Florentin Smarandache

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• PhD & Postdocs in Mathematics
• Professor Emeritus at University of New Mexico

Graph classes categorize graphs based on shared properties or structures, and numerous such classes have been proposed over time. In 1965, Zadeh [43] introduced a framework for managing uncertainty, which later inspired Rosenfeld [28, 31] to develop fuzzy graph theory in 1975. A Neutrosophic Graph, as a generalization of a Fuzzy Graph, associates e...

This paper explores the application of neutrosophic logic to Partial Falsifiability, Water Logic, and Neither Nor Logic through a mathematical perspective. Neutrosophic logic, as an extension of classical logic, introduces truth, indeterminacy, and falsehood as independent components, offering a framework to handle uncertainty more effectively [34]...

An International Journal in Information Science and Engineering Abbreviated key-title: Neutrosophic sets syst. (Online) Abbreviation: NSS Established: 2013 Frequency: Continuously online and monthly as volumes Language: English

Neutrosophic statistical analysis has garnered attention for its ability to incorporate the degree of indeterminacy in the analysis of imprecise and interval data under uncertainty—an aspect often neglected by classical statistics, fuzzy statistical analysis, and interval statistics. Recently, Woodall et al. (2025) have sparked critical discussions...

This paper focuses on the study of Soft Sets, a concept that has led to the development of various extensions, including Double-Framed Soft Sets [5, 6, 10, 21], Hypersoft Sets [1], SuperHyperSoft Sets [36], ForestSoft Sets, TreeSoft Sets [3], IndetermSoft Sets [35], and IndetermHyperSoft Sets [35]. These extensions have been actively explored in re...

Journal Information Publisher: University of New Mexico, United States Title proper: Neutrosophic Sets and Systems Founder: Florentin Smarandache Other variant title: An International Journal in Information Science and Engineering Abbreviated key-title: Neutrosophic sets syst. (Online) Abbreviation: NSS Established: 2013

Neutrosophy, as a new branch of philosophy, represents a significant departure from and extension of traditional philosophical perspectives. It emphasizes the study of neutralities—those ideas, states, or conditions that exist between opposites—and their interactions with ideational spectra. This short essay compares Neutrosophy to other philosophi...

The world, as we experience it, is suffused with indeterminacy. From the behavior of subatomic particles to the complexities of human thought and social systems, ambiguity and uncertainty are not exceptions but intrinsic properties of reality. Classical frameworks—whether grounded in strict probabilities, binary logic, or clear set membership—often...

This paper introduces the development of an almost unbiased estimator for estimating the unknown population median of the primary variable. The proposed estimator leverages neutrosophic auxiliary information and employs simple random sampling without replacement (SRSWOR). In order to establish the efficacy of the proposed method, we derive the math...

The main aim of this manuscript is to establish for the first time the reduction formulae of 2-refined neutrosophic indefinite integrals (shortly, RF2RNII) which have double indeterminacy I1 and I2 respectably. We then verify reduction formulae with appropriate examples and also apply reduction formulae to evaluate definite integrals in 2-refined n...

The pursuit of human knowledge has traditionally been organized into distinct disciplines—such as physics, philosophy, biology, and mathematics. This compartmentalization, while useful in many contexts, fails to capture the complexity and fluidity of reality. Neutrosophic Transdisciplinarity, a framework rooted in the principles of neutrosophy, off...

Philosophy has always tried to illuminate the complexity of existence, but often faced its own paradoxes. The emergence of philosophical schools and concepts, along with their arguments, reflect the dynamic interaction of ideas. This short note explores some nuanced principles governing philosophical thought, summarized in the proposed “philosophic...

An outline of Mathematica proof of Benzmuller's criticism of Godel's ontological arguments and an alternative semi-ontological argument of proving the existence of God. complete manuscript is not included here./ (appendix of an article, submitted to a journal for review)

Neutrosophic Sets and Systems (NSS) is an American academic journal, published continuously online as individual papers, and monthly as volumes, by The University of New Mexico, 705 Gurley Ave., Gallup Campus, NM 87301, United States, that has been created for publications of advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, ne...

this article describes a mathematical model of third party effectiveness as intermediary in a conflict situation, by virtue of chemical analogy

Seismic hazard analysis plays a vital role in evaluating the potential earthquake risk in a given region. Northeast India is one of the most seismically active zones due to its tectonic positioning at the collision boundary of the Indian and Eurasian plates. This study aims to implement a comprehensive Seismic Hazard Assessment (SHA) framework usin...

