Research

17 pages, 491 KiB

Open AccessEditor’s ChoiceArticle

On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

by Bilal A. Rather, Shariefuddin Pirzada, Tariq A. Naikoo and Yilun Shang

Mathematics 2021, 9(5), 482; https://doi.org/10.3390/math9050482 - 26 Feb 2021

Cited by 36 | Viewed by 2571

Given a commutative ring R with identity

1 \neq 0

, let the set

Z (R)

denote the set of zero-divisors and let

Z^{*} (R) = Z (R) ∖ {0}

be the set of [...] Read more.

Given a commutative ring R with identity

1 \neq 0

, let the set

Z (R)

denote the set of zero-divisors and let

Z^{*} (R) = Z (R) ∖ {0}

be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by

Γ (R)

, is a simple graph whose vertex set is

Z^{*} (R)

and each pair of vertices in

Z^{*} (R)

are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs

Γ (Z_{n})

for

n = p^{N_{1}} q^{N_{2}}

, where

p < q

are primes and

N_{1}, N_{2}

are positive integers. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

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7 pages, 237 KiB

Open AccessArticle

On Maximal Distance Energy

by Shaowei Sun, Kinkar Chandra Das and Yilun Shang

Mathematics 2021, 9(4), 360; https://doi.org/10.3390/math9040360 - 11 Feb 2021

Cited by 1 | Viewed by 1292

Abstract

Let G be a graph of order n. If the maximal connected subgraph of G has no cut vertex then it is called a block. If each block of graph G is a clique then G is called clique tree. The distance [...] Read more.

Let G be a graph of order n. If the maximal connected subgraph of G has no cut vertex then it is called a block. If each block of graph G is a clique then G is called clique tree. The distance energy

E_{D} (G)

of graph G is the sum of the absolute values of the eigenvalues of the distance matrix

D (G)

. In this paper, we study the properties on the eigencomponents corresponding to the distance spectral radius of some special class of clique trees. Using this result we characterize a graph which gives the maximum distance spectral radius among all clique trees of order n with k cliques. From this result, we confirm a conjecture on the maximum distance energy, which was given in Lin et al. Linear Algebra Appl 467(2015) 29-39. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

7 pages, 721 KiB

Open AccessArticle

The Mean Values of Character Sums and Their Applications

by Jiafan Zhang and Yuanyuan Meng

Mathematics 2021, 9(4), 318; https://doi.org/10.3390/math9040318 - 05 Feb 2021

Cited by 5 | Viewed by 1236

Abstract

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a [...] Read more.

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

7 pages, 257 KiB

Open AccessArticle

On Homogeneous Combinations of Linear Recurrence Sequences

by Marie Hubálovská, Štěpán Hubálovský and Eva Trojovská

Mathematics 2020, 8(12), 2152; https://doi.org/10.3390/math8122152 - 03 Dec 2020

Viewed by 1524

Abstract

Let

{(F_{n})}_{n \geq 0}

be the Fibonacci sequence given by

F_{n + 2} = F_{n + 1} + F_{n}

, for

n \geq 0

, where

F_{0} = 0

and

F_{1} = 1

. [...] Read more.

Let

{(F_{n})}_{n \geq 0}

be the Fibonacci sequence given by

F_{n + 2} = F_{n + 1} + F_{n}

, for

n \geq 0

, where

F_{0} = 0

and

F_{1} = 1

. There are several interesting identities involving this sequence such as

F_{n}^{2} + F_{n + 1}^{2} = F_{2 n + 1}

, for all

n \geq 0

. In 2012, Chaves, Marques and Togbé proved that if

{(G_{m})}_{m}

is a linear recurrence sequence (under weak assumptions) and

G_{n + 1}^{s} + \dots + G_{n + ℓ}^{s} \in {(G_{m})}_{m}

, for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ł and the parameters of

{(G_{m})}_{m}

. In this paper, we shall prove that if

P (x_{1}, \dots, x_{ℓ})

is an integer homogeneous s-degree polynomial (under weak hypotheses) and if

P (G_{n + 1}, \dots, G_{n + ℓ}) \in {(G_{m})}_{m}

for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ℓ, the parameters of

{(G_{m})}_{m}

and the coefficients of P. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

5 pages, 306 KiB

Open AccessArticle

On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function

by Štěpán Hubálovský and Eva Trojovská

Mathematics 2020, 8(10), 1796; https://doi.org/10.3390/math8101796 - 16 Oct 2020

Viewed by 1328

Abstract

Let

F_{n}

be the nth Fibonacci number. The order of appearance

z (n)

of a natural number n is defined as the smallest positive integer k such that

F_{k} \equiv 0 (mod n)

. In this [...] Read more.

