arXiv subjects

Duaa Abdullah

Publications and source records attributed to Duaa Abdullah.

At least 19 recordsLinked to original sources

arXiv2026

$\sigma$-Irregularity of Trees with Prescribed Maximum Degree: A Majorization--Duality--Stability Framework

Trees of maximum $\sigma$-irregularity subject to a prescribed maximum degree $\Delta$ have been characterized for $\Delta=4$ and $\Delta=5$, with the extension to arbitrary $\Delta\geqslant 3$. This paper develops a unified framework for this class of problems built on three pillars. A majorization-theoretic reformulation that decomposes $\sigma$-extremization into a Schur-convex optimization over degree sequences and a rearrangement type optimization over tree realizations, valid for every $\Delta\geqslant 3$. We establish a duality between the maximization and minimization problems, alongside the general $\Delta$ closed form for $\sigma_{\max}(n,\Delta)$. A stability result showing that the optimality gap between extremal and near extremal trees is bounded independently of $n$ and grows as $\Theta(\Delta^3)$.

math.GM↗

arXiv2026

Quantum Fractional Revival and Entanglement Entropy in Unitary Cayley Graphs

This paper extends the theory of quantum fractional revival (QFR) on unitary Cayley graphs $X=(V(\mathbb{Z}_n),E(S))$ in several directions that remained unresolved in previous work. First, we investigate QFR with respect to the Laplacian matrix Hamiltonian in addition to the adjacency matrix Hamiltonian. In particular, we prove that for regular graphs the two models differ only by a global phase factor, and we determine the conditions under which the Laplacian framework independently admits QFR. Second, for unitary Cayley graphs of order $n=2p$, where $p$ is an odd prime, we derive an explicit closed-form expression for the minimum revival time, $t^{*}=\frac{2\pi}{p},$ and show that the associated revival amplitudes are given by \[ \alpha=\cos\!\left(\frac{2\pi}{p}\right), \qquad \beta=-i\sin\!\left(\frac{2\pi}{p}\right). \] Third, we provide a complete characterization of strongly cospectral vertex pairs in $X=(V(\mathbb{Z}_n),E(S))$ through the arithmetic structure of $\mathbb{Z}_n$, establishing that strong cospectrality is equivalent to antipodality whenever $n$ is twice a prime. Finally, we compute the von Neumann entanglement entropy generated by QFR for all admissible graphs, thereby obtaining a collection of quantum information measures and proving that the entropy depends solely on the revival amplitudes $|\alpha|$ and $|\beta|$.

math.CO↗

arXiv2026

Degree Variance and the Fuzzy Sigma Index in Fuzzy Graphs

The sigma index of a graph, defined as the population variance of its degree sequence, is a fundamental measure of structural irregularity. In this paper, we introduce and systematically investigate its natural extension to fuzzy graphs, termed the fuzzy sigma index $$ \sigma^*(\Gamma) = \frac{1}{n} \sum_{v \in V(\Gamma)} \left( d_\Gamma(v) - \frac{2\,\mathrm{ew}}{n}\right)^2, $$ where $d_\Gamma(v)$ denotes the fuzzy degree of a vertex $v$, and $\mathrm{ew}$ represents the fuzzy size of the fuzzy graph $\Gamma=(V,\nu, \mu)$. We establish several fundamental properties of this topological index. In particular, we derive sharp lower and upper bounds. Analyze the behavior of $\sigma^*(\Gamma)$ under standard fuzzy graph operations. This work provides a foundation for further study of variance-based topological indices in fuzzy graph theory.