In [6], the relationship between hyperrealism and antirealism in physics is proposed. Hyperrealism-Antirealism describes a spectrum where theories transition from being empirically grounded (realist) to speculative frameworks dominated by untestable assumptions, thereby detaching from observable reality. This paper presents a concise mathematical f...

Operations research uses scientific methods that take the language of mathematics as a basis and uses the computer, without which it would not be possible to achieve numerical solutions to the problems raised. It is concerned with applying scientific methods to complex issues in the management and direction of large systems in various fields and...

The applications of Neutrosophic theory are reflected in various spheres of real life. The comprehensive and flexible nature of these neutrosophic sets makes the researchers to develop neutrosophic decision-making models. This research work proposes neutrosophic based deep learning decision model to determine the optimal match between the players a...

Neutrosophic Statistics (Smarandache in Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, InfoLearnQuest, 156 p., 2007; Smarandache in Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability, Sitech, Craiova, Romania, 140 p., 2013; Smarandache in Introduction to Neutrosophic Statistics, Sitech...

In this paper, we present the notion of neutrosophic ideals in gamma-near-rings. Subsequently, an examination of basic properties related to these ideals is conducted. We also provide characterizations of neutrosophic ideals. Furthermore, we establish a neutrosophic ideal by utilizing a collection of ideals. Finally, we introduce an application.

This comprehensive review delves into the wide-ranging uses of neutrosophic statistics across various domains, including Applied Sciences, Sensor technology, Astrophysics, Bioinformatics, Materials Science, Nanotechnology, and Microelectronics. Through the presentation of illustrative instances, the research underscores the efficacy of Neutrosophic...

Previously, we already discussed on creation and dis-creation process of charge and matter, presented at a conference in Prague, 2019 [1], see also [2]. By acknowledging recent development in plasma research, for instance as conducted in Safire project [3], in the present article allow us to offer further remark on creation and dis-creation of char...

Let us start by hypothesizing our solar system, not as a collection of isolated planets orbiting a star, but as a vast, intricate quantum system. Our previous works explored the possibility of applying low-temperature physics, specifically the Bogoliubov-de Gennes (BdG) equations, to cosmological scales [1, 2]. If we consider the BdG equations, typ...

Conflicts characterized by minimal or absent trust present a significant challenge for resolution. Traditional methods, such as GRIT (Graduated Reciprocation in Tension-reduction), often rely on a degree of pre-existing willingness to engage, which may be lacking in deeply entrenched disputes [1-3]. This article explores the critical role of a thir...

This article explores a key challenge in mathematics: how to solve systems of neutrosophic linear equations, where uncertainty and indeterminacy complicate traditional approaches. The research focuses on determining how symbolic calculus can offer an effective solution to these systems, which incorporate non-absolutely defined values, typical of th...

Professor Florentin Smarandache grounded the neutrosophic set theory (NST) to deal with uncertainty, indeterminacy, and inconsistent information. NSTs represent a powerful extension of traditional set theory, providing a more nuanced approach to handling various forms of uncertainty and vagueness. NSTs are particularly useful in situations involvin...

The chapter suggests a novel framework for analyzing social situations using neutrosophic logic/, which includes indeterminacy with truth and falsehood. Our approach demonstrates how neutrosophic sets/measures and statistics could be used to model the complexities of social norms and behaviors. The suggested framework, which uses mathematical formu...

Net Present Value (NPV) is a cornerstone of financial decision-making, allowing us to assess the profitability of certain investment by discounting future cash flows to their present-day value. However, what if we need to project the future value of an investment, taking into account inflation and the inherent uncertainty of the future? This articl...

In this volume, I delve into a diverse array of topics, spanning mathematics, physics, philosophy, artificial intelligence, and even touching upon social dynamics, literature, arts, criminal justice, and history. A significant portion of the book is dedicated to the ongoing development and exploration of neutrosophy and its related concepts. Togeth...

In this volume, I delve into a wide spectrum of topics, spanning mathematics, physics, philosophy, artificial intelligence, and social dynamics. A significant portion of the book is dedicated to neutrosophy and its extensions, including refined and quadripartitioned neutrosophic sets, neutrosophic determinants, and their applications in decision-ma...

