Cartesian Product Research Articles

Multi-attribute decision-making using q-rung orthopair fuzzy Zagreb index

The q-rung orthopair fuzzy set (q-ROFS), an extension of intuitionistic and Pythagorean fuzzy sets, offers greater flexibility in representing vague information with two possible outcomes, yes or no. The fuzzy Zagreb index is an important graph parameter, widely used in fields such as network theory, spectral graph theory, mathematics, and molecular chemistry. In this paper, the first and second Zagreb indices for q-rung orthopair fuzzy graphs (q-ROFGs) are introduced, and bounds for these indices are established, including their behavior in regular q-ROFGs. Additionally, it is explored, how various graph operations such as union, Cartesian product, direct product, and lexicographical product affect the first Zagreb index. Furthermore, a new approach is presented that combines Multiple-Attribute Decision-Making (MADM) with graph-based models to improve decision-making, particularly in vaccine selection. The methodology constructs a bipartite graph for each attribute, where virologists assign membership and non-membership values to vaccines. The Zagreb index is used to measure the importance of each vaccine, and a weighted aggregation technique normalizes the scores. The final ranking is derived from a computed score function. The results demonstrate the effectiveness of the approach in providing a systematic and mathematically rigorous framework for multi-attribute decision-making, with rank correlation analysis confirming its robustness compared to existing methods such as q-ROF PROMETHEE, q-ROF VICOR, q-ROF TOPSIS, q-ROFWG, and q-ROFWA.

Edge Co-Even Domination

Domination is one of those important parameters in graph theory which has a very wide range of applications. There are various types of domination depending on the structure of dominating sets. In this study, a new domination parameter called edge co-even domination number is introduced and denoted by γ_coe^' (G). Some basic graphs such as path, cycle, complete, complete bipartite, star, regular, wheel and their complement graphs are examined of this definition. In addition, some results of this parameter are found under graph operations, such as corona and cartesian product.

Monotonicity properties for Bernoulli percolation on layered graphs— A Markov chain approach

Monotonicity properties for Bernoulli percolation on layered graphs— A Markov chain approach

New Results About Aggregation Functions of Quasi-Pseudometric Modulars

In recent studies, Bibiloni-Femenias, Miñana, and Valero characterized the functions that aggregate a family of (quasi-)(pseudo)metric modulars defined over a fixed set X into a single one. In this paper, we adopt a related but different approach to examine those functions that allow us to define a (quasi-)(pseudo)metric modular in the Cartesian product of (quasi-)(pseudo)metric modular spaces. We base our research on the recent development of a general theory of aggregation functions between quantales. This enables us to shed light between the two different ways of aggregation (quasi-)(pseudo)metric modulars.

Tripled Fixed Points, Obtained by Ran-Reunrings Theorem for Monotone Maps in Partially Ordered Metric Spaces

Using the deep result of Ran & Reunrings, we generalize existing results for tripled fixed points. In contrast to the previously known results for tripled fixed points of maps with or without the mixed monotone property in partially ordered complete metric spaces, we demonstrate that it is possible to obtain results for the existence and uniqueness of such points for arbitrary maps with a type of monotonicity, with the partial ordering in the Cartesian product arising from the maps itself. We prove theorems that ensure the existence and uniqueness for tripled fixed points for maps with different types of monotone properties. We obtain sufficient conditions for the existence and uniqueness of systems of three nonlinear matrix equations. The obtained results are illustrated by solving systems of matrix equations.

The minimum orientable genus of the repeated Cartesian product of graphs

Determining the minimum genus of a graph is a fundamental optimisation problem in the study of network design and implementation as it gives a measure of non-planarity of graphs. In this paper, we are concerned with determining the smallest value of g such that a given graph G has an embedding on the orientable surface of genus g. In particular, we consider the Cartesian product of graphs since this is a well studied graph operation which is often used for modeling interconnection networks. The s-cube Qi(s) is obtained by taking the repeated Cartesian product of i complete bipartite graphs Ks,s. We determine the genus of the Cartesian product of the 2r-cube with the repeated Cartesian product of cycles and of the Cartesian product of the 2r-cube with the repeated Cartesian product of paths.

