M-polar Fuzzy Graph Research Articles

Certain Types of Vertices in m-Polar Fuzzy Graphs

In this study, we explore the concept of m-polar fuzzy (mPF) detour g-eccentric nodes within m-polar fuzzy graphs (mPFGs). We delve into the idea of mPF detour g-interior nodes and mPF detour g-boundary nodes, examining their significance and properties. Additionally, we establish the relationship between mPF detour g-boundary vertices and mPF cut vertices.

Inverse Graphs in m-Polar Fuzzy Environments and Their Application in Robotics Manufacturing Allocation Problems with New Techniques of Resolvability

The inverse in crisp graph theory is a well-known topic. However, the inverse concept for fuzzy graphs has recently been created, and its numerous characteristics are being examined. Each node and edge in m-polar fuzzy graphs (mPFG) include m components, which are interlinked through a minimum relationship. However, if one wants to maximize the relationship between nodes and edges, then the m-polar fuzzy graph concept is inappropriate. Considering everything we wish to obtain here, we present an inverse graph under an m-polar fuzzy environment. An inverse mPFG is one in which each component’s membership value (MV) is greater than or equal to that of each component of the incidence nodes. This is in contrast to an mPFG, where each component’s MV is less than or equal to the MV of each component’s incidence nodes. An inverse mPFG’s characteristics and some of its isomorphic features are introduced. The α-cut concept is also studied here. Here, we also define the composition and decomposition of an inverse mPFG uniquely with a proper explanation. The connectivity concept, that is, the strength of connectedness, cut nodes, bridges, etc., is also developed on an inverse mPF environment, and some of the properties of this concept are also discussed in detail. Lastly, a real-life application based on the robotics manufacturing allocation problem is solved with the help of an inverse mPFG.

Solution of road network problem with the help of m-polar fuzzy graph using isometric and antipodal concept

The isometry in crisp graph theory is a well-known fact. But, isometry under a fuzzy environment was developed recently and studied many facts. In a m-polar fuzzy graph, we have to think m components for each node and edge. Since, in our consideration, we consider m components for each nodes as well as edges, therefore we can not handle this type of situation using fuzzy model as their is a single components for this concept. Again, we can not apply bipolar or intuitionistic fuzzy graph model as each edges or nodes have just two components. Thus, these mPFG models give more efficient fuzziness results than other fuzzy model. Also, it is very interesting to develop and analyze such types of mPFGs with examples and related theorems. Considering all those things together, we have presented isometry under a m-polar fuzzy environment. In this paper, we have discussed the isometric m-polar fuzzy graph along with many exciting facts about it. Metric space properties have also been implemented on m-polar fuzzy isometric graph. We also have initiated a generalized fuzzy graph, namely antipodal m-polar fuzzy graphs, along with several issues. The degree of it is also presented along with edge regularity properties. We also give a relation between m-polar fuzzy antipodal graphs and their underlying crisp graphs. Its properties have also been discussed on m-polar fuzzy odd as well as even cycles, complete graphs, etc. Finally, a real-life application on a road network system in a m-polar fuzzy environment using the mu-distance concept is also presented.

Integrity on m-Polar Fuzzy Graphs and Its Application

Integrity for crisp graph theory is a well-defined topic. However, the integrity concept for fuzzy graphs has only recently been defined and extensively researched. However, in m-polar fuzzy graphs (mPFG), each node as well as edges has m components. So, defining integrity in the mPF environment needs a new concept. As in the m-polar fuzzy environment, each node and edge has m components, so we have more flexibility to address the uncertainty rather than fuzzy as well as other uncertain environments. In this article, we developed a brand-new idea known as node integrity on mPFG and went in-depth on a few of their related properties. We have thoroughly covered some of their related properties as well as a brand-new idea called dominating integrity on mPFG. Different types of integrity on mPFG such as node integrity, dominating integrity, and edge integrity are discussed thoroughly along with some of its interesting facts have been introduced. Under isomorphism, their properties have also been studied. We also discussed the interrelation between them. A new type of mPFG called efficient mPFG which is directly related to dominating integrity concept has also been introduced. Several facts about efficient mPFG have also been studied here along with details descriptions. Finally, a real-world mobile network application that is directly related to the integrity of the mPFG concept has been discussed.

M-Polar Fuzzy Graphs and Deep Learning for the Design of Analog Amplifiers

The design of analog circuits is a complex and repetitive process aimed at finding the best design variant. It is characterized by uncertainty and multivariate approaches. The designer has to make different choices to satisfy a predefined specification with required parameters. This paper proposes a method for facilitating the design of analog amplifiers based on m-polar fuzzy graphs theory and deep learning. M-polar fuzzy graphs are used because of their flexibility and the possibility to model different real-life multi-attribute problems. Deep learning is applied to solve a regression task and to predict the membership functions of the m-polar fuzzy graph vertices (the solutions), taking on the role of domain experts. The performance of the learner is high since the obtained errors are very small: Root Mean Squared Error is from 0.0032 to 0.0187, Absolute Error is from 0.022 to 0.098 and Relative Error is between 0.27% and 1.57%. The proposed method is verified through the design of three amplifiers: summing amplifier, subtracting amplifier, and summing/subtracting amplifier. The method can be used for improving the design process of electronic circuits with the possibility of automating some tasks.