Various set concepts are widely recognized for effectively a ddressing u ncertainty, i ncluding fuzzy sets, neutrosophic sets, plithogenic sets, rough sets, and soft sets. These concepts have been further extended through hyperstructures (based on powersets) and superhyperstructures (based on n-th powersets, which are sets with repeated power set s...

International Journal in Information Science and Engineering

This research explores Neutrosophy, a philosophical approach that focuses on identifying common elements between opposing concepts and analyzing the differences between similar concepts. In this context, Common Parts to Uncommon Things are studied, which are manifested when elements such as and share aspects in their intersection, and Uncommon Part...

In a world shaped by contradictions, uncertainties, and evolving paradigms, Neutrosophic Philosophy emerges as a groundbreaking framework that transcends binary thinking. Rooted in the study of neutralities, contradictions, and their dynamic interplay, this philosophy redefines classical logic, epistemology, and ontology, offering a comprehensive a...

The present article is dedicated to Robert N. Boyd, PhD, with whom we have discussed several exotic subjects in physics, including interstellar travel, med beds for future medicine, and the Pleiadian council. While we appreciate and admire his vast experience and involvement in several high-profile experiments, we respectfully disagree with his use...

Airports like Schiphol maintain exceptionally high safety standards, employing rigorous protocols and advanced technologies. Yet, a persistent paradox exists: despite these efforts, airport accidents, including those resulting in human fatalities, continue to occur, often following a logarithmic stable Pareto distribution. This "fat tail" phenomeno...

The global economy's reliance on energy supply and distribution necessitates a shift towards clean and renewable energy sources. This article explores how the economies of energy producers and consumers can be better modeled through the lens of autopoiesis, a concept developed by Humberto Maturana and further elaborated by Fritjof Capra (cf. Web of...

As many readers may know, graph theory is a fundamental branch of mathematics that examines networks consisting of nodes and edges, with a focus on their paths, structures, and properties [157]. A Fuzzy Graph extends this concept by assigning a membership degree between 0 and 1 to each edge and vertex, capturing the level of uncertainty. Expanding...

The present article is dedicated to Robert N. Boyd, PhD, with whom we have discussed several exotic subjects in physics, including interstellar travel, med beds for future medicine, and the Pleiadeian council. While we appreciate and admire his vast experience and involvement in several high-profile experiments, we respectfully disagree with his us...

In [6], the relationship between hyperrealism and antirealism in physics is proposed. Hyperrealism-Antirealism describes a spectrum where theories transition from being empirically grounded (realist) to speculative frameworks dominated by untestable assumptions, thereby detaching from observable reality. This paper presents a concise mathematical f...

The Neutrosophic Set offers a robust mathematical framework for handling uncertainty by incorporating three key membership functions: truth, indeterminacy, and falsity [36]. To address the growing complexity of real-world problems, advanced extensions such as the HyperNeutrosophic Set and the SuperHyperNeu-trosophic Set have been developed. These h...

The purpose of this paper is to introduce the important and useful notions of inverse and direct systems in the category of neutrosophic modules and present some of fundamental properties of them.

This paper explores the application of neutrosophic logic to Partial Falsifiability, Water Logic, and Neither Nor Logic through a mathematical perspective. Neutrosophic logic, as an extension of classical logic, introduces truth, indeterminacy, and falsehood as independent components, offering a framework to handle uncertainty more effectively [39]...

Concepts such as Fuzzy Sets [23, 47], Neutrosophic Sets [32, 33], and Plithogenic Sets [35] have been extensively studied for addressing uncertainty, with diverse applications across numerous fields. Building on the Plithogenic Set, the HyperPlithogenic Set and SuperHyperPlithogenic Set have also gained recognition [15]. A Plithogenic Cubic Set int...

Concepts such as Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets have been widely investigated for tackling uncertainty, with numerous applications explored across various domains. As extensions of the Plithogenic Set, the HyperPlithogenic Set and the SuperHyperPlithogenic Set are also recognized. A Symbolic Plithogenic Set (SPS) is a structure...

Concepts such as Fuzzy Sets [30, 72], Neutrosophic Sets [53, 55], Rough Sets [37], and Plithogenic Sets [59] have been extensively studied to address uncertainty, with diverse applications across various fields. Recently, TreeFuzzy, TreeNeutrosophic, TreePlithogenic, and TreeRough Sets have been defined [15]. This work examines their extensions: Fo...

