Energy of Pythagorean Fuzzy Graphs with Applications

Developing an intuitionistic fuzzy rough new correlation coefficient approach for enhancing robotic vacuum cleaner

An intuitionistic fuzzy rough model is a powerful tool for dealing with complex uncertainty and imprecision in graph-based models, combining the strengths of intuitionistic fuzzy sets and rough sets. In this research, a correlation coefficient is an established tool for finding the strength of the relationship between two intuitionistic fuzzy rough graphs since correlation coefficients are very capable of processing and interpreting data. Furthermore, an intuitionistic fuzzy rough environment is integrated with attribute decision-making based on correlation coefficients. In order to measure the correlation between two intuitionistic fuzzy rough graphs, this suggests utilising the correlation coefficient concept and weighted correlation coefficient. In order to identify decision-making issues that are supported by intuitionistic fuzzy rough preference relations, the Laplacian energy and new correlation coefficient of intuitionistic fuzzy rough graphs are calculated in this study. We propose a new approach to computing the relative position loads of establishments by adjusting the correlation coefficient between one personality's intuitionistic fuzzy rough preference relation and the other items, as well as the undecided corroboration of the intuitionistic fuzzy rough preference relation. This paper determines the ranking order of all alternatives and the best one by using the correlation coefficient between each option and the ideal choice. In the meantime, the appropriate example improves decision-making for robotic vacuum cleaners by effectively handling uncertain and imprecise data, thereby optimising cleaning performance.

Maximum degree and minimum degree spectral radii of some graph operations

New results relating to the maximum and minimum degree spectral radii of generalized splitting and shadow graphs have been constructed on the basis of any regular graph, referred as base graph. In particular, we establish the relations of extreme degree spectral radii of generalized splitting and shadow graphs of any regular graph.

Some New Bounds for the Inverse Sum Indeg Energy of Graphs

Let G be a (molecular) graph with n vertices, and di be the degree of its i-th vertex. Then, the inverse sum indeg matrix of G is the n×n matrix C(G) with entries cij=didjdi+dj, if the i-th and the j-th vertices are adjacent and 0 otherwise. Let μ1≥μ2≥…≥μn be the eigenvalues of C arranged in order. The inverse sum indeg energy of G, εisi(G) can be represented as ∑j=1n|μi|. In this paper, we establish several novel upper and lower sharp bounds on μ1 and εisi(G) via some other graph parameters, and describe the structures of the extremal graphs.

Investigation of the Main Energies of Picture Fuzzy Graph and its Applications

Picture fuzzy graph, belonging to fuzzy graphs family, has good capabilities at times when we are faced with problems that cannot be expressed by fuzzy graphs and intuitionistic fuzzy graphs. When an element membership is not clear, neutrality is a good option that can be well-supported by a picture fuzzy graph. The previous definitions limitations in fuzzy graph energy have led us to offer new definitions in picture fuzzy graphs. In this article, we expanded the energy concept on the picture fuzzy graph and sought to use this concept in modeling issues related to this graph and solving some problems including the neutrality state. We were able to show that neutrality, as part of total energy, is effective in energy-based decisions. This is noticeable in some types of energy and is more pronounced. We were looking for a way to rank the available options using the picture fuzzy graph and its Laplacian energy/energy in decision making. We studied some types of energy including Laplacian and skew Laplacian in both picture fuzzy graphs and picture fuzzy digraphs, and discussed some of its properties. We discussed some energy boundaries in this graph, and finally, the applications of energy were presented.

Lifetime prolongation of a wireless charging sensor network using a mobile robot via linear Diophantine fuzzy graph environment

Edge networking plays a major part in issues with computer networks and issues with the path. In this article, in linear Diophantine fuzzy (LDF) graphs, we present special forms of linear Diophantine fuzzy bridges, cut-vertices, cycles, trees, forests, and introduce some of their characteristics. Also, one of the most researched issues in linear Diophantine fuzzy sets (LDFS) and systems is the minimum spanning tree (MST) problem, where the arc costs have linear Diophantine fuzzy (LDF) values. In this work, we focus on an MST issue on a linear Diophantine fuzzy graph (LDFG), where each arc length is allocated a linear Diophantine fuzzy number (LDFG) rather than a real number. The LDFN can reflect the uncertainty in the LDFG’s arc costs. Two critical issues must be addressed in the MST problem with LDFG. One issue is determining how to compare the LDFNs, i.e., the cost of the edges. The other question is how to calculate the edge addition to determine the cost of the LDF-MST. To overcome these difficulties, the score function representation of LDFNs is utilized and Prim’s method is a well-known approach for solving the minimal spanning tree issue in which uncertainty is ignored, i.e., precise values of arc lengths are supplied. This technique works by providing more energy to nodes dependent on their position in the spanning tree. In addition, an illustrated example is provided to explain the suggested approach. By considering a mobile charger vehicle that travels across the sensor network on a regular basis, charging the batteries of each sensor node.

