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The topological index is a useful tool in mathematical chemistry that connects a numerical value to a network structure. The Sombor index is well-known for establishing such connections. This work presents the generalization of the Sombor index from a fuzzy point of view. We introduce the Sombor index for fuzzy graphs ( S O (F G) ). Some crucial bounds of S O (F G) are established for numerous graph operations, including Cartesian product, composition, join, and union. In addition, we find the relationship between the Sombor index and other topological indices. Moreover, we present an application of the Sombor index for fuzzy graphs to identify the most alarming type of cancer in India, which clearly justifies the generalization of the Sombor index. [ABSTRACT FROM AUTHOR]

The eternal 1-secure set, strategically protects graphical structures against attacks by placing sentinels within dominating sets. This study adapts this concept to real-world scenarios, like emergency response systems, by addressing challenges such as roadblocks and traffic congestion. We introduce the secure set and the eternal 1-secure set for fuzzy graphs based on edge membership levels J. Our analysis examines the relationship between the security number of a fuzzy graph G at level J and the domination number of its spanning subgraph at level J, demonstrating the potential of the eternal 1-secure set to enhance security measures in dynamic environments. [ABSTRACT FROM AUTHOR]

This review article consolidates and tabulates research on the product operations done in fuzzy, intuitionistic fuzzy, and neutrosophic graphs. This article encompasses the previous product discussions on fuzzy graphs and their extensions. This article aims to list the origin, structural properties, applications, etc. done by the researchers and academicians using the product behavior of two graphs on the fuzzified environment. This review provides a clear understanding of enhancements of product approach on graphs from fuzzy to neutrosophic kind. [ABSTRACT FROM AUTHOR]

In the field of system analysis, uncertainties pose significant challenges, demanding robust methodologies for decision-making. This study proposes a novel approach that integrates graph theory, fuzzy set theory, and linguistic data to effectively address uncertainties. Motivated by the imperative to enhance decision making authenticity amidst uncertainties, we introduce the hesitant fuzzy linguistic graph (HFLG) model. This model leverages graphical representation to present hesitant fuzzy linguistic values (HFLVs), providing decision-makers with an intuitive framework conducive to informed decision-making. Through a comprehensive examination of the HFLG model’s principles and operations, this study elucidates its utility in navigating uncertainties inherent in decision-making processes. Furthermore, a detailed case study illustrates the practical application of the HFLG model, highlighting its effectiveness across diverse real-world scenarios. The integration of graph theory, fuzzy set theory, and linguistic data in the HFLG model offers a valuable framework for addressing uncertainty, thereby advancing decision-making methodologies in complex environments.

Abstract In recent years, the study of q-rung orthopair fuzzy graphs (or q-ROFGs) has gained significant attention due to their ability to model complex and uncertain relationships. One crucial aspect in analyzing such graphs is the computation of their energy, which provides insights into their structural properties and dynamics. q-rung orthopair fuzzy sets were introduced, as a method for adjusting the range of indication of decision information by modifying a parameter q. This article presents a comprehensive approach for computing the energy of q-ROFGs. The new concepts of Zagreb energy of q-ROFGs are proposed and some bounds are determined. To demonstrate the applicability of the developed approach and show its viability, a numerical model related to the selection of the most suitable warehouse location is provided. Lastly, a comparative analysis is done, and the proposed approach is found to be compatible with existing models.

Abstract The concept of a crisp graph is essential in the study of outerplanar graphs because outerplanar graphs are a unique type of planar graphs containing special characteristics. One of the core concepts of crisp graphs, the notion of a subgraph, is utilized primarily to solve difficult data problems. In this paper, researching outerplanar graphs in fuzzy graphs is the main aim of the work. By allowing edges connecting vertices to have varying levels of membership or fuzziness, a fuzzy graph is a mathematical framework that develops on the concept of crisp graphs. Because of their capacity to describe unclear or imprecise relationships between items, fuzzy outerplanar graphs might have potential applications. The subgraphs of fuzzy outerplanar graphs are revealed by eliminating specific vertices or edges from the fuzzy graphs. Furthermore, maximum, and maximal fuzzy outerplanar subgraphs defined for both vertex and edge deletion are examined using examples. Some of the connections between the concepts discussed are represented as theorems and conclusions. An application for the layout of a bypass road is discussed using the proposed concepts.