Graph theory examines networks composed of nodes (vertices) and their connections (edges). A graph class is defined by shared structural properties governed by specific rules or constraints. This paper explores uncertain graph models, with a focus on Pythagorean, Fermatean, and Complex Turiyam Neu-trosophic Graphs, which extend Neutrosophic Graphs...

A fuzzy set generalizes classical set theory by assigning each element a membership value within [0, 1], allowing for the representation of partial or uncertain membership. It is well established that fuzzy sets can be further extended to Hyperfuzzy sets and SuperHyperfuzzy sets. However, as these concepts have been introduced only recently, their...

One of the most powerful tools in graph theory is the classification of graphs into distinct classes based on shared properties or structural features. Over time, many graph classes have been introduced, each aimed at capturing specific behaviors or characteristics of a graph. Neutrosophic Set Theory, a method for handling uncertainty, extends fuzz...

Making an intuitive decision in any business field without using adequate methodologies could lead to the selection of wrong solutions and eventually could cause a loss. The Multiple-Criteria Decision-Making (MCDM) methods could be beneficial in avoiding such an occurrence. The main target of this paper is to introduce such an MCDM technique that s...

A Neutrosophic Set is a mathematical framework that represents degrees of truth, indeterminacy, and falsehood to address uncertainty in membership values [41, 42]. In contrast, a Plithogenic Set extends this concept by incorporating attributes, their possible values, and the corresponding degrees of appurtenance and contradiction [50]. Among the re...

Concepts such as Fuzzy Sets [28,57], Neutrosophic Sets [42,44], and Plithogenic Sets [48] have been extensively studied to address uncertainty, finding diverse applications across various fields. The Soft Set provides a framework that associates each parameter with subsets of a universal set, enabling flexible approximations [31]. The TreeSoft Set...

Constelatii Diamantine este una dintre cele mai bune reviste literare romanesti ale zilelor noastre. Are o redacție internațională și colaboratori din diaspora românească din întreaga lume. Poezii, proză, eseuri, ilustrații. ***** Constelatii Diamantine is one of the best Romanian literary journals of today. It has an international editorial board...

Interval graphs represent vertices as intervals on the real line, with edges denoting overlapping intervals, while proper interval graphs prevent one interval from being fully contained within another. This paper explores interval and proper interval graphs within the frameworks of fuzzy, neutrosophic, and Turiyam Neutrosophic graphs. We examine ho...

The Neutrosophic Set provides a flexible mathematical framework for handling uncertainty by incorporating three distinct membership functions: truth, indeterminacy, and falsity [50]. In response to the growing complexity of real-world problems, advanced extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set have been introd...

Decision making, especially in real-world contexts, often involves complexities that go beyond simple linear logic. We are often faced with situations where the available choices present a series of conflicting criteria. This is a dilemma often described in the classic Indonesian proverb "buah simalakama" (English term: simalakama fruit, with probl...

Neutrosophic Sets are conceptual frameworks designed to address uncertainty. A Neutrosophic TwoFold Algebra is a hybrid algebraic structure defined over a neutrosophic set, combining classical algebraic operations with neutrosophic components. Concepts such as Hyperalgebra and Superhyperalgebra extend classical Algebra using Power Sets and-th power...

Decision making, especially in real-world contexts, often involves complexities that go beyond simple linear logic. We are often faced with situations where the available choices present a series of conflicting criteria. This is a dilemma often described in the classic Indonesian proverb "buah simalakama" (simalakama fruit, implying : if eaten, fat...

This book is the sixth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Building upon the foundational contributions of previous volumes, this edition focuses on the exploration and development of Various New Uncertai...

This book is the fifth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of Various SuperHyperConcepts, building on the foundational advancements introduced in previous...

In sciences, it is known that normal distribution is often assumed, but there are fields where kurtosis or skewness effect happens for instance in financial markets. While there are debates on efficient market hypothesis (EMH), from practical view point, there is the black swan hypothesis of Nassim N. Taleb. We shall consider therefore how to consi...

This study explores the contributions and influence of Mihály Bencze, born on November 20, 1954, in Săcele (Brașov County, Romania), on education and culture in Romania. Bencze is a prominent figure in the fields of mathematics, pedagogy, and literature, dedicating his life to promoting excellence in mathematics education and preserving Hungarian c...