math.GM↗

arXiv2026

Theta-Relations Among Degree-Based Tree Indices

In this paper, degree-based topological indices play a key role in the structural analysis of graphs in this paper and have significant uses in chemical graph theory. We investigate the connections between three such tree indices: the Albertson, Sombor, and Sigma indices. We show that the quadratic degree deviation, measured by the Sigma index, tightly controls the Sombor index of a tree by establishing sharp two-sided bounds. We demonstrate that the Sombor and Sigma indices are asymptotically equivalent up to constant factors as a direct result. A pure $\Theta$-relationship between the Sombor index and the Albertson index is derived by taking into account extremal trees with a fixed degree sequence. This finding demonstrates that, in extremal configurations, quadratic degree interactions and absolute degree disparities scale appropriately. Overall, our data suggest that the Sombor index functions as an intermediate descriptor, capturing both global degree dispersion and local edge irregularity. From a structural standpoint, these findings clarify the relationship between vertex-based and edge-based irregularity measurements in trees.

math.CO↗

arXiv2026

Integrating Linear Regression and Multi-Criteria Decision Making for Assessing Financial Statement Risks in Manufacturing Firms

Evaluating the financial performance of manufacturing firms requires consideration of both the time value of money and the relative importance of multiple decision criteria. Conventional approaches relying solely on deterministic discounting often fail to account for interactions among economic, operational, and managerial factors. This study proposes an integrated framework that combines time-discounted economic analysis with linear regression to evaluate control system efficiency. A theoretical discounting model is first developed to convert costs and benefits occurring at different times into present-value terms using compound interest functions. The model accommodates one-time expenditures, time-proportional costs, and complex cost structures arising during system development and commissioning. To empirically assess how discounted economic performance is influenced by multiple criteria, linear regression serves as the approximation method.

econ.TH↗

arXiv2026

On Extremal Family Trees $(\mathcal{T}_n)_{n\geqslant 3}$ Beyond Caterpillars and Greedy Constructions

This paper investigates topological indices for the greedy tree $\mathcal{T}_\mathscr{D}$ associated with a graphic degree sequence $\mathscr{D} = (d_1 \geqslant d_2 \geqslant \dots \geqslant d_n)$ of a tree. A fundamental challenge in the study of topological indices lies in establishing precise bounds, as such findings illuminate intrinsic relationships among diverse indices. We investigate the extremal properties of the graph invariant $\sigma$ over the family $\mathcal{T}_n$ of all trees on $n \ge 3$ vertices. Specifically, we compare the minimum values of $\sigma$ attained in restricted subclasses -- including caterpillar trees and greedy trees -- with the global minimum over $\mathcal{T}_n$. We prove that caterpillar trees do not achieve the minimum value of $\sigma$ among all trees, whereas greedy trees attain values no smaller than this global minimum. Moreover, we show that certain trees, which are neither caterpillars nor greedy trees, have $\sigma$-values strictly between the global minimum over $\mathcal{T}_n$ and the minimum among caterpillar trees. These results highlight structural limitations of these common tree classes in extremal problems and offer new insights into the role of non-caterpillar, non-greedy trees in minimizing graph invariants.

math.GM↗

arXiv2026

Analysis of the Density of Words under Morphism $\{a,b\}$

In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from Fibonacci word. Fibonacci words over the alphabet $\{a,b\}$, we define a novel \emph{power} operation that yields a formal linear combination in the free abelian group generated by all finite words.

math.GM↗

arXiv2025

Bounds on the Albertson Index for Trees with Given Degree Sequences

In this paper, we presents novel and sharp bounds on the Albertson index of trees, revealing deep connections between degree sequences and graph irregularity where the Albertson index of Caterpillar tree satisfy \[ \operatorname{irr}(G)=\left( {{d_n} - 1} \right)^2 + \left( {d_1 - 1} \right)^2 + \sum\limits_{i = 2}^{n - 1} {\left( {{d_i} - 1} \right)\left( {{d_i} - 2} \right)} +\sum_{i=1}^{n-1}|d_i-d_{i+1}|. \] We derive powerful inequalities that precisely characterize the minimum and maximum values of the Albertson index, incorporating intricate dependencies on vertex degrees, edge counts, and the average of elements in degree sequence $\mathscr{D}=(d_1,d_2,\dots,d_n)$ where $d_n\geqslant d_{n-1}\geqslant \dots\geqslant d_2\geqslant d_1$. Our results not only improve existing extremal bounds but also uncover striking relationships between the structure of trees and their irregularity measurements. These advances open new avenues for the analysis of graph irregularity and contribute essential tools for the study of degree-based topological indices in combinatorial graph theory.