Randic and reciprocal randic spectral radii and energies of some graph operations

The largest absolute eigenvalue of a matrix A associated to the graph G is called the A-Spectral Radius of the graph G, and A-energy of the graph G is defined as the absolute sum of all its eigenvalues. In the present article, we compute Randic energies, Reciprocal Randic energies, Randic spectral radii and Reciprocal Randic radii of s-shadow and s-splitting graph of G. We actually relate these energies and Spectral Radii of new graphs with the energies and Spectral Radii of original graphs.

New concepts of pentapartitioned neutrosophic graphs and applications for determining safest paths and towns in response to COVID-19

Pentapartitioned neutrosophic sets are a generalization of the single-valued and quadri-partitioned single-valued neutrosophic sets, and utilizes five symbol-valued neutrosophic logic. In this paper, we introduce some novel concepts regarding pentapartitioned neutrosophic graphs (PPNGs), and emphasize the effectiveness at interpreting extremely heterogeneous data that are prevalent in our daily life, particularly data gathered from various different sources which are becoming increasingly common place in the current times. The applicability of the proposed PPNG is demonstrated by applying the PPNGs on a potential real-life scenario on responding to the spread of COVID-19, where PPNGs are used to determine the safest path of travel and the safest place to stay to minimize the chances of getting infected. Both of this information have proven to be vital aspects in the efforts to combat the spread of the COVID-19 pandemic while providing the necessary support to the domestic economies, most of which are currently in recession due to the adverse effects brought upon by the pandemic. Hence, the PPNGs are applicable to all countries around the world and can be used under any circumstances such as pandemics or even in regular situations to optimize the travelling time and distance.

{0,1}-Brauer Configuration Algebras and Their Applications in Graph Energy Theory

The energy E(G) of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determining appropriated bounds of these types of energies for significant classes of graphs, digraphs and matrices, provided that, in general, finding out their exact values is a problem of great difficulty. In this paper, the trace norm of a {0,1}-Brauer configuration is introduced. It is estimated and computed by associating suitable families of graphs and posets to Brauer configuration algebras.

Novel dynamic fuzzy Decision-Making framework for COVID-19 vaccine dose recipients

Introduction

The vaccine distribution for the COVID-19 is a multicriteria decision-making (MCDM) problem based on three issues, namely, identification of different distribution criteria, importance criteria and data variation. Thus, the Pythagorean fuzzy decision by opinion score method (PFDOSM) for prioritising vaccine recipients is the correct approach because it utilises the most powerful MCDM ranking method. However, PFDOSM weighs the criteria values of each alternative implicitly, which is limited to explicitly weighting each criterion. In view of solving this theoretical issue, the fuzzy-weighted zero-inconsistency (FWZIC) can be used as a powerful weighting MCDM method to provide explicit weights for a criteria set with zero inconstancy. However, FWZIC is based on the triangular fuzzy number that is limited in solving the vagueness related to the aforementioned theoretical issues.

Objectives

This research presents a novel homogeneous Pythagorean fuzzy framework for distributing the COVID-19 vaccine dose by integrating a new formulation of the PFWZIC and PFDOSM methods.

Methods

The methodology is divided into two phases. Firstly, an augmented dataset was generated that included 300 recipients based on five COVID-19 vaccine distribution criteria (i.e., vaccine recipient memberships, chronic disease conditions, age, geographic location severity and disabilities). Then, a decision matrix was constructed on the basis of an intersection of the 'recipients list' and 'COVID-19 distribution criteria'. Then, the MCDM methods were integrated. An extended PFWZIC was developed, followed by the development of PFDOSM.

Results

(1) PFWZIC effectively weighted the vaccine distribution criteria. (2) The PFDOSM-based group prioritisation was considered in the final distribution result. (3) The prioritisation ranks of the vaccine recipients were subject to a systematic ranking that is supported by high correlation results over nine scenarios of the changing criteria weights values.

