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

Deadline for manuscript submissions: closed (5 July 2020) | Viewed by 44550

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Special Issue Editors

Prof. Dr. Jose M. Rodriguez

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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, Collections and Topics in MDPI journals

Prof. Dr. José M. Sigarreta

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Guest Editor

Faculty of Mathematics. Autonomous University of Guerrero-Acapulco Campus, Calle Carlos E. Adame 54, Garita, Acapulco CP-39650, Guerrero, Mexico
Interests: discrete mathematics; alliances in graphs; conformable and non-conformable calculus; geometry; topological indices
Special Issues, Collections and Topics 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 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

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

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

Published Papers (14 papers)

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34 pages, 8775 KiB

Open AccessArticle

Closed Knight’s Tours on (m,n,r)-Ringboards

by Wasupol Srichote, Ratinan Boonklurb and Sirirat Singhun

Symmetry 2020, 12(8), 1217; https://doi.org/10.3390/sym12081217 - 25 Jul 2020

Viewed by 3830

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 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 > 2 r

, an

(m, n, r)

-ringboard or

(m, n, r)

-annulus-board is defined to be an

m \times 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, n \geq 3

and

m, n > 2 r

has a closed knight’s tour if and only if (a)

m = n = 3

and

r = 1

or (b)

m, n \geq 7

and

r \geq 3

. 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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10 pages, 307 KiB

Open AccessArticle

Trees with Minimum Weighted Szeged Index Are of a Large Diameter

by Risto Atanasov, Boris Furtula and Riste Škrekovski

Symmetry 2020, 12(5), 793; https://doi.org/10.3390/sym12050793 - 9 May 2020

Cited by 2 | Viewed by 2100

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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8 pages, 432 KiB

Open AccessArticle

The Differential on Graph Operator Q(G)

by Ludwin A. Basilio, Jair Castro Simon, Jesús Leaños and Omar Rosario Cayetano

Symmetry 2020, 12(5), 751; https://doi.org/10.3390/sym12050751 - 6 May 2020

Cited by 4 | Viewed by 2777

Abstract

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 [...] Read more.

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

\partial (S)

is defined as

| B (S) | - | S |

. The differential of G, denoted by

\partial (G)

, is the maximum value of

\partial (S)

for all subsets

S \subseteq 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

\partial (G)

and

\partial (Q (G))

. Besides, we exhibit some results relating the differential

\partial (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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30 pages, 8697 KiB

Open AccessArticle

Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag–Leffler Memories Involving the Truncated M-Derivative

by Jesús Emmanuel Solís-Pérez and José Francisco Gómez-Aguilar

Symmetry 2020, 12(4), 626; https://doi.org/10.3390/sym12040626 - 15 Apr 2020

Cited by 15 | Viewed by 2747

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 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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10 pages, 291 KiB

Open AccessArticle

On the Metric Dimension of Arithmetic Graph of a Composite Number

by Shahid ur Rehman, Muhammad Imran and Imran Javaid

Symmetry 2020, 12(4), 607; https://doi.org/10.3390/sym12040607 - 11 Apr 2020

Cited by 6 | Viewed by 2581

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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17 pages, 826 KiB

Open AccessArticle

On the Generalized Hermite–Hadamard Inequalities via the Tempered Fractional Integrals

by Pshtiwan Othman Mohammed, Mehmet Zeki Sarikaya and Dumitru Baleanu

Symmetry 2020, 12(4), 595; https://doi.org/10.3390/sym12040595 - 8 Apr 2020

Cited by 89 | Viewed by 3182

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 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)

11 pages, 964 KiB

Open AccessArticle

Modeling Alcohol Concentration in Blood via a Fractional Context

by Omar Rosario Cayetano, Alberto Fleitas Imbert, José Francisco Gómez-Aguilar and Antonio Fernando Sarmiento Galán

Symmetry 2020, 12(3), 459; https://doi.org/10.3390/sym12030459 - 13 Mar 2020

Cited by 5 | Viewed by 2346

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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11 pages, 1727 KiB

Open AccessArticle

Entropy Generation in a Mass-Spring-Damper System Using a Conformable Model

by Jorge M. Cruz-Duarte, J. Juan Rosales-García and C. Rodrigo Correa-Cely

Symmetry 2020, 12(3), 395; https://doi.org/10.3390/sym12030395 - 4 Mar 2020

Cited by 7 | Viewed by 6813

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.

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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13 pages, 267 KiB

Open AccessArticle

On the Inverse Degree Polynomial

by Paul Bosch, José Manuel Rodríguez, Omar Rosario and José María Sigarreta

Symmetry 2019, 11(12), 1490; https://doi.org/10.3390/sym11121490 - 7 Dec 2019

Viewed by 2058

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.

(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)

16 pages, 2281 KiB

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

by Thamarai Subramaniam, Hamid A. Jalab, Rabha W. Ibrahim and Nurul F. Mohd Noor

Symmetry 2019, 11(11), 1392; https://doi.org/10.3390/sym11111392 - 10 Nov 2019

Cited by 12 | Viewed by 3424

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 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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9 pages, 1803 KiB

Open AccessArticle

A Multi-Stage Homotopy Perturbation Method for the Fractional Lotka-Volterra Model

by Martin P. Arciga-Alejandre, Jorge Sanchez-Ortiz, Francisco J. Ariza-Hernandez and Gabriel Catalan-Angeles

Symmetry 2019, 11(11), 1330; https://doi.org/10.3390/sym11111330 - 24 Oct 2019

Cited by 5 | Viewed by 2484

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 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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21 pages, 2919 KiB

Open AccessFeature PaperArticle

A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework

by Wojciech Przemysław Hunek and Łukasz Wach

Symmetry 2019, 11(10), 1322; https://doi.org/10.3390/sym11101322 - 22 Oct 2019

Cited by 3 | Viewed by 2754

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 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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11 pages, 238 KiB

Open AccessArticle

New Hermite–Hadamard Type Inequalities Involving Non-Conformable Integral Operators

by Juan E. Nápoles Valdés, José M. Rodríguez and José M. Sigarreta

Symmetry 2019, 11(9), 1108; https://doi.org/10.3390/sym11091108 - 3 Sep 2019

Cited by 46 | Viewed by 2765

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.

(This article belongs to the Special Issue Recent Advances in Discrete and Fractional Mathematics)

Review

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20 pages, 965 KiB

Open AccessReview

Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods

by Siow W. Jeng and Adem Kilicman

Symmetry 2020, 12(6), 959; https://doi.org/10.3390/sym12060959 - 5 Jun 2020

Cited by 14 | Viewed by 3870

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 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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