Symmetry

Research

Jump to: Review

32 pages, 968 KB

Open AccessArticle

Bounds for General Zagreb Indices and Improved Topological Coindices with QSPR Benchmarking on Octane Isomers

by Suha Wazzan and Abdu Alameri

Symmetry 2026, 18(7), 1139; https://doi.org/10.3390/sym18071139 - 3 Jul 2026

Viewed by 195

Abstract

Topological descriptors play an important role in chemical graph theory and QSPR/QSAR studies by relating molecular structure to measurable physicochemical properties. Among standard benchmark families, octane isomers are frequently used to evaluate the behavior of degree-based descriptors because of their rich branching patterns [...] Read more.

Topological descriptors play an important role in chemical graph theory and QSPR/QSAR studies by relating molecular structure to measurable physicochemical properties. Among standard benchmark families, octane isomers are frequently used to evaluate the behavior of degree-based descriptors because of their rich branching patterns and well-documented physicochemical data. Although many studies have examined topological indices for octane isomers, comparatively fewer works have focused on topological coindices and derived coindex-based descriptors. In this work, we study several known topological coindices and four derived descriptors, denoted by

K J_{1}, K J_{2}, K J_{3},

and

K J_{4}

, for comparative analysis on the octane-isomer benchmark. We also present a unified treatment of lower and upper bounds for the first and second

(α, β)

-general Zagreb indices, together with their reduced and expanded variants, in terms of basic graph parameters, such as the minimum degree, maximum degree, order, and size. These bounds cover a range of familiar special cases, including classical Zagreb, forgotten, Sombor, and Randić-type indices, thereby placing several known descriptors within a common framework. For the application part, the original octane-isomer analysis is retained as a controlled benchmark for the proposed descriptors. In addition, an expanded QSPR experiment is added using a Zenodo molecular dataset containing 90 organic compounds and nine physicochemical endpoints. SMILES strings were converted into hydrogen-suppressed molecular graphs, graph-theoretical descriptors were computed, and ordinary least squares, ridge regression, and PLS(2) models were evaluated using an 80:20 train/test split and five-fold cross-validation. The expanded results show strong or useful performance for selected endpoints, especially critical volume, molecular volume, standard Gibbs free energy of formation, and logarithmic water solubility, whereas some temperature-related endpoints remain less stable. The results therefore support the usefulness of degree-based and coindex-based descriptors as compact exploratory QSPR variables while also emphasizing the need for cautious interpretation, redundancy analysis, and external validation on broader chemical families. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

► Show Figures

Figure 1

45 pages, 15055 KB

Open AccessArticle

A Novel Differential Geometry Methodology for Curvature and Shape Characterization of Ellipsoids: Shape Transition Symmetry Breaking and Mechanistic Insights into Self-Assembly Curvature Driving Force

by David Uriel Zamora Cisneros, Matthew J. Harrington, Noémie-Manuelle Dorval Courchesne and Alejandro D. Rey

Symmetry 2026, 18(6), 1022; https://doi.org/10.3390/sym18061022 - 13 Jun 2026

Viewed by 238

Abstract

This paper develops, implements, and uses a novel methodology that integrates global and local geometric modeling of ellipsoids and transforms curvatures and shape descriptors into energy metrics that provide curvature-driven mechanistic insight into self-assembly driving force pathways associated with fiber and film formation. [...] Read more.