The linear Diophantine fuzzy graph (LDFG) notion serves as a new mathematical approach for the ambiguity and uncertainty modeling in decision-making issues. An LDFG eliminates the strict limitations of various existing graphs. The energy concept in graph theory is one of the most attractive topics that is very important in biological and chemical sciences. The article aims at developing the notion of fuzzy graphs (FG) towards LDFGs, and, also, we extend the energy notion of an FG to the energy of an LDFG and use the concept of energy to model problems linked to the LDFG. To fulfill such a purpose, we make an LDFG and investigate the effectiveness of that part by calculating the concept of energy on this LDFG. We define the LDFG adjacency matrix (AM) concept and the energy of an LDFG. Also, we introduce the new Laplacian energy (LE) concept of an LDFG and investigate its properties. Finally, an application of the LDFG energy to find the most effective component in the hospital information system has been presented.

About 70 years ago, the reconstruction conjecture was proposed as an interesting problem in graph theory. In this paper, we extend the concept of reconstruction to fuzzy graphs and introduce the new concept of recognizable properties. We then explore various recognizable properties of fuzzy graphs, including regularity, degree sequence, strength, and block. Additionally, we identify some reconstructable fuzzy graphs. Finally, in conclusion, the reconstruction of fuzzy graphs can be applied to recover old maps by analyzing old photos. [ABSTRACT FROM AUTHOR]

Competition graphs, which are mathematical representations of competitive relationships among entities, have found applications in various domains. This article introduces and explores several types of graphs, including fuzzy incidence competition graphs, fuzzy incidence neighborhood graphs, p-competition fuzzy incidence graphs, m-step fuzzy incidence competition graphs, and m-step fuzzy incidence neighborhood graphs. We establish sufficient conditions under which an edge is independent strong or independent incidence strong. Relation between competition graphs of isomorphic fuzzy incidence digraphs is established. Through rigorous analysis, several intriguing results concerning these graphs are derived. Additionally, at the end of the article, we evaluate a group of researchers using fuzzy incidence graphs to identify the most influential member within the group. [ABSTRACT FROM AUTHOR]

Image retrieval involves searching for images relevant to a user-provided query image. In this paper, we aim to develop a graph-based model with deep representations for Content-Based Image Retrieval (CBIR). Inspired by recent advancements in deep learning, we initially employ a fine-tuned Convolutional Neural Network (CNN) to capture deep semantic features for a specific target image database. Utilizing these learned features, we then introduce an graph-based ranking method for online retrieval. This model's constructed graph is designed to characterize the geometrical structure of the data manifold, facilitating an efficient ranking process. Finally, based on user-provided feedback regarding relevant and irrelevant images, we update the retrieval system in both the deep learning framework and the graph-based ranking model in an offline manner. Extensive simulations confirm the efficiency and effectiveness of our proposed model. [ABSTRACT FROM AUTHOR]

Graph-based document models have good capabilities to reveal inter-dependencies among unstructured text data. Natural language processing (NLP) systems that use such models as an intermediate representation have shown good performance. This paper proposes a novel fuzzy graph-based document model and to demonstrate its effectiveness by applying fuzzy logic tools for text summarization. The proposed system accepts a text document as input and identifies some of its sentence level features, namely sentence position, sentence length, numerical data, thematic word, proper noun, title feature, upper case feature, and sentence similarity. The fuzzy membership value of each feature is computed from the sentences. We also propose a novel algorithm to construct the fuzzy graph as an intermediate representation of the input document. The Recall-Oriented Understudy for Gisting Evaluation (ROUGE) metric is used to evaluate the model. The evaluation based on different quality metrics was also performed to verify the effectiveness of the model. The ANOVA test confirms the hypothesis that the proposed model improves the summarizer performance by 10% when compared with the state-of-the-art summarizers employing alternate intermediate representations for the input text. [ABSTRACT FROM AUTHOR]

