Symmetry

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11 pages, 272 KiB

Open AccessArticle

On the Evaluation of Rectangular Matrix Permanents: A Symmetric and Combinatorial Analysis

by Ahmet Zahid Küçük

Symmetry 2025, 17(4), 507; https://doi.org/10.3390/sym17040507 - 27 Mar 2025

Viewed by 76

Abstract

This paper presents a combinatorial perspective on evaluating the permanent for a rectangular matrix. It proves that the permanent can be computed using the permanents of its largest square submatrices. The proof employs a structured combinatorial method and reveals a connection to the [...] Read more.

This paper presents a combinatorial perspective on evaluating the permanent for a rectangular matrix. It proves that the permanent can be computed using the permanents of its largest square submatrices. The proof employs a structured combinatorial method and reveals a connection to the subset-sum problem, known as the grid shading problem. Furthermore, this study uncovers an inherent symmetry in the distribution of terms, highlighting structured patterns within permanent computation. This perspective bridges combinatorial principles with matrix theory, offering new insights into their interplay. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

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16 pages, 294 KiB

Open AccessArticle

A New Family of Multipartition Graph Operations and Its Applications in Constructing Several Special Graphs

by Qiuping Li, Liangwen Tang, Qingyun Liu and Mugang Lin

Symmetry 2025, 17(3), 467; https://doi.org/10.3390/sym17030467 - 20 Mar 2025

Viewed by 180

Abstract

A new family of graph operations based on multipartite graph with an arbitrary number of parts is defined and their applications are explored in this paper. The complete spectra of graphs derived from multipartite graphs are determined. Because the adjacency matrix of the [...] Read more.

A new family of graph operations based on multipartite graph with an arbitrary number of parts is defined and their applications are explored in this paper. The complete spectra of graphs derived from multipartite graphs are determined. Because the adjacency matrix of the multipartite graph is symmetric, we can use it to generate an unlimited number of special symmetric graphs. Methods for generating countless new families of integral graphs using these multipartite graph operations have been presented. By applying these multipartite graph operations, we can construct infinitely many orderenergetic graphs from orderenergetic or non-orderenergetic graphs. Additionally, infinite pairs of equienergetic and non-cospectral graphs can be generated through these new operations. Moreover, this kind of graph operation can also be used to construct other special graphs related to eigenvalues and energy. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

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13 pages, 536 KiB

Open AccessArticle

Symmetric Point Sets with Few Intersection Numbers in PG(r,q)

by Stefano Innamorati

Symmetry 2025, 17(2), 179; https://doi.org/10.3390/sym17020179 - 24 Jan 2025

Viewed by 503

Abstract

In 1965, H. Retkin and E. Stein defined a symmetric point set as a set of points with the same intersection numbers. In this paper, we perform a detailed analysis of symmetric point sets of the finite projective space which shows that the [...] Read more.

In 1965, H. Retkin and E. Stein defined a symmetric point set as a set of points with the same intersection numbers. In this paper, we perform a detailed analysis of symmetric point sets of the finite projective space which shows that the class of symmetric sets is very broad including caps, two-character sets and transitive sets. We derive necessary conditions for the existence of such sets. Since the most studied sets are caps and two-character sets and not much seems to be known in the general case of sets, which are different from caps, with more than two intersection numbers, by using incidence-preserving group actions, symmetric point sets with few intersection numbers are provided. The results indicate that any finite projective space contains symmetric sets with few intersection numbers. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

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15 pages, 425 KiB

Open AccessArticle

Sequence of Bounds for Spectral Radius and Energy of Digraph

by Jietong Zhao, Saira Hameed, Uzma Ahmad, Ayesha Tabassum and Leila Asgharsharghi

Symmetry 2024, 16(10), 1386; https://doi.org/10.3390/sym16101386 - 18 Oct 2024

Cited by 1 | Viewed by 1151

Abstract

The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph

D

having at [...] Read more.

The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph

D

having at least one doubly adjacent vertex in terms of indegree is proposed. Particularly, it is exhibited that

ρ (D) \geq α^{j} = \sqrt{\frac{\sum_{p = 1}^{m} {(χ_{j + 1}^{(p)})}^{2}}{\sum_{p = 1}^{m} {(χ_{j}^{(p)})}^{2}}}

, such that equality is attained iff

D = \overset{\leftrightarrow}{G} +

{

DE \notin

Cycle}, where each component of associated graph is a k-regular or

(k_{1}, k_{2})

semiregular bipartite. By utilizing the sequence of lower bounds of the spectral radius of

D

, the sequence of upper bounds of energy of

D

, where the sequence decreases when

\sqrt{\frac{e}{U}} \leq α^{j}

and increases when

\sqrt{\frac{e}{U}} > α^{j}

, are also proposed. All of the obtained inequalities are elaborated using examples. We also discuss the monotonicity of these sequences. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

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17 pages, 284 KiB

Open AccessArticle

A Note on Incomplete Fibonacci–Lucas Relations

by Jingyang Zhong, Jialing Yao and Chan-Liang Chung

Symmetry 2023, 15(12), 2113; https://doi.org/10.3390/sym15122113 - 24 Nov 2023

Cited by 2 | Viewed by 1056

Abstract

We define the incomplete generalized bivariate Fibonacci p-polynomials and the incomplete generalized bivariate Lucas p-polynomials. We study their recursive relations and derive an interesting relationship through their generating functions. Subsequently, we prove an incomplete version of the well-known Fibonacci–Lucas relation and [...] Read more.