Selection of Sampling and Surrogate Modeling Methods for State-Point Evaluations of an AGN-201M Reactor

Nuclear reactor digital twins (DTs) have been proposed for use as a safeguards technology to efficiently monitor new and novel reactors as they come online. A safeguards DT needs to be capable of detecting misuse and diversion as they occur, requiring physics models to be accurate and efficient. Mathematical surrogate models are capable of achieving the necessary efficiency and can largely maintain the accuracy of higher-order models given a quality training sample. The Multiphysics Object-Oriented Simulation Environment (MOOSE) code framework is specifically equipped to generate training samples and create surrogate models using full-order reactor physics models. Utilizing an operational AGN-201M reactor’s specifications, two surrogate types were trained on samples of variable size, and using Cartesian products, Latin hypercube sampling, and quadrature sampling, each was compared and evaluated on accuracy when compared to a full-order Monte Carlo model. Both surrogate types were able to capture reactivity changes within 0.05 $ of the Monte Carlo model while reducing the computation costs by eight orders of magnitude.

Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations

The Wiener polarity index of a graph <i>G</i>, is the number of unordered pairs of vertices that are at distance 3 in <i>G</i>. This index can reflect the specific distance relation between vertices in the graph, and provides a new way for the study of graph structure. In this paper, the graph entropy based on Wiener polarity index defined. Based on the above definition of graph entropy, it compares the graph entropy of path and balanced double star graphs based on Wiener polarity index. The expressions of graph entropy based on Wiener polarity index for trees with diameter <i>d ≥ 3</i> are studied under four graph operations: tensor product, strong product, Cartesian product and composite graph.

An operations and relations on the Cartesian Product of Interval Valued Intuitionistic Fuzzy Matrices

In this paper, we study an operations and relations on the cartesian product over interval valued intuitionistic fuzzy matrices are introduced and its some properties are explored. We prove some equality based on the operation and the relation over interval valued intuitionistic fuzzy matrices. Finally, we introducing some cartesian formulas 1, 2, 3, 4, 5 in cartesian product of interval valued intuitionistic fuzzy matrices.

Quantumlike Product States Constructed from Classical Networks.

Can complex classical systems be designed to exhibit superpositions of tensor products of basis states, thereby mimicking quantum states? We exhibit a one-to-one map between the product basis of quantum states comprising an arbitrary number of qubits and the eigenstates of a construction comprising classical oscillator networks. Specifically, we prove the existence of this map based on Cartesian products of graphs, where the graphs depict the layout of oscillator networks. We show how quantumlike gates can act on the classical networks to allow quantumlike operations in the state space.

CART-ANOVA-Based Transfer Learning Approach for Seven Distinct Tumor Classification Schemes with Generalization Capability.

Background/Objectives: Deep transfer learning, leveraging convolutional neural networks (CNNs), has become a pivotal tool for brain tumor detection. However, key challenges include optimizing hyperparameter selection and enhancing the generalization capabilities of models. This study introduces a novel CART-ANOVA (Cartesian-ANOVA) hyperparameter tuning framework, which differs from traditional optimization methods by systematically integrating statistical significance testing (ANOVA) with the Cartesian product of hyperparameter values. This approach ensures robust and precise parameter tuning by evaluating the interaction effects between hyperparameters, such as batch size and learning rate, rather than relying solely on grid or random search. Additionally, it implements seven distinct classification schemes for brain tumors, aimed at improving diagnostic accuracy and robustness. Methods: The proposed framework employs a ResNet18-based knowledge transfer learning (KTL) model trained on a primary dataset, with 20% allocated for testing. Hyperparameters were optimized using CART-ANOVA analysis, and statistical validation ensured robust parameter selection. The model's generalization and robustness were evaluated on an independent second dataset. Performance metrics, including precision, accuracy, sensitivity, and F1 score, were compared against other pre-trained CNN models. Results: The framework achieved exceptional testing accuracy of 99.65% for four-class classification and 98.05% for seven-class classification on the source 1 dataset. It also maintained high generalization capabilities, achieving accuracies of 98.77% and 96.77% on the source 2 datasets for the same tasks. The incorporation of seven distinct classification schemes further enhanced variability and diagnostic capability, surpassing the performance of other pre-trained models. Conclusions: The CART-ANOVA hyperparameter tuning framework, combined with a ResNet18-based KTL approach, significantly improves brain tumor classification accuracy, robustness, and generalization. These advancements demonstrate strong potential for enhancing diagnostic precision and informing effective treatment strategies, contributing to advancements in medical imaging and AI-driven healthcare solutions.