Fuzzy graph models occur specifically in the dynamic processes of physical, biological, and social systems. Data transmission reliability, routing optimization, and network failure detection are all aided by connectivity analysis of fuzzy graph models. The cyclic connectivity analysis in terms of cycles in fuzzy graphs has grown into a prominent field of study. As strong cycles between vertices serve as a critical path for information flow, we propose a connectivity metric that considers the significance of strong cycles, providing a comprehensive approach in the area of connectivity. In this paper we extends the average connectivity concepts in terms of the cycles in fuzzy graphs. Also, the average cyclic connectivity of subgraphs, complete fuzzy graphs, θ -fuzzy graphs, and other related topics are examined. An algorithm for finding the new connectivity measure is provided and a practical implementation in real world scenarios is also discussed. [ABSTRACT FROM AUTHOR]

Fuzzy graphs find numerous applications in real life. One of the extensions of fuzzy graphs is Fermatean fuzzy graphs. Here, we introduce the concepts of Fermatean fuzzy set to the domain of graph theory and obtain a novel type of graph, referred to as Fermatean fuzzy graph (FFG ). The article establishes fundamental terms such as strong FFG , complete FFG , regular FFG , path, degree, total degree, homomorphism, and isomorphism of FFG , as well as the complement of FFG . Alongside the introduction of these concepts, their important properties, theorems, and illustrative examples are defined and discussed. Finally, an application has surfaced suggesting the utilization of FFG s to scrutinize the key factors affecting the productivity of Tata Motors, paving the way for a comprehensive analysis of the company's operational efficiency using score function. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of hesitant bipolar-valued fuzzy graph (HBVFG), which captures the two opposing perspectives, namely the positive and negative opinions. The novelty, importance and implications of this concept are illustrated by some results, examples, and graphical representations. There are, respectively, some theoretical terms of graphs such as partial directed hesitant bipolar-valued fuzzy subgraph (HBVFSG), directed HBVFSG, directed spanning HBVFSG, strong directed HBVFG and complete directed HBVFG which are introduced. The operations, such as Cartesian, direct, lexicographical, and strong products, are also defined between two HBVFGs with examples. The mapping relations, such as homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism, are derived with an example. The applications of directed HBVFGs with algorithms for finding the optimal path in a network and the dominant node and influence of index with the self-persistence degree of a node in a social network are presented. For each problem, an algorithm is developed and its effectiveness is demonstrated by examples. The proposed concept is assessed in terms of theory and practice. The benefits of the proposed solution are highlighted and a clear comparison is made with the existing methods. [ABSTRACT FROM AUTHOR]

In crisp graph theory, there are numerous topological indices accessible, including the Misbalance Prodeg Index, which is one of the most well-known degree-based topological indexes. In crisp graphs, both vertices and edges have membership values of 1 or 0, whereas in fuzzy graphs, both vertices and edges have different memberships. This is an entire contrast to the crisp graph. In this paper, we introduce the Fuzzy Misbalance Prodeg Index of a fuzzy graph, which is a generalized form of the Misbalance Prodeg Index of a graph. We find bounds of this index and find bounds of certain classes of graphs such as path graph, cycle graph, complete graph, complete bipartite graph, and star graph. We give an algorithm to find the Fuzzy Misbalance Prodeg Index of a graph for the model of multi-criteria decision-making is established. We present applications from daily life in multi-criteria decision-making. We apply our obtained model of the Fuzzy Misbalance Prodeg Index for the multicriteria decision-making to the choice of the best supplier and we also show the graphical analysis of our index with the other indices that show how our index is better than other existing indices. [ABSTRACT FROM AUTHOR]