This paper develops, implements, and uses a novel methodology that integrates global and local geometric modeling of ellipsoids and transforms curvatures and shape descriptors into energy metrics that provide curvature-driven mechanistic insight into self-assembly driving force pathways associated with fiber and film formation. Global parameterization, based on eccentricities, and local parameterization, based on mean and Gaussian curvatures, are mapped into shape and curvedness parameterizations that clearly distinguish critical shape effects from curvedness effects in contrast to classical mean and Gaussian curvature methodologies. Finally, the mechanistic insights are illustrated using a liquid membranology model, transforming the geometric descriptors into bending energy densities that point the way to fiber and film assembly pathways. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

► Show Figures

Figure 1

26 pages, 330 KB

Open AccessArticle

On r-Circulant Matrices with Higher-Order Fibonacci Numbers

by Can Kızılateş, Erkan Kayataş and Wei-Shih Du

Symmetry 2026, 18(6), 1011; https://doi.org/10.3390/sym18061011 - 12 Jun 2026

Viewed by 284

Abstract

In this paper, we introduce and investigate a new class of r-circulant matrices whose entries are generated by higher-order Fibonacci numbers. Explicit representations of the eigenvalues of these matrices are derived by means of the Binet formula together with the structural properties [...] Read more.

In this paper, we introduce and investigate a new class of r-circulant matrices whose entries are generated by higher-order Fibonacci numbers. Explicit representations of the eigenvalues of these matrices are derived by means of the Binet formula together with the structural properties of r-circulant matrices. Based on these representations, a closed-form expression for the determinant is obtained. In addition, several summation identities involving higher-order Fibonacci numbers are established, including formulas for partial sums, sums of squares, and weighted sums. These identities play a fundamental role in the derivation of the norm expressions and spectral estimates of the matrices. Furthermore, several matrix norms, including the Euclidean (Frobenius) norm, the 1-norm, the ∞-norm, and the spectral norm, are investigated in detail. Lower and upper bounds for the spectral norm are obtained for both cases

| r | \geq 1

and

| r | < 1

by employing Hadamard product techniques and classical norm inequalities. Finally, numerical examples are presented to illustrate and validate the theoretical results. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

19 pages, 1136 KB

Open AccessArticle

Canal Hypersurfaces Generated by Pseudo-Null Curves with Bishop Frame in Lorentz–Minkowski 4-Space

by Ahmet Kazan, Sema Kazan, Sümeyye Gür Mazlum, Emel Karaca, Mustafa Altın and Luca Grilli

Symmetry 2026, 18(6), 935; https://doi.org/10.3390/sym18060935 - 29 May 2026

Viewed by 245

Abstract

In this paper, we deal with the canal hypersurfaces that are formed as the envelope of a family of pseudo-hyperspheres or pseudo-hyperbolic hyperspheres with centers lying on a pseudo-null curve with Bishop vector fields in four-dimensional Lorentz–Minkowski space. We give main theorems which [...] Read more.

In this paper, we deal with the canal hypersurfaces that are formed as the envelope of a family of pseudo-hyperspheres or pseudo-hyperbolic hyperspheres with centers lying on a pseudo-null curve with Bishop vector fields in four-dimensional Lorentz–Minkowski space. We give main theorems which contain the parametric expressions of these canal hypersurfaces along with their Gaussian, mean, and principal curvatures and important geometric characterizations. We also provide these characterizations for tubular hypersurfaces. Finally, we construct an example to allow for better understanding and comprehension of the results. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

► Show Figures

Figure 1

22 pages, 405 KB

Open AccessArticle

Multiple Points with Increasing Multiplicities on a Fixed Projective Set

by Edoardo Ballico

Symmetry 2026, 18(5), 877; https://doi.org/10.3390/sym18050877 - 21 May 2026

Viewed by 189

Abstract

Take a finite subset S of an n-dimensional projective space. We study the Hilbert function of the multiples

m S

of S, mainly when S is general or at least very general. We recall several classical conjectures on this problem, raise new [...] Read more.