Let

F_{n}

be the nth Fibonacci number. The order of appearance

z (n)

of a natural number n is defined as the smallest positive integer k such that

F_{k} \equiv 0 (mod n)

. In this paper, we shall find all positive solutions of the Diophantine equation

z (φ (n)) = n

, where

φ

is the Euler totient function. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

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8 pages, 252 KiB

Open AccessArticle

Repdigits as Product of Fibonacci and Tribonacci Numbers

by Dušan Bednařík and Eva Trojovská

Mathematics 2020, 8(10), 1720; https://doi.org/10.3390/math8101720 - 07 Oct 2020

Cited by 5 | Viewed by 1677

Abstract

In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a [...] Read more.

In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a Tribonacci number (both with the same indexes). To work on this problem, our approach is to combine lower bounds from the Baker’s theory with reduction methods (based on the theory of continued fractions) due to Dujella and Pethö. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

6 pages, 249 KiB

Open AccessArticle

Algebraic Numbers of the form α^T with α Algebraic and T Transcendental

by Štěpán Hubálovský and Eva Trojovská

Mathematics 2020, 8(10), 1687; https://doi.org/10.3390/math8101687 - 01 Oct 2020

Cited by 1 | Viewed by 1288

Abstract

Let

α \neq 1

be a positive real number and let

P (x)

be a non-constant rational function with algebraic coefficients. In this paper, in particular, we prove that the set of algebraic numbers of the form [...] Read more.

Let

α \neq 1

be a positive real number and let

P (x)

be a non-constant rational function with algebraic coefficients. In this paper, in particular, we prove that the set of algebraic numbers of the form

α^{P (T)}

, with T transcendental, is dense in some open interval of

R

. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

10 pages, 245 KiB

Open AccessArticle

On Special Spacelike Hybrid Numbers

by Anetta Szynal-Liana and Iwona Włoch

Mathematics 2020, 8(10), 1671; https://doi.org/10.3390/math8101671 - 01 Oct 2020

Cited by 6 | Viewed by 1758

Abstract

Hybrid numbers are generalizations of complex, hyperbolic and dual numbers. A hyperbolic complex structure is frequently used in both pure mathematics and numerous areas of physics. In this paper we introduce a special kind of spacelike hybrid number, namely the [...] Read more.

Hybrid numbers are generalizations of complex, hyperbolic and dual numbers. A hyperbolic complex structure is frequently used in both pure mathematics and numerous areas of physics. In this paper we introduce a special kind of spacelike hybrid number, namely the

F (p, n)

-Fibonacci hybrid numbers and we give some of their properties. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

6 pages, 211 KiB

Open AccessArticle

The Italian Domination Numbers of Some Products of Directed Cycles

by Kijung Kim

Mathematics 2020, 8(9), 1472; https://doi.org/10.3390/math8091472 - 01 Sep 2020

Cited by 4 | Viewed by 2005

Abstract

An Italian dominating function on a digraph D with vertex set

V (D)

is defined as a function

f : V (D) \to {0, 1, 2}

such that every vertex [...] Read more.

An Italian dominating function on a digraph D with vertex set

V (D)

is defined as a function

f : V (D) \to {0, 1, 2}

such that every vertex

v \in V (D)

with

f (v) = 0

has at least two in-neighbors assigned 1 under f or one in-neighbor w with

f (w) = 2

. In this article, we determine the exact values of the Italian domination numbers of some products of directed cycles. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

8 pages, 436 KiB

Open AccessArticle

On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

by Pavel Trojovský

Mathematics 2020, 8(8), 1387; https://doi.org/10.3390/math8081387 - 18 Aug 2020

Cited by 3 | Viewed by 2203

Abstract

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about [...] Read more.