Conclusion

The findings of this study are expected to ensuring equitable protection against COVID-19 and thus help accelerate vaccine progress worldwide.

On Laplacian energy of picture fuzzy graphs in site selection problem

In the present work, the adjacency matrix, the energy and the Laplacian energy for a picture fuzzy graph/directed graph have been introduced along with their lower and the upper bounds. Further, in the selection problem of decision making, a methodology for the ranking of the available alternatives has been presented by utilizing the picture fuzzy graph and its energy/Laplacian energy. For the shake of demonstrating the implementation of the introduced methodology, the task of site selection for the hydropower plant has been carried out as an application. The originality of the introduced approach, comparative remarks, advantageous features and limitations have also been studied in contrast with intuitionistic fuzzy and Pythagorean fuzzy information.

A note on different types of product of neutrosophic graphs

Fuzzy set and neutrosophic set are two efficient tools to handle the uncertainties and vagueness of any real-world problems. Neutrosophic set is more capable than fuzzy set to deal the uncertainties of a real-life problem. This research paper introduces some new concept of single-valued neutrosophic graph (SVNG). We have also presented some different operations on SVNG such as rejection, symmetric difference, maximal product, and residue product with appropriate examples, and some of their important theorems are also described. Then, we have described the concept of total degree of a neutrosophic graph with some interesting examples. We have also presented an efficient approach to solve a decision-making problem using SVNG.

q-Rung orthopair fuzzy graphs under Hamacher operators

A q-rung orthopair fuzzy set (q-ROFS) is more practical and powerful than intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS) to model uncertainty in various decision-making problems. In this research article, we introduce the notion of q-rung orthopair fuzzy Hamacher graphs (q-ROFHGs). We utilize the Hamacher operators because they are flexible and parameterized in decision making. We determine the energy of q-ROFHGs as well as the energy of splitting and shadow q-ROFHGs. In addition, we propose the Randić energy of q-ROFHG and its some substantial results. Further, we present the idea of q-rung orthopair fuzzy Hamacher digraphs (q-ROFHDGs). We solve a decision-making numerical example related to the selection of best housing society for investment by calculating the energy and Randić energy of q-ROFHDGs and an algorithm to exhibit the applicability of the presented concepts in decision making. Finally, we present the conclusion.

Multi-criteria group decision making with Pythagorean fuzzy soft topology

Pythagorean fuzzy set (PFS) introduced by Yager (2013) is the extension of intuitionistic fuzzy set (IFS) introduced by Atanassov (1983). PFS is also known as IFS of type-2. Pythagorean fuzzy soft set (PFSS), introduced by Peng et al. (2015) and later studied by Guleria and Bajaj (2019) and Naeem et al. (2019), are very helpful in representing vague information that occurs in real world circumstances. In this article, we introduce the notion of Pythagorean fuzzy soft topology (PFS-topology) defined on Pythagorean fuzzy soft set (PFSS). We define PFS-basis, PFS-subspace, PFS-interior, PFS-closure and boundary of PFSS. We introduce Pythagorean fuzzy soft separation axioms, Pythagorean fuzzy soft regular and normal spaces. Furthermore, we present an application of PFSSs to multiple criteria group decision making (MCGDM) using choice value method in the real world problems which yields the optimum results for investment in the stock exchange. We also render an application of PFS-topology in medical diagnosis using TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution). The applications are accompanied by Algorithms, flow charts and statistical diagrams.

A Study on Domination in Vague Incidence Graph and Its Application in Medical Sciences

Fuzzy graphs (FGs), broadly known as fuzzy incidence graphs (FIGs), have been acknowledged as being an applicable and well-organized tool to epitomize and solve many multifarious real-world problems in which vague data and information are essential. Owing to unpredictable and unspecified information being an integral component in real-life problems that are often uncertain, it is highly challenging for an expert to illustrate those problems through a fuzzy graph. Therefore, resolving the uncertainty accompanying the unpredictable and unspecified information of any real-world problem can be done by applying a vague incidence graph (VIG), based on which the FGs may not engender satisfactory results. Similarly, VIGs are outstandingly practical tools for analyzing different computer science domains such as networking, clustering, and also other issues such as medical sciences, and traffic planning. Dominating sets (DSs) enjoy practical interest in several areas. In wireless networking, DSs are being used to find efficient routes with ad-hoc mobile networks. They have also been employed in document summarization, and in secure systems designs for electrical grids; consequently, in this paper, we extend the concept of the FIG to the VIG, and show some of its important properties. In particular, we discuss the well-known problems of vague incidence dominating set, valid degree, isolated vertex, vague incidence irredundant set and their cardinalities related to the dominating, etc. Finally, a DS application for VIG to properly manage the COVID-19 testing facility is introduced.

