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Symmetry in Combinatorics and Discrete Mathematics, 2nd Edition

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Dear Colleagues,

Many interesting symmetric properties appear in combinatorial numbers, including binomial coefficients, Stirling numbers, and Bernoulli numbers. Such symmetric identities can be interpreted and studied from combinatorial, arithmetical, geometrical, or analytical aspects. This Special Issue will present articles exploring new interpretations, applications, and symmetrical and asymmetrical relations in combinatorics and discrete mathematics.

Prof. Dr. Takao Komatsu
Guest Editor

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Research

7 pages, 656 KB

Open AccessArticle

On the

CF

-Connectedness of the k-Power of P_n

by Michal Staš

Symmetry 2026, 18(4), 617; https://doi.org/10.3390/sym18040617 - 5 Apr 2026

Viewed by 302

Abstract

The study of graph connectivity is a central topic in graph theory, with

CF

-connectedness being a specialized property of interest. A connected graph is

CF

-connected if, in every optimal drawing, there exists a path between every pair of vertices such that [...] Read more.

The study of graph connectivity is a central topic in graph theory, with

CF

-connectedness being a specialized property of interest. A connected graph is

CF

-connected if, in every optimal drawing, there exists a path between every pair of vertices such that no edges cross. This paper explores the

CF

-connectedness of the k-power of

P_{n}

, denoted

P_{n}^{k}

, where

P_{n}

is a path on n vertices,

P_{n}^{k}

is a graph on the same vertex set as

P_{n}

, and an edge

{u, v}

exists in

P_{n}^{k}

if the distance between u and v on

P_{n}

is at most k. The paper concludes with a controversial drawing of the graph

P_{10}^{5}

with only 16 crossings, which refutes the truth of Zheng et al.’s conjecture that the upper bound

cr (P_{n}^{5}) \leq 4 n - 23

holds with equality for all

n \geq 8

. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics, 2nd Edition)

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20 pages, 346 KB

Open AccessArticle

Symmetry and Attention Dynamics in Ducci-Generated Jacobsthal Circulant Matrices

by Bahar Kuloğlu, Taras Goy and Engin Özkan

Symmetry 2026, 18(3), 520; https://doi.org/10.3390/sym18030520 - 18 Mar 2026

Viewed by 430

Abstract

A Ducci sequence generated by the vector

A = (a_{1}, a_{2}, \dots, a_{n}) \in Z^{n}

is defined by

(A, D_{A}, D_{A}^{2}, D_{A}^{3}, \dots)

[...] Read more.

A Ducci sequence generated by the vector

A = (a_{1}, a_{2}, \dots, a_{n}) \in Z^{n}

is defined by

(A, D_{A}, D_{A}^{2}, D_{A}^{3}, \dots)

, where the Ducci map

D : Z^{n} \to Z^{n}

is given by

D_{A} = (| a_{2} - a_{1} |, | a_{3} - a_{2} |, \dots, | a_{n} - a_{n - 1} |, | a_{1} - a_{n} |) .

In this paper, we examine the impact of iterative Ducci transformations on Jacobsthal numbers and construct circulant and skew-circulant matrices generated by the resulting sequences. Their properties are investigated through matrix norms (Euclidean (Frobenius), spectral, and ℓ_p), determinants, and eigenvalues. To extend the classical analysis, we incorporate the Convolutional Block Attention Module (CBAM) from deep learning and interpret the structured matrices as simulated image inputs. By analyzing channel-attention vectors and their variances, we assess how successive Ducci transformations influence attention distribution. The first-order transformation produces greater variance in attention weights, indicating enhanced feature discrimination, whereas higher-order transformations promote a more balanced distribution. The results highlight how Ducci transformations influence attention variance in structured matrices. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics, 2nd Edition)

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