math.GM↗

arXiv2025

Assessing Financial Statement Risks among $\mathrm{MCDM}$ Techniques

In this paper, to determine the financial risks faced by an industrial company, assessing the relative importance of these risks and identifying the years most exposed to financial risk using modern multi-criteria decision-making techniques. Applied to AL-Ahliah Vegetable Oil Company, the research utilizes the Analytical Hierarchy Process and Simple Additive Weighting to analyze financial ratios from 2008 to 2017.

econ.TH↗

arXiv2025

Determining the Qibla Direction by Astronomical and Geometrical Methods

This paper investigates the determination of the Qibla direction using both astronomical and geometrical approaches. The study reviews historical and classical methods employed by Muslim scholars and astronomers including the use of instruments such as the astrolabe and compass. It further explores spherical trigonometry techniques to precisely calculate the Qibla azimuth from any given location on Earth. The research clarifies geometric constructions and presents a computational model implemented in C++ to facilitate accurate Qibla determination. This interdisciplinary analysis underscores the rich tradition of Islamic astronomy and geometry in solving practical religious requirements, providing both theoretical frameworks and practical algorithms for modern application.

physics.hist-ph↗

arXiv2025

The Engineering and Programming Methods Used in Manufacture of Astrolabes and Errors Resulting

In this study, we first reviewed the traditional astrolabe design methods and identified potential sources of manufacturing error. We then proposed an analytical approach using computer assistance to develop designs for the astrolabe components. This approach marks a pioneering step toward designing and producing a physical astrolabe model aided by computer technology. Our goal was to revive this significant heritage instrument while leveraging modern techniques and software to produce astrolabe models free from traditional manufacturing inaccuracies.

math.HO↗

arXiv2025

On the Asymptotic Palindrome Density of Fibonacci Infinite Words

In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism $0 \to 01$, $1 \to 0$, we define a derived ternary word $\mathbb{Y}$ and establish new results relating its density components $\mathrm{dens}(\lambda,n)$, $\mathrm{dens}(\alpha,n)$, and $\mathrm{dens}(\beta,n)$, deriving explicit formulae and bounds on their behavior. We further prove a general density theorem for infinite words with paired subwords, showing that the associated palindromic prefix density is bounded above by $\frac{1}{\varphi_1}$, where $\varphi_1 = (1 + \sqrt{5})/2$ is the golden ratio. Our approach connects the structure of Fibonacci and Thue--Morse sequences with precise asymptotic and combinatorial interpretations for the observed densities.

math.CO↗

arXiv2025

The Orientalists' Stance Towards Arabic Sciences (Especially Arabic Astronomy)

In this paper, we highlight the influence of Arab/Islamic civilization in the field of the history of astronomy on European historians. We also aim to elucidate the stance of Orientalists toward the study of Arab sciences and to clarify their orientations, with a particular focus on astronomy, while revealing the significant role played by Arab scholars in this domain and the impact of their contributions-especially astronomical tables (zij)-on Western astronomers. Furthermore, we have clarified the mechanisms of transmission of Arab sciences, particularly astronomy, from Arab scholars to Western scholars, and the role of Arab astronomers in Western civilization. In addition, we address the contributions of Arab scholars to the development of astronomy and the perspective of Orientalists, particularly David King, regarding this matter. We also underscore the importance of Orientalists' works in analyzing Arab/Islamic scholarly output, identifying its influence on the West in the field of astronomy, and demonstrating how Western scholars benefited from translations of Arabic books in this discipline. In this paper, we adopt the historical retrieval methodology, by referencing previously documented astronomical information and contributions, with an emphasis on the processes of transmission of these sciences from the Arabs to the West.