Certain Properties of Vague Graphs with a Novel Application

Fuzzy graph models enjoy the ubiquity of being present in nature and man-made structures, such as the dynamic processes in physical, biological, and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems that are often uncertain, for an expert, it is highly difficult to demonstrate those problems through a fuzzy graph. Resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem can be done using a vague graph (VG), with which the fuzzy graphs may not generate satisfactory results. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGs. The objective of this paper is to present certain types of vague graphs (VGs), including strongly irregular (SI), strongly totally irregular (STI), neighborly edge irregular (NEI), and neighborly edge totally irregular vague graphs (NETIVGs), which are introduced for the first time here. Some remarkable properties associated with these new VGs were investigated, and necessary and sufficient conditions under which strongly irregular vague graphs (SIVGs) and highly irregular vague graphs (HIVGs) are equivalent were obtained. The relation among strongly, highly, and neighborly irregular vague graphs was established. A comparative study between NEI and NETIVGs was performed. Different examples are provided to evaluate the validity of the new definitions. A new definition of energy called the Laplacian energy (LE) is presented, and its calculation is shown with some examples. Likewise, we introduce the notions of the adjacency matrix (AM), degree matrix (DM), and Laplacian matrix (LM) of VGs. The lower and upper bounds for the Laplacian energy of a VG are derived. Furthermore, this study discusses the VG energy concept by providing a real-time example. Finally, an application of the proposed concepts is presented to find the most effective person in a hospital.

Spherical Fuzzy Graphs with Application to Decision-Making

In a network model, the evaluation information given by decision makers are occasionally of types: yes, abstain, no, and refusal. To deal with such problems, we use mathematical models based on picture fuzzy sets. The spherical fuzzy model is more versatile than the picture fuzzy model as it broadens the space of uncertain and vague information, due to its outstanding feature of vast space of participation of acceptable triplets. Graphs are a mathematical representation of networks. Thus to deal with many real-world phenomena represented by networks, spherical fuzzy graphs can be used to model different practical scenarios in a more flexible manner than picture fuzzy graphs. In this research article, we discuss two operations on spherical fuzzy graphs (SFGs), namely, symmetric difference and rejection; and develop some results regarding their degrees and total degrees. We describe certain concepts of irregular SFGs with several important properties. Further, we present an application of SFGs in decision making.

A Study on Hypergraph Representations of Complex Fuzzy Information

The paradigm shift prompted by Zadeh’s fuzzy sets in 1965 did not end with the fuzzy model and logic. Extensions in various lines have produced e.g., intuitionistic fuzzy sets in 1983, complex fuzzy sets in 2002, or hesitant fuzzy sets in 2010. The researcher can avail himself of graphs of various types in order to represent concepts like networks with imprecise information, whether it is fuzzy, intuitionistic, or has more general characteristics. When the relationships in the network are symmetrical, and each member can be linked with groups of members, the natural concept for a representation is a hypergraph. In this paper we develop novel generalized hypergraphs in a wide fuzzy context, namely, complex intuitionistic fuzzy hypergraphs, complex Pythagorean fuzzy hypergraphs, and complex q-rung orthopair fuzzy hypergraphs. Further, we consider the transversals and minimal transversals of complex q-rung orthopair fuzzy hypergraphs. We present some algorithms to construct the minimal transversals and certain related concepts. As an application, we describe a collaboration network model through a complex q-rung orthopair fuzzy hypergraph. We use it to find the author having the most outstanding collaboration skills using score and choice values.