Air pollution from vehicle emissions, industrial activities, and medical facilities poses significant health risks in urban areas, underscoring the necessity for robust air quality index (AQI) monitoring. This paper presents a novel method for AQI prediction by integrating a fuzzy centre merge graph with an optimal value-based fuzzy inference system (FCMG-OP-FIS) and machine learning (ML). Traditional ML techniques encounter difficulties when converting regression datasets into classification formats, particularly when unable to label the dataset using the traditional method. The proposed FCMG-OP-FIS model efficiently converts regression data into a classification framework. Unlike traditional AQI prediction methods that rely solely on pollutant data, this approach incorporates both pollutant and meteorological data to improve prediction accuracy. The innovative fuzzy centre merge graph (FCMG) balances the dataset for optimal solutions and facilitates input grouping for Simulink, simplifying rule management. The FCMG-OP-FIS model generates a regression output for AQI, which is subsequently classified into levels (healthy, moderate, or unhealthy) using IF-THEN rules. To enhance accuracy further, a random forest classifier (RFC) is trained on the FCMG-OP-FIS classified output data. The regression output of the FCMG-OP-FIS model is validated using metrics such as RMSE (0.48), MSE (0.23), MAE (0.23), and MAPE (1.77%). Additionally, the classification output from the RFC model employs advanced validation techniques including stratified shuffle validation, grid search cross-validation, and confusion matrix analysis, achieving an accuracy rate of 99%, with the F1 score, precision, and recall over all at 99%. These results demonstrate the effectiveness of the proposed model in accurately labelling data for classification and predicting AQI through ML, highlighting its potential for practical application in environmental monitoring and management.

In crisp graph theory, there are numerous topological indices accessible, including the Misbalance Prodeg Index, which is one of the most well-known degree-based topological indexes. In crisp graphs, both vertices and edges have membership values of $ 1 $ or $ 0 $, whereas in fuzzy graphs, both vertices and edges have different memberships. This is an entire contrast to the crisp graph. In this paper, we introduce the Fuzzy Misbalance Prodeg Index of a fuzzy graph, which is a generalized form of the Misbalance Prodeg Index of a graph. We find bounds of this index and find bounds of certain classes of graphs such as path graph, cycle graph, complete graph, complete bipartite graph, and star graph. We give an algorithm to find the Fuzzy Misbalance Prodeg Index of a graph for the model of multi-criteria decision-making is established. We present applications from daily life in multi-criteria decision-making. We apply our obtained model of the Fuzzy Misbalance Prodeg Index for the multi-criteria decision-making to the choice of the best supplier and we also show the graphical analysis of our index with the other indices that show how our index is better than other existing indices.

Fuzzy graph theory has an important role in modeling uncertainty in the real environment. In this work, a new approach is proposed concerning the graph coloring problem of a fuzzy graph using Gröbner basis, and a generalization of the classical crisp chromatic number of a graph. This approach is established on the successive coloring functions of the crisp graphs. Using this method, we can determine whether or not, for a fixed k , the given graph can be colored. If such a coloring exists, the number of distinct k -colorings and chromatic number in the given fuzzy graph can be obtained. As an application, we analyze the traffic lights problem by using this method. [ABSTRACT FROM AUTHOR]

Fuzzy graphs (FGs) are efficient mathematical models for analyzing many problems of daily life. Various problems in the fields of physics, telecommunication science, psychology and genetics can be modeled and solved with the help of fuzzy graph theory. One of the most widely used types of FG is vague graph (VG). VGs play an important role in various fields such as computer science, psychology, medicine, and political sciences and are used to find effective people in an organization or social institution. Today, connectivity indices are used in computer networks and economics to find the most suitable place for construction. These indices also have many applications in medical sciences, including ultrasound and radiology. Hence, in this research, we introduce connectivity index (CI), average connectivity index (ACI), and randic index (RI) in VGs with some examples. Likewise, some new vertices such as enhancing vertex (EV), reducing vertex (RV), and neutral vertex (NV) are described. Finally, an application of CI in the selecting town for building a school is given. [ABSTRACT FROM AUTHOR]