Take a finite subset S of an n-dimensional projective space. We study the Hilbert function of the multiples

m S

of S, mainly when S is general or at least very general. We recall several classical conjectures on this problem, raise new open questions, and prove some particular cases. An open question is if all

m S

have the expected Hilbert function. We find cases in which there are Zariski open subsets of sets S with maximal rank for all m and pairs

(n, # S)

for which no such open set exists. We start the study of the m-Terracini sets proving when the first one is nonempty for Veronese embeddings. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

20 pages, 1840 KB

Open AccessArticle

The Equitable Coloring of Circulant Graphs

by Xiaoyu Jin, Guiying Yan and Weihua Yang

Symmetry 2026, 18(5), 774; https://doi.org/10.3390/sym18050774 - 30 Apr 2026

Viewed by 449

Abstract

A proper vertex coloring is equitable if the sizes of any two color classes differ by at most one. Any graph G with maximum degree

Δ (G)

admits an equitable

(Δ (G) + 1)

-coloring, computable in [...] Read more.

A proper vertex coloring is equitable if the sizes of any two color classes differ by at most one. Any graph G with maximum degree

Δ (G)

admits an equitable

(Δ (G) + 1)

-coloring, computable in

O (Δ (G) n^{2})

time for n vertices. A circulant graph

G (n; D)

is the graph with vertex set

Z_{n}

and two vertices

x, y

are adjacent if

| x - y | \in \pm D mod n

. The partitioning problem in parallel decoding of multi-edge QC-LDPC codes can be interpreted as an equitable coloring problem. We prove some upper bounds for

χ_{=} (G (n; D))

and develop equitable coloring algorithms, including pattern-based periodic coloring and step-based coloring. The proposed methods typically use fewer than

Δ (G) + 1

colors and have computational complexity lower than

O (Δ (G) n^{2})

for circulant graphs

G (n; D)

with small

| D |

. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

► Show Figures

Figure 1

30 pages, 977 KB

Open AccessArticle

Field-Theoretic Derivation of the Constructal Law from Non-Equilibrium Thermodynamics

by Antonio F. Miguel

Symmetry 2026, 18(5), 732; https://doi.org/10.3390/sym18050732 - 24 Apr 2026

Cited by 1 | Viewed by 469

Abstract

Traditional analyses of transport phenomena rely on prescribed geometric boundaries, yet natural flow systems dynamically evolve their architecture to maximize access to currents. To address this disparity, we propose a field-theoretic framework for the constructal law that treats physical geometry as a dynamic [...] Read more.

Traditional analyses of transport phenomena rely on prescribed geometric boundaries, yet natural flow systems dynamically evolve their architecture to maximize access to currents. To address this disparity, we propose a field-theoretic framework for the constructal law that treats physical geometry as a dynamic state variable, represented by a time-dependent conductivity tensor. Using a variational approach grounded in non-equilibrium thermodynamics, we derive a general tensor evolution equation. Within this framework, macroscopic flow architecture emerges deterministically from the continuous competition between non-linear flux-induced accretion, linear entropic relaxation, and spatial smoothing. Scaling analysis reduces this dynamic to a tri-parameter dimensionless phase space: a morphogenic number driving structural growth, a structural diffusion number governing spatial coherence, and a stochastic intensity number providing the microscopic seeds for symmetry breaking. Our principal result is the analytical prediction of a critical bifurcation. When the local morphogenic number strictly exceeds unity, the system escapes its stable, isotropic configuration and branches into highly conductive, anisotropic architectures. We demonstrate the predictive validity and trans-scalar applicability of this continuum theory by mapping it to highly diverse phase transitions, successfully capturing phenomena ranging from microscopic aerosol agglomeration and microbial resistance, to macroscopic coral plasticity and crystal growth instabilities, and finally to the astrophysical launching of relativistic jets from black holes. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

► Show Figures

Figure 1

15 pages, 1087 KB

Open AccessArticle

Third-Order Differential Subordination for Analytic Functions Involving the Lommel Function of the First Kind

by Suha Hammad, Mohammad El-Ityan, Tariq Al-Hawary, Ibtisam Aldawish and Feras Yousef

Symmetry 2026, 18(4), 642; https://doi.org/10.3390/sym18040642 - 10 Apr 2026

Viewed by 637

Abstract

This study investigates third-order differential subordination and its influence on classes of analytic functions associated with the Lommel function of the first kind. By employing a newly defined operator

L_{w}^{j} f (z)

, we identify and characterize the admissible [...] Read more.