We define the incomplete generalized bivariate Fibonacci p-polynomials and the incomplete generalized bivariate Lucas p-polynomials. We study their recursive relations and derive an interesting relationship through their generating functions. Subsequently, we prove an incomplete version of the well-known Fibonacci–Lucas relation and make some extensions to the relation involving incomplete generalized bivariate Fibonacci and Lucas p-polynomials. An argument about going from the regular to the incomplete Fibonacci–Lucas relation is discussed. We provide a relation involving the incomplete Leonardo and the incomplete Lucas–Leonardo p-numbers as an illustration. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

11 pages, 299 KiB

Open AccessArticle

More Variations on Shuffle Squares

by Jarosław Grytczuk, Bartłomiej Pawlik and Mariusz Pleszczyński

Symmetry 2023, 15(11), 1982; https://doi.org/10.3390/sym15111982 - 26 Oct 2023

Cited by 1 | Viewed by 1749

Abstract

We study an abstract variant of squares (and shuffle squares) defined by a constraint graph G, specifying which pairs of words form a square. So, a shuffle G-square is a word that can be split into two disjoint subwords U and W (of [...] Read more.

We study an abstract variant of squares (and shuffle squares) defined by a constraint graph G, specifying which pairs of words form a square. So, a shuffle G-square is a word that can be split into two disjoint subwords U and W (of the same length), which are joined by an edge. This setting generalizes a recently introduced model of shuffle squares based on word symmetry and permutations. By using the probabilistic method, we provide a sufficient condition for a constraint graph G guaranteeing the avoidability of shuffle G-squares. By a more-elementary method (known as Rosenfeld counting), we prove that G-squares are avoidable over an alphabet of size

4 α

,

α > 1

, provided that the degree of every word of length n in G is at most

α^{n}

. We also introduce the concept of the cutting distance between words and state several conjectures involving this notion and various kinds of shuffle squares. We suspect that, for every

k ⩾ 2

, there is a constant

c_{k}

such that every even word can be turned into a shuffle square by cutting it in at most

c_{k}

places and rearranging the resulting pieces. We present some computational, as well as theoretical evidence in favor of this conjecture. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

17 pages, 312 KiB

Open AccessArticle

The p-Numerical Semigroup of the Triple of Arithmetic Progressions

by Takao Komatsu and Haotian Ying

Symmetry 2023, 15(7), 1328; https://doi.org/10.3390/sym15071328 - 29 Jun 2023

Cited by 4 | Viewed by 1228

Abstract

For given positive integers

a_{1}, a_{2}, \dots, a_{k}

with

gcd (a_{1}, a_{2}, \dots, a_{k}) = 1

, the denumerant [...] Read more.

For given positive integers

a_{1}, a_{2}, \dots, a_{k}

with

gcd (a_{1}, a_{2}, \dots, a_{k}) = 1

, the denumerant

d (n) = d (n; a_{1}, a_{2}, \dots, a_{k})

is the number of nonnegative solutions

(x_{1}, x_{2}, \dots, x_{k})

of the linear equation

a_{1} x_{1} + a_{2} x_{2} + \dots + a_{k} x_{k} = n

for a positive integer n. For a given nonnegative integer p, let

S_{p} = S_{p} (a_{1}, a_{2}, \dots, a_{k})

be the set of all nonnegative integer n’s such that

d (n) > p

. In this paper, by introducing the p-numerical semigroup, where the set

N_{0} \ S_{p}

is finite, we give explicit formulas of the p-Frobenius number, which is the maximum of the set

N_{0} \ S_{p}

, and related values for the triple of arithmetic progressions. The main aim is to determine the elements of the p-Apéry set. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

Review

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29 pages, 396 KiB

Open AccessReview

Reciprocal Symmetry via Inverse Series Pairs

by Wenchang Chu

Symmetry 2023, 15(5), 1086; https://doi.org/10.3390/sym15051086 - 15 May 2023

Viewed by 1532

Abstract

Reciprocal series are employed to systematically review convolution sums, orthogonality relations, recurrence relations and reciprocal formulae for several classical number sequences, such as binomial coefficients, Stirling numbers, Bernoulli numbers, and Euler numbers. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

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