This study introduces the novel concept of m-polar interval-valued fuzzy hypergraph (m-PIVFHG), an advancement that combines the strengths of fuzzy theory and hypergraph models to improve decision-making processes. m-PIVFHGs allow vertices to have degrees of membership across multiple polarities within sub-interval values of [0,1]. This offers better adaptability and precision than traditional models. This paper systematically explores the theoretical foundations of m-PIVFHGs, detailing their unique characteristics and presenting duality concepts with illustrative examples. It also defines various cut and level types specific to m-PIVFHGs and examines their properties. The practical utility of m-PIVFHGs is demonstrated through a real-world application aimed at optimizing decision-making in a university setting, showcasing significant improvements over existing fuzzy graph methodologies.

The Zagreb indices (ZIs) are important graph invariants that are used extensively in many different fields in mathematics and chemistry, such as network theory, spectral graph theory, fuzzy graph theory (FGT) and molecular chemistry, etc. The hyper-ZI is introduced especially for fuzzy graphs (FGs) in this study. The study computes this index's bounds for a variety of FG types, including paths, cycles, stars, complete FGs and partial fuzzy subgraphs. It is shown that isomorphic FGs produce the same values for this index. Moreover, interesting connections are established between the hyper-ZI and the second ZI for FGs. Moreover, bounds on this index are found for the following operations: direct product, Cartesian product, composition, join, union, strong product and semi-strong product of two FGs. In the end, the effectiveness of this index is compared with three other topological indices: hyper-ZI for crisp graphs, first ZI for FGs and F-index for FGs, in an analysis of the crime “Murder” in India. While the hyper-ZI for FGs, first ZI for FGs and F-index for FGs yield similar outcomes, the hyper-ZI for FGs demonstrates superior realism in detecting crimes in India compared to its crisp graph counterpart.

Abstract This paper introduces the concept of hesitant bipolar-valued fuzzy graph (HBVFG), which captures the two opposing perspectives, namely the positive and negative opinions. The novelty, importance and implications of this concept are illustrated by some results, examples, and graphical representations. There are, respectively, some theoretical terms of graphs such as partial directed hesitant bipolar-valued fuzzy subgraph (HBVFSG), directed HBVFSG, directed spanning HBVFSG, strong directed HBVFG and complete directed HBVFG which are introduced. The operations, such as Cartesian, direct, lexicographical, and strong products, are also defined between two HBVFGs with examples. The mapping relations, such as homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism, are derived with an example. The applications of directed HBVFGs with algorithms for finding the optimal path in a network and the dominant node and influence of index with the self-persistence degree of a node in a social network are presented. For each problem, an algorithm is developed and its effectiveness is demonstrated by examples. The proposed concept is assessed in terms of theory and practice. The benefits of the proposed solution are highlighted and a clear comparison is made with the existing methods.

In this paper, we introduce a c-prime fuzzy graph of a nearring with respect to a level ideal of a fuzzy ideal. We find a relation between properties of the fuzzy ideal and properties of the fuzzy graph. We introduce ideal symmetry of the fuzzy graph and obtain conditions under which the graph is ideal symmetric. We find a relation between nearring homomorphisms and graph homomorphisms. We investigate conditions required for the homomorphic image of a c-prime fuzzy ideal to be a c-prime fuzzy ideal. [ABSTRACT FROM AUTHOR]

The most critical task of a social network is to identify a central node. Numerous methods for determining centrality are documented in the literature. It contributes to online commerce by disseminating news, advertisements and other content via central nodes. Existing methods capture the node's direct reachability. This study introduces a novel method for quantifying centrality in a fuzzy environment. This measurement takes into account the reachability of nodes and their direct connections. Several critical properties have been demonstrated. A small Facebook network is used to illustrate the issue. Additionally, appropriate tables and graphs present a comparative study with existing methods for centrality measurement. [ABSTRACT FROM AUTHOR]