This study investigates third-order differential subordination and its influence on classes of analytic functions associated with the Lommel function of the first kind. By employing a newly defined operator

L_{w}^{j} f (z)

, we identify and characterize the admissible function classes that satisfy the corresponding third-order differential subordinations. These admissibility conditions enable the derivation of several key results, including a sandwich-type theorem obtained as a direct consequence of the established framework. The findings contribute to a broader understanding of analytic functions governed by higher-order differential constraints and highlight the significant role played by the Lommel function in shaping these geometric properties. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

16 pages, 397 KB

Open AccessArticle

Symmetry and Structural Analysis of Power Congruence Graphs over a Set of Moduli

by Ahmad Almutlg and Muhammad Awais Raza

Symmetry 2026, 18(4), 582; https://doi.org/10.3390/sym18040582 - 29 Mar 2026

Cited by 1 | Viewed by 837

Abstract

In this article, we introduce and investigate a novel class of graphs that are called Power Congruence Graph PCGs, which are defined over the vertex set V

= {0, 1, 2, \dots, n - 1}

where [...] Read more.

In this article, we introduce and investigate a novel class of graphs that are called Power Congruence Graph PCGs, which are defined over the vertex set V

= {0, 1, 2, \dots, n - 1}

where two vertices

a, b \in V

are adjacent if

a^{k} \equiv b^{k} (mod m)

for some modulus

m \in M_{p}

, where

M_{p} = {p, p^{2}, \dots, p^{t} ∣ p^{t} < n}

. We thoroughly characterize the structural features of these graphs, establishing that each PCG decomposes into a union of

d + 1

complete components, where

d = \frac{p - 1}{gcd (k, p - 1)}

. The component sizes are explicitly given for n, p, and k. This decomposition highlights symmetry patterns in the component arrangement, emphasizing connectedness and structural balance. We derive key graph-theoretic metrics such as degree distribution, size, chromatic number, clique number and domination number. We also compute the adjacency and Laplacian matrices, as well as their spectra and associated graph energies to better understand the structural similarities and differences among PCGs with different exponents and prime moduli. This paper offers a systematic framework for comprehending power congruence based graph constructs, integrating number theory with structural and spectral graph theory and illustrating the natural symmetry that underpins these combinatorial structures. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

► Show Figures

Figure 1

Review

Jump to: Research

259 pages, 2069 KB

Open AccessReview

A Review on Solving Sylvester-Type Equations

by Qing-Wen Wang and Jiale Gao

Symmetry 2026, 18(6), 984; https://doi.org/10.3390/sym18060984 - 6 Jun 2026

Viewed by 245

Abstract

The solution theory of Sylvester-type equations finds wide applications in control theory, robotics, and image processing. This paper systematically surveys, classifies and summarizes the existing research results of three classes of Sylvester-type equations: matrix equations, tensor equations, and operator equations. It extracts nine [...] Read more.

The solution theory of Sylvester-type equations finds wide applications in control theory, robotics, and image processing. This paper systematically surveys, classifies and summarizes the existing research results of three classes of Sylvester-type equations: matrix equations, tensor equations, and operator equations. It extracts nine mainstream research methods and clarifies the internal correlations among these methods, as well as their applicable equation types. Combined with four prior review articles focusing on special cases of Sylvester-type equations, this work establishes a comprehensive framework for solving such equations. It not only provides a systematic theoretical foundation and a clear research thread for subsequent researchers but also offers valuable methodological insights for further investigations in related fields. Full article

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

Journal Menu

Journal Browser

Mathematics: Feature Papers 2026

Share This Special Issue

Editor

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Related Special Issue

Published Papers (10 papers)

Research

Review

Further Information

Guidelines

MDPI Initiatives

Follow MDPI