Connectivity indices of m-polar fuzzy network model, with an application to a product manufacturing problem

Connectivity is among the most essential concerns in graph theory and its applications. We consider this issue in a framework that stems from the combination of m-polar fuzzy set theory with graphs. We introduce two measurements of connectedness of m-polar fuzzy graphs that we call their connectivity and average connectivity indices. Examples are given, and the theoretical performance of these concepts is investigated. Particularly, we are concerned with the effect of deleting a vertex or an edge from an m-polar fuzzy graph, on its connectivity and average connectivity indices. We also establish bounding expressions for the connectivity index in complete m-polar fuzzy graphs, complete bipartite m-polar fuzzy graphs, and wheel m-polar fuzzy graphs. Moreover, we introduce some special types of vertices called m-polar fuzzy connectivity reducing vertices, m-polar fuzzy connectivity enhancing vertices, and m-polar fuzzy connectivity neutral vertices. Our theoretical contribution is applied to a product manufacturing problem that takes advantage of multi-polar uncertain information. The justification for our application is systematized using an algorithm. Finally, we compare the proposed method to existing methodologies to demonstrate its feasibility and applicability.

A Novel Study of Graphs Based on m -Polar Cubic Structures

By combining the notions of interval-valued m-polar fuzzy graphs and m-polar fuzzy graphs, the notion of m-polar cubic graphs is first introduced. Then, the degree of a vertex in m-polar cubic graphs and complete m-polar cubic graphs is defined. After that, the concepts of direct product and strong product of m-polar cubic graphs are given. Moreover, weak isomorphism and co-weak isomorphism are defined, and examples are given to prove that weak isomorphism and co-weak isomorphism are not an isomorphism. Finally, the notion of complement m-polar cubic graphs and weak complement m-polar cubic graphs is presented.

Menger’s theorem for m-polar fuzzy graphs and application of m-polar fuzzy edges to road network

The main objective of this research article is to classify different types of m-polar fuzzy edges in an m-polar fuzzy graph by using the strength of connectedness between pairs of vertices. The identification of types of m-polar fuzzy edges, including α-strong m-polar fuzzy edges, β-strong m-polar fuzzy edges and δ-weak m-polar fuzzy edges proved to be very useful to completely determine the basic structure of m-polar fuzzy graph. We analyze types of m-polar fuzzy edges in strongest m-polar fuzzy path and m-polar fuzzy cycle. Further, we define various terms, including m-polar fuzzy cut-vertex, m-polar fuzzy bridge, strength reducing set of vertices and strength reducing set of edges. We highlight the difference between edge disjoint m-polar fuzzy path and internally disjoint m-polar fuzzy path from one vertex to another vertex in an m-polar fuzzy graph. We define strong size of an m-polar fuzzy graph. We then present the most celebrated result due to Karl Menger for m-polar fuzzy graphs and illustrate the vertex version of Menger’s theorem to find out the strongest m-polar fuzzy paths between affected and non-affected cities of a country due to an earthquake. Moreover, we discuss an application of types of m-polar fuzzy edges to determine traffic-accidental zones in a road network. Finally, a comparative analysis of our research work with existing techniques is presented to prove its applicability and effectiveness.

An investigation on m-polar fuzzy threshold graph and its application on resource power controlling system

The threshold graph is a well studied topic. But, the fuzzy threshold graph is defined recently and investigated many properties. In a fuzzy threshold graph, only one threshold is considered for every vertex and edge. In a m-polar fuzzy graph, each vertex and each edge has a m number of membership values. So, defining a threshold graph for m-polar fuzzy graph is not easy and needs some new ideas. By considering m thresholds each for each component out of m components of membership values of a vertex or edge, we defined m-polar fuzzy threshold graph (mPFTG). Many interesting properties are also presented on mPFTG. The mPFTG is also decomposed in a very unique way and this decomposition generates m different fuzzy threshold graphs and the properties of decomposed graphs are explained. The important parameters viz. m-polar fuzzy threshold dimension and mPF threshold partition number are also defined and investigated thoroughly. Finally, a real-life application using mPFTG on a resource power controlling system is presented.