Containers in fuzzy graphs are studied in this paper. A container in a graph is a family of internally disjoint paths between a pair of vertices. Equivalently, a container in a fuzzy graph is a collection of internally disjoint strongest paths. Strongest paths often provide maximum contribution to a network's traffic flow. Therefore analysis of strongest paths and containers helps examining the dynamics and complexities of networks. Apart from certain structural properties of containers, two classic results on graph degrees are also generalized to fuzzy setup. Spanning containers of blocks, fuzzy trees and locamin cycles are discussed. An algorithm to identify the maximum and minimum capacity of containers and an application related to human trafficking are also proposed. [ABSTRACT FROM AUTHOR]

Connectivity measures real energization state of manifold applications and user interference in many criteria-based network systems. Randic index is a parameter of this type that can be used to measure the combined power of molecular graph or graphical network system. The concept of Randic index has been applied in a connected system in Indonesia tourism sectors under some uncertain conditions. First, Randic index of fuzzy graph (FG) and fuzzy subgraph are introduced and investigated their properties. Second, different upper and lower bounds of Randic index and their isomorphic properties in fuzzy graphs are exhibited. Third, the Randic index in directed fuzzy graph (FDG) is introduced. Due to presence of similar contribution of vertices, many formulas for calculating Randic index of regular fuzzy graphs are presented. Fourth, some similarity and distinction of Randic index with connectivity index (CI) and Wiener index (WI) in fuzzy graph are analyzed. Fifth, an algorithm and a flowchart are proposed to determine Randic index in a fuzzy graph. Finally, an application is depicted to mention the order of important crossroads and stoppages between Balai Kemambang (BKB) and Bukit Kendalisada (BKS) in Indonesia tourism sectors. [ABSTRACT FROM AUTHOR]

This study introduces a ground-breaking approach to analyzing dihedral groups through the lens of fuzzy graph theory, significantly enhancing computational efficiency in group theory. By extending the fuzzy topological indices to polynomial forms, this research drastically reduces the calculation time for these indices from hours to mere seconds. A notable feature of this work is the innovative use of polynomial regression, a machine learning technique, to generate polynomials for adjacency and degree matrices within the fuzzy conjugate graph of a dihedral group. This method not only simplifies calculations but also incorporates an error analysis component, ensuring accuracy and reliability. The integration of fuzzy graph theory with polynomial regression in this context is a pioneering step, offering valuable insights into the structural attributes of dihedral groups. This research goes beyond traditional methods, highlighting the effectiveness of machine learning in deciphering complex patterns in group theory. The findings and techniques presented hold great promise for future applications in graph theory and group theory, offering a novel perspective for understanding and analyzing intricate graph structures. This study stands as a significant contribution to the field, potentially revolutionizing the way complex mathematical problems are approached and solved.

This article presents a set of generalized $ \wp $-dependent formulae for fuzzy and crisp versions of various graph indices, including the first and second Zagreb indices, harmonic index and Randic index. These formulae are applied to the identity graph of the commutative ring $ Z_{\wp} $, and the resulting indices are calculated by using MATLAB software for 20 prime numbers. The generated data were applied for machine learning by using Python and Jupyter notebook to investigate the relationship between fuzzy and crisp indices. The article also includes the relationship between fuzzy and crisp indices in the form of six-degree polynomials and an error analysis.

The energy of a graph is related to its spectrum, which is equal to the total of the latent values of the pertinent adjacency matrix. In this research work, we proposed some of the features and the energy of the Seidel Laplacian of a fuzzy graph. Also, the lower and upper bounds for the energy of the Seidel Laplacian of a fuzzy graph were studied with suitable illustrative examples.