In sciences, it is known that normal distribution is often assumed, but there are fields where kurtosis or skewness effect happens for instance in financial markets. So we shall consider where does it come from? At this point we can consider for example Minsky instability hypothesis, but at the same time from practical view point, there is the blac...

This book is the sixth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Building upon the foundational contributions of previous volumes, this edition focuses on the exploration and development of Various New Uncertai...

This book is the fifth volume in the series of Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of Various SuperHyperConcepts, building on the foundational advancements introduced in previous...

Hypergraphs extend traditional graphs by allowing edges to connect multiple nodes, while superhypergraphs further generalize this concept to represent even more complex relationships. Neural networks, inspired by biological systems, are widely used for tasks such as pattern recognition, data classification, and prediction. Graph Neural Networks (GN...

Body-Mind-Soul-Spirit Fluidity is a concept rooted in psychology and phenomenology, offering significant insights into human decision-making and well-being. Similarly, in social analysis and social sciences, frameworks such as PDCA, DMAIC, SWOT, and OODA have been established to enable structured evaluation and effective problem-solving. Furthermor...

Journal Information Publisher: University of New Mexico, United States

This article provides a concise yet comprehensive review of the groundbreaking two-volume work, Supermathematics. Bases (2nd edition, 2012), authored by Professor Mircea Eugen Şelariu. By integrating centric and eccentric mathematics, Şelariu introduces a novel field of research with far-reaching applications. His pioneering approach stands as a si...

This paper builds on the foundational advancements introduced in [22, 29-32]. The Neutrosophic Set provides a flexible mathematical framework for managing uncertainty by utilizing three membership functions: truth, indeterminacy, and falsity. Recent extensions, such as the HyperNeutrosophic Set and the SuperHy-perNeutrosophic Set, have been develop...

This paper builds upon the foundation established in [50, 51]. The Neutrosophic Set provides a robust mathematical framework for handling uncertainty, defined by three membership functions: truth, indeterminacy, and falsity. Recent developments have introduced extensions such as the Hyperneutrosophic Set and SuperHyper-neutrosophic Set to tackle in...

This paper builds upon the foundational work presented in [38-40]. The Neutrosophic Set provides a comprehensive mathematical framework for managing uncertainty, defined by three membership functions: truth, indeterminacy, and falsity. Recent advancements have introduced extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Se...

This paper builds upon the foundational advancements introduced in [14, 25-27]. The Neutrosophic Set offers a versatile mathematical framework for addressing uncertainty through its three membership functions: truth, indeterminacy, and falsity. Extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set have been recently propos...

This paper builds upon the foundational advancements introduced in [26,39-43]. The Neutrosophic Set provides a versatile mathematical framework for addressing uncertainty through its three membership functions: truth, indeterminacy, and falsity [84]. Extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set have been recently...

This book is the fourth volume in the series Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of the HyperUncertain Set, building on the foundational advancements introduced in previous volume...

This work investigates the evolution of traditional set theory to address complex and ambiguous real-world phenomena. It introduces hierarchical hyperstructures and superhyperstructures, where superhyper-structures are formed by iteratively applying power sets to create nested abstractions. The focus is placed on three foundational set-based framew...

The Neutrosophic Set is a mathematical framework designed to manage uncertainty, characterized by three membership functions: truth (T), indeterminacy (I), and falsity (F). In recent years, extensions such as the Hyperneutrosophic Set and SuperHyperneutrosophic Set have been introduced to address more complex scenarios. This paper proposes new conc...

This paper delves into the advancements of classical set theory to address the complexities and uncertainties inherent in real-world phenomena. It highlights three major extensions of traditional set theory Fuzzy Sets [288], Neutrosophic Sets [237], and Plithogenic Sets [243]-and examines their further generalizations into Hyperfuzzy [106], HyperNe...

This paper is a continuation of the work presented in [35]. The Neutrosophic Set provides a mathematical framework for managing uncertainty, characterized by three membership functions: truth, indeterminacy, and falsity. Recent advancements have introduced extensions such as the Hyperneutrosophic Set and SuperHyper-neutrosophic Set to address more...

This book is the fourth volume in the series Collected Papers on Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. This volume specifically delves into the concept of the HyperUncertain Set, building on the foundational advancements introduced in previous volume...

