquare matrices, up to now applied solely to the integer-order plants. On such foundation a new set of stability-related tools is introduced, among them the key role played by so-called control zeros. Control zeros constitute an extension of transmission zeros for nonsquare fractional-order LTI MIMO systems under inverse model control. Based on the sets of stable control zeros a minimum-phase behavior is specified because of the stability of newly defined perfect control law described in the non-integer-order framework. The whole theory is complemented by pole-free fractional-order perfect control paradigm, a special case of fractional-order perfect control strategy. A significant number of simulation examples confirm the correctness and research potential proposed in the paper methodology. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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Open AccessArticle
New Hermite–Hadamard Type Inequalities Involving Non-Conformable Integral Operators
Symmetry 2019, 11(9), 1108; https://doi.org/10.3390/sym11091108 - 03 Sep 2019
Cited by 6
Abstract
At present, inequalities have reached an outstanding theoretical and applied development and they are the methodological base of many mathematical processes. In particular, Hermite– Hadamard inequality has received considerable attention. In this paper, we prove some new results related to Hermite–Hadamard inequality via [...] Read more.
At present, inequalities have reached an outstanding theoretical and applied development and they are the methodological base of many mathematical processes. In particular, Hermite– Hadamard inequality has received considerable attention. In this paper, we prove some new results related to Hermite–Hadamard inequality via symmetric non-conformable integral operators. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)

Review

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Open AccessReview
Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods
Symmetry 2020, 12(6), 959; https://doi.org/10.3390/sym12060959 - 05 Jun 2020
Cited by 3
Abstract
Rough volatility models are recently popularized by the need of a consistent model for the observed empirical volatility in the financial market. In this case, it has been shown that the empirical volatility in the financial market is extremely consistent with the rough [...] Read more.
Rough volatility models are recently popularized by the need of a consistent model for the observed empirical volatility in the financial market. In this case, it has been shown that the empirical volatility in the financial market is extremely consistent with the rough volatility. Currently, fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form and therefore, we must rely on numerical methods to obtain a solution. In this paper, we will be giving a short introduction to option pricing theory (Black–Scholes model, classical Heston model and its characteristic function), an overview of the current advancements on the rough Heston model and numerical methods (fractional Adams–Bashforth–Moulton method and multipoint Padé approximation method) for solving the fractional Riccati equation. In addition, we will investigate on the performance of multipoint Padé approximation method for the small u values in D α h ( u i / 2 , x ) as it plays a huge role in the computation for the option prices. We further confirm that the solution generated by multipoint Padé (3,3) method for the fractional Riccati equation is incredibly consistent with the solution generated by fractional Adams–Bashforth–Moulton method. Full article
(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)
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