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence

{(T_{n}^{(k)})}_{n \geq 0}

. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

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10 pages, 274 KiB

Open AccessArticle

Some Diophantine Problems Related to k-Fibonacci Numbers

by Pavel Trojovský and Štěpán Hubálovský

Mathematics 2020, 8(7), 1047; https://doi.org/10.3390/math8071047 - 30 Jun 2020

Cited by 1 | Viewed by 1697

Abstract

Let

k \geq 1

be an integer and denote

{(F_{k, n})}_{n}

as the k-Fibonacci sequence whose terms satisfy the recurrence relation [...] Read more.

Let

k \geq 1

be an integer and denote

{(F_{k, n})}_{n}

as the k-Fibonacci sequence whose terms satisfy the recurrence relation

F_{k, n} = k F_{k, n - 1} + F_{k, n - 2}

, with initial conditions

F_{k, 0} = 0

and

F_{k, 1} = 1

. In the same way, the k-Lucas sequence

{(L_{k, n})}_{n}

is defined by satisfying the same recursive relation with initial values

L_{k, 0} = 2

and

L_{k, 1} = k

. The sequences

{(F_{k, n})}_{n \geq 0}

and

{(L_{k, n})}_{n \geq 0}

were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that

F_{k, n}^{2} + F_{k, n + 1}^{2} = F_{k, 2 n + 1}

and

F_{k, n + 1}^{2} - F_{k, n - 1}^{2} = k F_{k, 2 n}

, for all

k \geq 1

and

n \geq 0

. In this paper, we shall prove that if

k > 1

and

F_{k, n}^{s} + F_{k, n + 1}^{s} \in {(F_{k, m})}_{m \geq 1}

for infinitely many positive integers n, then

s = 2

. Similarly, that if

F_{k, n + 1}^{s} - F_{k, n - 1}^{s} \in {(k F_{k, m})}_{m \geq 1}

holds for infinitely many positive integers n, then

s = 1

or

s = 2

. This generalizes a Marques and Togbé result related to the case

k = 1

. Furthermore, we shall solve the Diophantine equations

F_{k, n} = L_{k, m}

,

F_{k, n} = F_{n, k}

and

L_{k, n} = L_{n, k}

. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

10 pages, 334 KiB

Open AccessArticle

A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers

by Ana Paula Chaves and Pavel Trojovský

Mathematics 2020, 8(6), 1010; https://doi.org/10.3390/math8061010 - 20 Jun 2020

Cited by 4 | Viewed by 2272

Abstract

The sequence of the k-generalized Fibonacci numbers

{(F_{n}^{(k)})}_{n}

is defined by the recurrence

F_{n}^{(k)} = \sum_{j = 1}^{k} F_{n - j}^{(k)}

beginning with the k [...] Read more.

The sequence of the k-generalized Fibonacci numbers

{(F_{n}^{(k)})}_{n}

is defined by the recurrence

F_{n}^{(k)} = \sum_{j = 1}^{k} F_{n - j}^{(k)}

beginning with the k terms

0, \dots, 0, 1

. In this paper, we shall solve the Diophantine equation

F_{n}^{(k)} = {(F_{m}^{(l)})}^{2} + 1

, in positive integers

m, n, k

and l. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

8 pages, 259 KiB

Open AccessArticle

On the Growth of Some Functions Related to z(n)

by Pavel Trojovský

Mathematics 2020, 8(6), 876; https://doi.org/10.3390/math8060876 - 01 Jun 2020

Cited by 2 | Viewed by 1251

Abstract

The order of appearance

z : Z_{> 0} \to Z_{> 0}

is an arithmetic function related to the Fibonacci sequence

{(F_{n})}_{n}

. This function is defined as the smallest positive integer solution of the congruence [...] Read more.

The order of appearance

z : Z_{> 0} \to Z_{> 0}

is an arithmetic function related to the Fibonacci sequence

{(F_{n})}_{n}

. This function is defined as the smallest positive integer solution of the congruence

F_{k} \equiv 0 (\mod n)

. In this paper, we shall provide lower and upper bounds for the functions

\sum_{n \leq x} z (n) / n

,

\sum_{p \leq x} z (p)

and

\sum_{p^{r} \leq x} z (p^{r})

. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

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