A Novel Description on Edge-Regular q-Rung Picture Fuzzy Graphs with Application

Picture fuzzy model is a generalized structure of intuitionistic fuzzy model in the sense that it not only assigns the membership and nonmembership values in the form of orthopair ( μ , ν ) to an element, but it assigns a triplet ( μ , η , ν ) , where η denotes the neutral degree and the difference π = 1 - ( μ + η + ν ) indicates the degree of refusal. The q-rung picture fuzzy set( q -RPFS) provides a wide formal mathematical sketch in which uncertain and vague conceptual phenomenon can be precisely and rigorously studied because of its distinctive quality of vast representation space of acceptable triplets. This paper discusses some properties including edge regularity, total edge regularity and perfect edge regularity of q-rung picture fuzzy graphs (q-RPFGs). The work introduces and investigates these properties for square q-RPFGs and q-RPF line graphs. Furthermore, this study characterizes how regularity and edge regularity of q-RPFGs structurally relate. In addition, it presents the concept of ego-networks to extract knowledge from large social networks under q-rung picture fuzzy environment with algorithm.

q-Rung Orthopair Fuzzy Hypergraphs with Applications

The concept of q-rung orthopair fuzzy sets generalizes the notions of intuitionistic fuzzy sets and Pythagorean fuzzy sets to describe complicated uncertain information more effectively. Their most dominant attribute is that the sum of the q th power of the truth-membership and the q th power of the falsity-membership must be equal to or less than one, so they can broaden the space of uncertain data. This set can adjust the range of indication of decision data by changing the parameter q, q ≥ 1 . In this research study, we design a new framework for handling uncertain data by means of the combinative theory of q-rung orthopair fuzzy sets and hypergraphs. We define q-rung orthopair fuzzy hypergraphs to achieve the advantages of both theories. Further, we propose certain novel concepts, including adjacent levels of q-rung orthopair fuzzy hypergraphs, ( α , β ) -level hypergraphs, transversals, and minimal transversals of q-rung orthopair fuzzy hypergraphs. We present a brief comparison of our proposed model with other existing theories. Moreover, we implement some interesting concepts of q-rung orthopair fuzzy hypergraphs for decision-making to prove the effectiveness of our proposed model.

q-Rung Orthopair Fuzzy Competition Graphs with Application in the Soil Ecosystem

The q-rung orthopair fuzzy set is a powerful tool for depicting fuzziness and uncertainty, as compared to the Pythagorean fuzzy model. The aim of this paper is to present q-rung orthopair fuzzy competition graphs (q-ROFCGs) and their generalizations, including q-rung orthopair fuzzy k-competition graphs, p-competition q-rung orthopair fuzzy graphs and m-step q-rung orthopair fuzzy competition graphs with several important properties. The study proposes the novel concepts of q-rung orthopair fuzzy cliques and triangulated q-rung orthopair fuzzy graphs with real-life characterizations. In particular, the present work evolves the notion of competition number and m-step competition number of q-rung picture fuzzy graphs with algorithms and explores their bounds in connection with the size of the smallest q-rung orthopair fuzzy edge clique cover. In addition, an application is illustrated in the soil ecosystem with an algorithm to highlight the contributions of this research article in practical applications.

Planar Graphs under Pythagorean Fuzzy Environment

Graph theory plays a substantial role in structuring and designing many problems. A number of structural designs with crossings can be found in real world scenarios. To model the vagueness and uncertainty in graphical network problems, many extensions of graph theoretical ideas are introduced. To deal with such uncertain situations, the present paper proposes the concept of Pythagorean fuzzy multigraphs and Pythagorean fuzzy planar graphs with some of their eminent characteristics by investigating Pythagorean fuzzy planarity value with strong, weak and considerable edges. A close association is developed between Pythagorean fuzzy planar and dual graphs. This paper also includes a brief discussion on non-planar Pythagorean fuzzy graphs and explores the concepts of isomorphism, weak isomorphism and co-weak isomorphism for Pythagorean fuzzy planar graphs. Moreover, it presents a problem that shows applicability of the proposed concept.

Specific Types of Pythagorean Fuzzy Graphs and Application to Decision-Making

The purpose of this research study is to present some new operations, including rejection, symmetric difference, residue product, and maximal product of Pythagorean fuzzy graphs (PFGs), and to explore some of their properties. This research article introduces certain notions, including intuitionistic fuzzy graphs of 3-type (IFGs3T), intuitionistic fuzzy graphs of 4-type (IFGs4T), and intuitionistic fuzzy graphs of n-type (IFGsnT), and proves that every IFG(n−1)T is an IFGnT (for n ≥ 2). Moreover, this study discusses the application of Pythagorean fuzzy graphs in decision making.