math.HO↗

arXiv2025

A Study of NP-Completeness and Undecidable Word Problems in Semigroups

In this paper we explore fundamental concepts in computational complexity theory and the boundaries of algorithmic decidability. We examine the relationship between complexity classes \textbf{P} and \textbf{NP}, where $L \in \textbf{P}$ implies the existence of a deterministic Turing machine solving $L$ in polynomial time $O(n^k)$. Central to our investigation is polynomial reducibility. Also, we demonstrate the existence of an associative calculus $A(\mathfrak{T})$ with an algorithmically undecidable word problem, where for a Turing machine $\mathfrak{T}$ computing a non-recursive function $E(x)$, we establish that $q_1 01^x v \equiv q_0 01^i v \Leftrightarrow x \in M_i$ for $i \in \{0,1\}$, where $M_i = \{x \mid E(x) = i\}$. This connection between computational complexity and algebraic undecidability illuminates the fundamental limits of algorithmic solutions in mathematics.

cs.CC↗

arXiv2025

The Effect of Using Popular Mathematical Puzzles on The Mathematical Thinking of Syrian Schoolchildren

In this paper we provide a good overview of the problems and the background of mathematics education in Syrian schools. We aimed to study the effect of using popular mathematical puzzles on the mathematical thinking of schoolchildren, by conducting a paired experimental study (pre-test and post-test control group design) of the data we obtained through a sample taken from students of sixth-grade primary school students in Syria the Lady Mary School in Syria, in order to evaluate the extent of the impact of popular mathematical puzzles on students' ability to solve problems and mathematical skills, and then the skills were measured and the results were analyzed using a t-test as a tool for statistical analysis.

econ.TH↗

arXiv2025

The Role of Mathematical Folk Puzzles in Developing mathematical Thinking and Problem-Solving Skills

This paper covers a variety of mathematical folk puzzles, including geometric (Tangrams, dissection puzzles), logic, algebraic, probability (Monty Hall Problem, Birthday Paradox), and combinatorial challenges (Eight Queens Puzzle, Tower of Hanoi). It also explores modern modifications, such as digital and gamified approaches, to improve student involvement and comprehension. Furthermore, a novel concept, the "Minimal Dissection Path Problem for Polyominoes," is introduced and proven, demonstrating that the minimum number of straight-line cuts required to dissect a polyomino of N squares into its constituent units is $\mathrm{N}-1$. This problem, along with other puzzles, offers practical classroom applications that reinforce core mathematical concepts like area, spatial reasoning, and optimization, making learning both enjoyable and effective.

econ.TH↗

arXiv2025

Closed-Form Analysis and Extremal Bounds of Albertson and Sigma Indices in Trees with Prescribed Degree Sequences

This study explores the irregularity properties of trees with prescribed degree sequences by analyzing two prominent topological indices: the Albertson index and the sigma index. With a particular emphasis on caterpillar trees -frequently used to model molecular chains- we derive a closed-form expression for the Albertson index: \[ \mathrm{irr}(\mathscr{C}(n,m)) = m(m+1)n - 2m + 2, \quad \text{for } n \geq 3. \] Furthermore, we establish extremal bounds for both indices across tree families characterized by fixed degree sequences. The results yield a unified analytical framework for comparing linear and quadratic irregularity measures, and provide new structural insights relevant to applications in chemical graph theory and extremal graph analysis.

math.CO↗

arXiv2025

Optimal Behaviour in Extremal Bounds for $\sigma$-Irregularity

In this paper, we establishe the extremal bounds of the topological indices -- Sigma index -- focusing on analyzing the sharp upper bounds and the lower bounds of the Sigma index, which is known $\sigma(G)=\sum_{uv\in E(G)}(d_G(u)-d_G(v))^2$. We establish precise lower and upper bounds for the Sigma index, leveraging a non-increasing degree sequence $\mathscr{D} = (d_1, d_2, \dots, d_n)$, A fundamental challenge in the study of topological indices lies in establishing precise bounds, as such findings illuminate intrinsic relationships among diverse indices.

math.CO↗