Acyclic coloring of two operations of graphs

Abstract The coloring theory of graphs is a very important direction in graph theory. The graph coloring problem has a strong application background. Many practical problems such as timetabling problems, frequency allocation problems, traffic arrangements, circuit design, and storage problems can be transformed into graph coloring problems. A substantial amount of scholarly work has been dedicated to the exploration of acyclic coloring, resulting in a plethora of significant findings that significantly contribute to the theoretical framework of vertex coloring in graphs. An acyclic coloring of a graph G refers to a proper vertex coloring where the subgraph resulting from any pair of color classes does not encompass any cycles. We analyze acyclic coloring in splitting graphs and Cartesian products of paths and cycles by using the method of structural coloring, reductive proof, and mathematical induction. Moreover, we present the precise acyclic chromatic numbers for splitting graphs of cycles and Cartesian products of paths and cycles.

AI-Assisted Wearable Devices for Promoting Human Health and Strength Using Complex Interval-Valued Picture Fuzzy Soft Relations

Wearable technology allows users to monitor and track various fitness-related activities and data, including calories burned, distance traveled, heart rate, and sleep patterns. To choose best app for fitness, here introduced a new innovation in fuzzy algebra named as complex interval valued picture fuzzy soft relations (CIVPFSR) by defining the cartesian product (CP) of two complex interval valued picture fuzzy soft sets (CIVPFSS). Additionally, the many kinds of CIVPFS relations are described, and their outcomes are also mentioned in this article. The CIVPFSS has a complex structure that includes multidimensional variables for membership, abstinence, and non-membership degrees. As a result, this research offers modeling techniques for wearable technology that are based on CIVPFSR. The score function for best human decision making is also formulated in this process. Lastly, a comparison of current structures is done to demonstrate the viability of the suggested work with the conclusion highlighting its potential to significantly advance wearable technology. Additionally, a topological space is developed for CIVPFSR, providing a geometric interpretation of these relations and offering insights into their behavior in multi-dimensional decision spaces. The basic operations, operators, related theorems, and results are discussed. Results concerning the interiors and closures are also addressed.

Connectivity in Cartesian products of fuzzy graphs: application to decision making

Connectivity in Cartesian products of fuzzy graphs: application to decision making

On homomorphism and Cartesian products of fuzzy PUP ideal on PUP algebra

Background In this paper, we explore the Cartesian product of fuzzy PUP ideals and subalgebras within a PUP (pseudo-UP)algebra X. We investigate how fuzzy pseudo-UP ideals are affected under PUP homomorphisms, particularly focusing on the image and inverse image of the fuzzy sets K α and K v , which generalize classical fuzzy sets. These fuzzy sets aim to quantify the level of ambiguity in fuzzy situations. Methods We introduced the concept of the Cartesian product of fuzzy PUP ideals and subalgebras of PUP algebra and analyzed their properties. Specifically, we studied the behavior of fuzzy pseudo-UP ideals under PUP homomorphism, using the fuzzy sets K α and K v to model ambiguity. We also examined the strongest fuzzy PUP relation on a fuzzy PUP subalgebra and ideal of PUP algebra. Results We proved that the Cartesian product of fuzzy PUP ideals forms a fuzzy pseudo-UP ideal. This result was characterized through its level sets, providing a deeper understanding of the interaction between fuzzy PUP ideals and their algebraic structure. Additionally, the strongest fuzzy PUP relation on a fuzzy PUP subalgebra was defined, and several important properties were derived. Conclusions This work extends the concept of fuzzy PUP ideals in the context of PUP algebras and explores how fuzzy homomorphism influence their structure. Our results contribute to a better understanding of fuzzy relations in algebraic settings and open the door for further research into the properties and applications of fuzzy ideals in PUP algebras.

Approximation and the Multidimensional Moment Problem

The aim of this paper is to apply polynomial approximation by sums of squares in several real variables to the multidimensional moment problem. The general idea is to approximate any element of the positive cone of the involved function space with sums whose terms are squares of polynomials. First, approximations on a Cartesian product of intervals by polynomials taking nonnegative values on the entire R2, or on R+2, are considered. Such results are discussed in Lμ1R2 and in CS1×S2-type spaces, for a large class of measures, μ, for compact subsets Si, i=1,2 of the interval [0,+∞). Thus, on such subsets, any nonnegative function is a limit of sums of squares. Secondly, applications to the bidimensional moment problem are derived in terms of quadratic expressions. As is well known, in multidimensional cases, such results are difficult to prove. Directions for future work are also outlined.