Hypergraphs extend traditional graphs by allowing edges (known as hyperedges) to connect more than two vertices, rather than just pairs. This paper explores fundamental problems and algorithms in the context of SuperHypergraphs, an advanced extension of hypergraphs enabling modeling of hierarchical and complex relationships. Topics covered include...

The accelerated progress of aquaponics offers a promising remedy for food production in arid regions, where success heavily hinges on sustaining optimal water quality parameters of aquaponic system. However, managing water parameters in large-scale aquaponic farms, given their complex and interconnected nature, poses significant challenges. Various...

Abstract Liquefaction of soil exposes buildings, bridges, and other vital infrastructure to structural failure, subsidence, and loss of bearing capacity during earthquakes. It can endanger safety and result in significant financial losses in seismically active areas. The objective of the present study is to identify the key factors influencing soi...

Big Science refers to large-scale, well-funded collaborative research designed to refine established theories and produce significant experimental or observational data. In contrast, Deep Science emphasizes reexamining fundamental principles, exploring alternative frameworks, and proposing transformative theories to tackle core scientific challenge...

Graph theory, a branch of mathematics, studies the relationships between entities using vertices and edges. Uncertain Graph Theory has emerged within this field to model the uncertainties present in real-world networks. Graph labeling involves assigning labels, typically integers, to the vertices or edges of a graph according to specific rules or c...

Journal Information Publisher: University of New Mexico, United States Title proper: Neutrosophic Sets and Systems Founder: Florentin Smarandache

Epistemology, the study of knowledge, has long grappled with the challenge of determining truth. While the combination of deductive and inductive reasoning has been a cornerstone of scientific inquiry, several limitations emerge in its application to contemporary epistemology. One significant issue lies within the core of falsificationism, a promin...

The study of uncertainty has been a significant area of research, with concepts such as fuzzy sets [87], fuzzy graphs [51], and neutrosophic sets [58] receiving extensive attention. In Neutrosophic Logic, indeterminacy often arises from real-world complexities. This paper explores the concept of locality as a key factor in determining indeterminacy...

This manuscript has been presented in a Centennial Anniversary of Prof Yu Lomsadze, 17-19th December 2024.

Graph characteristics are often studied through various parameters, with ongoing research dedicated to exploring these aspects. Among these, graph width parameters—such as treewidth—are particularly important due to their practical applications in algorithms and real-world problems. A hypergraph generalizes traditional graph theory by abstracting a...

In real world while addressing the challenges of ambiguity and vagueness, the problems call for a scientific approach to quantify the uncertainties. The negative binomial distribution, a cornerstone in classical probability theory, traditionally examines the probability of achieving the kth success in n trials, following failures, with precise obse...

In light of the great development witnessed by our contemporary world, it has become necessary to focus on scientific methods and use the quantitative method to reach more accurate decisions, appropriate to the surrounding circumstances and factors. The process of decision-making and choosing the optimal alternative depends on the type and quality...

It is well known that the term vagueness is spread in all aspects of our lives, this manuscript will clarify the meaning of independence (strict meaning of independence, the illusory meaning of independence, Oscillating between the true meaning of independence, and the fallacy of claiming independence and not applying it), this triplet actually ref...

In graph theory, the hypergraph [24] extends the traditional graph structure by allowing edges to connect multiple vertices, and this concept is further broadened by the superhypergraph [192,194]. Additionally, several types of uncertain graphs have been explored, including fuzzy graphs [151, 171], neutrosophic graphs [38, 39], turiyam graphs [68,...

In graph theory, the hypergraph [22] extends the traditional graph structure by allowing edges to connect multiple vertices, and this concept is further broadened by the superhypergraph [174,176]. Additionally, several types of uncertain graphs have been explored, including fuzzy graphs [136, 153], neutrosophic graphs [35, 36], and plithogenic grap...

Combinatorics is a branch of mathematics focused on counting, arranging, and combining elements within a set under specific rules and constraints. This field is particularly fascinating due to its ability to yield novel results through the integration of concepts from various mathematical domains. Its significance remains unchanged in areas that ad...

Graph theory is a fundamental branch of mathematics that studies networks made up of nodes (vertices) and connections (edges). A key concept in graph theory is the intersection graph, where vertices represent sets, and edges are drawn between vertices if their corresponding sets intersect. A circular-arc graph specifically models the intersections...