Research

11 pages, 282 KiB

Open AccessArticle

Distance Fibonacci Polynomials by Graph Methods

by Dominik Strzałka, Sławomir Wolski and Andrzej Włoch

Symmetry 2021, 13(11), 2075; https://doi.org/10.3390/sym13112075 - 03 Nov 2021

Cited by 2 | Viewed by 1283

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, [...] Read more.

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

13 pages, 366 KiB

Open AccessArticle

A Single-Key Variant of LightMAC_Plus

by Haitao Song

Symmetry 2021, 13(10), 1818; https://doi.org/10.3390/sym13101818 - 29 Sep 2021

Viewed by 957

Abstract

LightMAC_Plus proposed by Naito (ASIACRYPT 2017) is a blockcipher-based MAC that has beyond the birthday bound security without message length in the sense of PRF (Pseudo-Random Function) security. In this paper, we present a single-key variant of LightMAC_Plus that has beyond the birthday [...] Read more.

LightMAC_Plus proposed by Naito (ASIACRYPT 2017) is a blockcipher-based MAC that has beyond the birthday bound security without message length in the sense of PRF (Pseudo-Random Function) security. In this paper, we present a single-key variant of LightMAC_Plus that has beyond the birthday bound security in terms of PRF security. Compared with the previous construction LightMAC_Plus1k of Naito (CT-RSA 2018), our construction is simpler and of higher efficiency. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

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11 pages, 579 KiB

Open AccessArticle

Effect of a Ring onto Values of Eigenvalue–Based Molecular Descriptors

by Izudin Redžepović, Slavko Radenković and Boris Furtula

Symmetry 2021, 13(8), 1515; https://doi.org/10.3390/sym13081515 - 18 Aug 2021

Cited by 4 | Viewed by 1538

Abstract

The eigenvalues of the characteristic polynomial of a graph are sensitive to its symmetry-related characteristics. Within this study, we have examined three eigenvalue–based molecular descriptors. These topological molecular descriptors, among others, are gathering information on the symmetry of a molecular graph. Furthermore, they [...] Read more.

The eigenvalues of the characteristic polynomial of a graph are sensitive to its symmetry-related characteristics. Within this study, we have examined three eigenvalue–based molecular descriptors. These topological molecular descriptors, among others, are gathering information on the symmetry of a molecular graph. Furthermore, they are being ordinarily employed for predicting physico–chemical properties and/or biological activities of molecules. It has been shown that these indices describe well molecular features that are depending on fine structural details. Therefore, revealing the impact of structural details on the values of the eigenvalue–based topological indices should give a hunch how physico–chemical properties depend on them as well. Here, an effect of a ring in a molecule on the values of the graph energy, Estrada index and the resolvent energy of a graph is examined. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

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16 pages, 297 KiB

Open AccessArticle

On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene

by Umar Ali, Hassan Raza and Yasir Ahmed

Symmetry 2021, 13(8), 1374; https://doi.org/10.3390/sym13081374 - 28 Jul 2021

Cited by 4 | Viewed by 1841

Abstract

The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and

2 n

squares, denoted by

L_{n}^{6, 4, 4}

. In this paper, by using [...] Read more.

The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and

2 n

squares, denoted by

L_{n}^{6, 4, 4}

. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of

L_{n}^{6, 4, 4}

consisting of the eigenvalues of symmetric tri-diagonal matrices

L_{A}

and

L_{S}

of order

4 n + 1

. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

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17 pages, 334 KiB

Open AccessArticle

Some Properties of the Arithmetic–Geometric Index

by Edil D. Molina, José M. Rodríguez, José L. Sánchez and José M. Sigarreta

Symmetry 2021, 13(5), 857; https://doi.org/10.3390/sym13050857 - 12 May 2021

Cited by 12 | Viewed by 1623

Abstract

Recently, the arithmetic–geometric index (AG) was introduced, inspired by the well-known and studied geometric–arithmetic index (GA). In this work, we obtain new bounds on the arithmetic–geometric index, improving upon some already known bounds. In particular, we show families of graphs where such bounds [...] Read more.

Recently, the arithmetic–geometric index (AG) was introduced, inspired by the well-known and studied geometric–arithmetic index (GA). In this work, we obtain new bounds on the arithmetic–geometric index, improving upon some already known bounds. In particular, we show families of graphs where such bounds are attained. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

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14 pages, 278 KiB

Open AccessArticle

On the Generalized Laplace Transform

by Paul Bosch, Héctor José Carmenate García, José Manuel Rodríguez and José María Sigarreta

Symmetry 2021, 13(4), 669; https://doi.org/10.3390/sym13040669 - 13 Apr 2021

Cited by 6 | Viewed by 1922

Abstract

In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution [...] Read more.

In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

14 pages, 293 KiB

Open AccessArticle

Mathematical Properties of Variable Topological Indices

by José M. Sigarreta

Symmetry 2021, 13(1), 43; https://doi.org/10.3390/sym13010043 - 30 Dec 2020

Cited by 12 | Viewed by 1682

Abstract

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices

A_{α}

and

B_{α}

, defined [...] Read more.

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices

A_{α}

and

B_{α}

, defined for each graph

H = (V (H), E (H))

by

A_{α} (H) = \sum_{i j \in E (H)} f {(d_{i}, d_{j})}^{α}

and

B_{α} (H) = \sum_{i \in V (H)} h {(d_{i})}^{α}

, where

d_{i}

denotes the degree of the vertex i and

α

is any real number. Many important topological indices can be obtained from

A_{α}

and

B_{α}

by choosing appropriate symmetric functions and values of

α

. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of

A_{α}

(respectively,

B_{α}

) involving

A_{β}

(respectively,

B_{β}

). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

12 pages, 1467 KiB

Open AccessArticle

Transitivity on Minimum Dominating Sets of Paths and Cycles

by Juan C. Hernández-Gómez, Gerardo Reyna-Hérnandez, Jesús Romero-Valencia and Omar Rosario Cayetano

Symmetry 2020, 12(12), 2053; https://doi.org/10.3390/sym12122053 - 11 Dec 2020

Cited by 4 | Viewed by 1505

Abstract

Transitivity on graphs is a concept widely investigated. This suggest to analyze the action of automorphisms on other sets. In this paper, we study the action on the family of

γ

-sets (minimum dominating sets), the graph is called

γ

-transitive if given [...] Read more.

Transitivity on graphs is a concept widely investigated. This suggest to analyze the action of automorphisms on other sets. In this paper, we study the action on the family of

γ

-sets (minimum dominating sets), the graph is called

γ

-transitive if given two

γ

-sets there exists an automorphism which maps one onto the other. We deal with two families: paths

P_{n}

and cycles

C_{n}

. Their

γ

-sets are fully characterized and the action of the automorphism group on the family of

γ

-sets is fully analyzed. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

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15 pages, 6878 KiB

Open AccessArticle

Icosahedral Polyhedra from D₆ Lattice and Danzer’s ABCK Tiling

by Abeer Al-Siyabi, Nazife Ozdes Koca and Mehmet Koca

Symmetry 2020, 12(12), 1983; https://doi.org/10.3390/sym12121983 - 30 Nov 2020

Cited by 3 | Viewed by 1725

Abstract

It is well known that the point group of the root lattice D₆ admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H₃, its roots, and weights are determined in terms of those of D [...] Read more.

It is well known that the point group of the root lattice D₆ admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H₃, its roots, and weights are determined in terms of those of D₆. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D₆ determined by a pair of integers (m₁, m₂) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H₃, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m₁, m₂) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H₃, and those related transformations in 6D space with D₆ symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D₆. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

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