Fuzzy graphs are of great significance in the modeling and analysis of complex systems characterized by uncertain and imprecise information. Among various types of fuzzy graphs, cubic fuzzy graphs stand out due to their ability to represent the membership degree of both vertices and edges using intervals and fuzzy numbers, respectively. The study of connectivity in fuzzy graphs depends on understanding key concepts such as fuzzy bridges, cutnodes and trees, which are essential for analyzing and interpreting intricate networks. Mastery of these concepts enhances decision-making, optimization and analysis in diverse fields including transportation, social networks and communication systems. This paper introduces the concepts of partial cubic fuzzy bridges and partial cubic fuzzy cutnodes and presents their relevant findings. The necessary and sufficient conditions for an edge to be a partial cubic fuzzy bridge and cubic fuzzy bridge are derived. Furthermore, it introduces the notion of cubic fuzzy trees, provides illustrative examples and discusses results relevant to cubic fuzzy trees. The upper bonds for the number of partial cubic fuzzy bridges in a complete CFG is calculated. As an application, the concept of partial cubic fuzzy bridges is used to identify cities most severely affected by traffic congestion resulting from accidents. [ABSTRACT FROM AUTHOR]

The energy of a graph is related to its spectrum, which is equal to the total of the latent values of the pertinent adjacency matrix. In this research work, we proposed some of the features and the energy of the Seidel Laplacian of a fuzzy graph. Also, the lower and upper bounds for the energy of the Seidel Laplacian of a fuzzy graph were studied with suitable illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an equiprime fuzzy graph of a nearring with respect to the level ideal of a fuzzy ideal. We interrelate graph theoretical properties of the graph and ideal theoretical properties of nearring. We show that the properties like vertex cut, connectedness of the graph depend on the properties of the fuzzy ideal. We define ideal symmetry of the graph and find conditions for the graph to be ideal symmetric. If the fuzzy ideal is equiprime then we show that the level set induces a fuzzy clique. We find conditions required for the level set to be the vertex cover of the graph. We find interrelation between equiprime fuzzy graph and fuzzy graph of nearring with respect to level ideal. We study properties of the graph under neaaring homomorphism. We prove that the connectedness of the graph in homomorphic image depends on properties of ideal. We obtain conditions required for homomorphic image of an equiprime fuzzy ideal to be an equiprime fuzzy ideal. [ABSTRACT FROM AUTHOR]

This paper aims to show an optimum solution for the real-time examination timetabling problem with a fixed number of sessions and a set of difficulty level of each course. The uncertainty and haziness around the problem are taken into account by defining the weighted precedence general vague graph. In this paper, the scheduling part of the examination timetabling problem is approached through novel greedy coloring technique for the weighted precedence general vague graph. As per necessity, the new variant of vertex coloring, namely semi coloring of a graph is defined, such as, for the given graph G(V, E) and the positive integers r ≤ |E| and k < χ(G), the graph G can have the proper k−coloring by removing any r edges from the graph. Finally, the greedy algorithm set out for obtaining the semi coloring of the weighted precedence general vague graph and the minimum number of edge removal. [ABSTRACT FROM AUTHOR]

As a critical field of theoretical chemistry, chemical graph theory works as a key part in the model topology calculation and prediction based on chemical molecular structures. The paper can contribute to the research in the following aspects: (1) We propose the concept of bipolar fuzzy influence graph, using positive and negative membership functions to describe positive and negative uncertainties respectively; (2) Based on the new concept, we give a hypothesis to explain the memory of water. The hypothesis is based on two salient facts of the bipolar fuzzy influence graph. Initially, the non-collapsibility of the positive and negative membership functions ensures that the information after using fuzzy coding will not change with the material layer. Then, the influence pairs connect the vertices and edges between different components in the graph through bipolar uncertain influence. At the end of this paper, the conditions for the establishment of the hypothesis and the facts to be verified in the future are analyzed. [ABSTRACT FROM AUTHOR]

This article presents a set of generalized P-dependent formulae for fuzzy and crisp versions of various graph indices, including the first and second Zagreb indices, harmonic index and Randic index. These formulae are applied to the identity graph of the commutative ring ZP, and the resulting indices are calculated by using MATLAB software for 20 prime numbers. The generated data were applied for machine learning by using Python and Jupyter notebook to investigate the relationship between fuzzy and crisp indices. The article also includes the relationship between fuzzy and crisp indices in the form of six-degree polynomials and an error analysis. [ABSTRACT FROM AUTHOR]