Radio number of the cartesian product of a tree and a complete graph

A radio labeling of a graph \( G \) is a mapping \( f : V(G) \to \{0, 1, 2, \dots\} \) such that \( |f(u)-f(v)| \geq \text{diam}(G) + 1 - d(u,v) \) for every pair of distinct vertices \( u,v \) of \( G \), where \( \text{diam}(G) \) is the diameter of \( G \) and \( d(u,v) \) is the distance between \( u \) and \( v \) in \( G \). The radio number \( \text{rn}(G) \) of \( G \) is the smallest integer \( k \) such that \( G \) admits a radio labeling \( f \) with \( \max\{f(v) : v \in V(G)\} = k \). In this paper, we give a lower bound for the radio number of the Cartesian product of a tree and a complete graph and give two necessary and sufficient conditions for the sharpness of the lower bound. We also give three sufficient conditions for the sharpness of the lower bound. We determine the radio number of the Cartesian product of a level-wise regular tree and a complete graph which attains the lower bound. The radio number of the Cartesian product of a path and a complete graph derived in [B. M. Kim, W. Hwang, and B. C. Song, Radio number for the product of a path and a complete graph, J. Comb. Optim., 30 (2015), 139--149] can be obtained using our results in a short way.

Fuzzy PTM subalgebra of PTM - algebra

Background This article introduces the concept of fuzzy pseudo-TM subalgebra (FPTMSA) of pseudo-TM algebra (PTMA). This study explores the relationship between FPTMSAs and their level sets within the context of PTMA. It further investigates the homomorphic behavior of FPTMSAs under various algebraic operations, thereby providing a foundation for further exploration of the algebraic structure of fuzzy pseudo-ideals within PTMAs. Methods The concept of level sets for FPTMSAs is defined and examined, highlighting their role in characterizing these subalgebras within a PTMA. This study explores the behavior of FPTMSAs under homomorphisms, considering both homomorphic images (HI) and inverse images (II) of FPTMSAs in PTMAs. In addition, the composition of epimorphic images and inverse images of FPTMSAs is discussed. The study also considers the properties of FPTMSAs under Cartesian products (CPs), demonstrating that the Cartesian product of two FPTMSAs remains an FPTMSA. Results This study establishes several key properties of FPTMSAs in PTMAs, particularly focusing on their behavior under homomorphisms, epimorphisms, and Cartesian products. Specifically, this study demonstrates that the Cartesian product of two FPTMSAs results in another FPTMSA. Additionally, this study examines the relationships between the homomorphic image and inverse image of FPTMSAs and presents further algebraic results related to these concepts. Conclusions The article concludes by highlighting the significant role of FPTMSAs in the structure of PTMAs, particularly in their interaction with homomorphisms, epimorphisms, and Cartesian products. The results suggest that the FPTMSAs are robust under various algebraic operations. Future work will extend these concepts to intuitionistic fuzzy pseudo-TM subalgebra within PTMAs with the goal of uncovering new algebraic insights and results.

Fundamentals of Fermatean Neutrosophic Soft Set with Application in Decision Making Problem

The classical set theory based on crisp sets is not able to deal with uncertainties which is a common feature of various real-world problems. This problem is solved using modified forms of sets such as fuzzy sets, Intuitionistic fuzzy sets, neutrosophic sets, soft sets and hypersoft sets and others along with their hybrids. In this paper, a modified hybrid of soft set named Fermatean Neutrosophic Soft set ($FrNSS$) is established. Basic entities of set theory including subsets, null set, universal set along with different operators are defined. With respect to these operators, the algebraic structures as monoid, semigroup and semiring are defined. Also, fermatean neutrosophic soft topological space and the cartesian product of fermatean neutrosophic soft sets and fermatean neutrosophic soft relation are defined to establish an application of this hybrid structure to decision-making problems.

Stress in Directed Graphs: A Generalization of Graph Stress

In graph theory, centrality measures are used to identify the most important or influential nodes within a network. Stress centrality is one such measure, which helps quantify how “stressed” a node is within the overall graph structure based on the number of shortest paths that pass through it. Stress centrality provides a more thorough assessment of a node’s relevance than other centrality metrics since it considers not only the direct connections of a node but also the indirect effects through its neighboring nodes. Furthermore, stress centrality can be applied to biological, technical, and social networks, among other kinds of networks. The idea of stress has been expanded from undirected graphs to directed graphs in our study. The stress on a vertex v in D is half of the number of geodesics passing through the vertex v, denoted by st(v). This definition reduces to the case of stress of undirected graph whenever the digraph is symmetric. We have some findings on the stress on a digraph as well as the stress on a vertex. A few standard digraphs’ stresses are obtained, including the stress on any vertex in the cartesian product of digraphs, and some characterization on stress regular digraphs is made.