In communication networks, strong connectivity between nodes is critical. The failure of strong connectivity between nodes may jeopardize the network’s stability. In fuzzy graphs, various dominating sets using strong edges are identified to avoid network stability. In this paper, the concept of bridge domination set and bridge domination number in fuzzy graphs is introduced. A few prominent properties of bridge domination numbers are chosen and analyzed using relevant examples. The bridge domination number of fuzzy trees, constant fuzzy cycles, and complete fuzzy and bipartite fuzzy graphs are identified. The use of bridge domination in a partial mesh topology to ensure network continuity is demonstrated in the event of a node failure.

The pandemic situation created by COVID’19 is ridiculous. It has made even the blood relations hide themselves from the infected person. The whole world was stunned by this situation. This is because of the uncertainty in the way in which this disease is spread. As an advancement of this disease, a few other variants like delta, omicron etc. also got spread. It is essential to find a solution to this situation . The variants Omicron and Delta are taken into consideration here. Though both the vibrant colours look alike, the symptoms and prevention methods changes for each of these vibrants. This work aims to make a study of the parameters responsible for these variants. As a result of this study, the parameters involved in the spread of these diseases are identified, and the prevention parameters are concluded. The major benefit of this comparatively study is to identify the parameters that are inconclusive, applying the concepts of fuzzy cognitive maps and neutrosophic cognitive maps is applied to bring out the result.

The aim of this article is to concentrate on the notion of closed neutrosophic domination (CND) number 𝛾𝑐𝑙 (𝐺) of a neutrosophic graph (NG) with using effective edge, furthermore we gain a few outcomes on this notion, the relation between 𝛾𝑐𝑙 (𝐺) and some other notions is acquired, eventually the notion of (CND) number of (join neutrosophic graphs) is came in.

The main goal behind mathematical cryptography is to keep messages and multimedia messages secret at a time when modern means of communication have spread and become very diverse. Fuzzy matrices as strong tools which was defined to deal with incomplete and uncertain data and many relationships in real life problems especially those which are related to images and graphs, may considered as important subjects for secret information and communication. The aim of this research paper is to present a new model and method for encrypting 2×2 fuzzy matrices using the basic concepts in neutrosophic number theory and El Gamal algorithm in cryptography, where we generalize El Gamal algorithm to become applicable to the ring of neutrosophic integer numbers that represents the studied fuzzy matrices. On the other hand, we study the applications of the novel algorithm to the encryption and decryption of some fuzzy relations represented in terms of fuzzy functions. 20 Mohammad Abobala, Ali Allouf, On A Novel Security Scheme for The Encryption and Decryption Of 2×2 Fuzzy Matrices with Rational Entries Based on The Algebra of Neutrosophic Integers and El-Gamal Crypto-System In addition, we illustrate many examples to clarify the validity of the new algorithm.

Graph theory has various applications in computer science, such as image segmentation, clustering, data mining, image capturing, and networking. Fuzzy graph (FG) theory has been widely adopted to handle uncertainty in graph-related problems. Interval-valued picture fuzzy graphs (IVPFGs) are a generalization of FGs, interval-valued FGs, intuitionistic fuzzy graphs (IFGs), and interval-valued IFGs. This paper introduces the concept of interval-valued picture fuzzy sets to graph theory and presents a new type of graph called the IVPFG. Within this framework, we define the degree, order, and size of IVPFGs. The paper further explores various operations on IVPFGs, including the Cartesian product, composition, join, and union. The paper delves into the properties of these operations, providing proofs and examples to support the findings. By studying the operations on IVPFGs, we can gain insights into their behavior and leverage this knowledge for solving graph-based problems in the presence of uncertainty. Also, an application regarding merging of community is provided.