Interval valued m-polar fuzzy planar graph and its application

In this article, a new idea of interval-valued m-polar fuzzy (IVmPF) graph is introduced and investigated some of it’s properties. Here, IVmPF multiset, interval-valued m-polar fuzzy (IVmPF) multi graph are presented. IVmPF planar value of an interval-valued m-polar fuzzy (IVmPF) planar graph along with degree of planarity value is also introduced to measure the planarity value of an interval-valued m-polar fuzzy (IVmPF) graph. Some related terms like complete IVmPF graph, strong IVmPF graph, strong edges, faces of IVmPF planar graph are presented. In this paper, IVmPF dual graph is also described which is closely related to IVmPF planar graph. Some important properties are studied on IVmPF dual graph. Lastly, a real life application on IVmPF planar graph has been discussed to show its practicability.

On m-Polar Interval-valued Fuzzy Graph and its Application

In this paper, the concept of the -polar fuzzy graph (-PFG) and interval-valued fuzzy graph (IVFG) is integrated and introduced an unprecedented kind of fuzzy graph designated as -polar interval-valued fuzzy graph (-PIVFG). Complement of the -PIVFG is defined and the failure of this definition in some cases are highlighted. Various examples are cited and then redefined the notation of complement such that it applies to all -PIVFGs. The other algebraic properties such as isomorphism, weak isomorphism, co-weak isomorphism of the -PIVFG are investigated. Moreover, some basic results on the isomorphic property of -PIVFG are proved. Finally, an application of -PIVFG is explored. Abbreviations: The following abbreviations are employed in this study: FS: Fuzzy set; FG: Fuzzy graph; IVFS: Interval-valued fuzzy sets; IVFG: Interval-valued fuzzy graph; -PFS: -polar fuzzy sets; -PFG: -polar fuzzy graph; -PIVFS: -polar interval-valued fuzzy sets; -PIVFG: -polar interval-valued fuzzy graph.

Unify Steiner Weiner Distance for Some Class of m-Polar Fuzzy Graphs

ABSTRACT Sometimes information in a network model is based on multi-agent, multi-attribute, multi-object, multi-polar information or uncertainty rather than a single bit. An m-polar fuzzy model is useful for such network models which gives more and more precision, flexibility, and comparability to the system as compared to the classical, fuzzy models. On the other, The Steiner tree problem in networks, and particularly in graphs, was formulated by Hakimi [25] and Levi [21] by definition minimal size connected tree sub graph that contains the vertices in S. Steiner trees have applications to multiprocessor computer networks. For example, it may be desired to connect a certain set of processors with a sub network that uses the fewest communication links. In this paper, we extend Steiner distance SWk (G) for m-polar fuzzy graphs and give this parameter for Join, composition and Cartesian product of two m-polar fuzzy graphs.

Complex multi-fuzzy context analysis at different granulation

The m-p olar fuzzy graph representation of concept lattice gives a way to deal with its periodic changes at a given phase of time. One of the suitable examples is opinion of reviewers towards acceptance and rejection of a manuscript used to change several times for a given journal. Dealing with these types of m-polar complex fuzzy attributes is a crucial task for the data analytic researchers. The first problem arises with its precise mathematical representation and second with its algebraic processing for knowledge discovery tasks. In this regard, the calculus of complex multi-fuzzy set and its granulation is introduced for handling complex multi-fuzzy context and its graphical traversal. The information obtained from the proposed methods is also analyzed with recently introduced mathematical model on m-polar fuzzy set as well as complex fuzzy concept lattice.

Fuzzy colouring of m-polar fuzzy graph and its application

Fuzzy colouring of m-polar fuzzy graph and its application

Application of Strong Arcs in m-Polar Fuzzy Graphs

Recently, m-polar fuzzy graph (mPF graph) becomes a growing research topic as the generalization of fuzzy graph. In this paper, at first mPF path, mPF cycle in an mPF graph are defined. The strength of connectedness of mPF path is introduced. Next, strongest and strong mPF path, mPF bridges, mPF cut nodes, mPF tree and mPF forests in an mPF graph are considered. Also, it is proved that an arc of mPF tree is strong mPF arc if and only if it is an mPF bridge. Finally, mPF end nodes in an mPF graph is defined and investigated some properties of it. An application of strongest path problem in 3-polar fuzzy graph is also given at the end.

Object and attribute oriented m-polar fuzzy concept lattice using the projection operator

In the current decade, descriptive analysis of uncertainty existing in m-polar fuzzy attributes is addressed as one of the crucial tasks. To deal with these types of attributes mathematical algebra of m-polar fuzzy graph and its concept lattice representation was introduced recently. In this process, a problem is addressed when an expert wants to discover some useful pattern based on maximal acceptance of m-polar fuzzy attributes (or objects) for solving the particular issue of a given problem. To deal with this problem, the current paper focuses on drawing the object and attribute based m-polar fuzzy concept lattice using the projection operator. One of the suitable examples for the proposed method is also given with illustration of object and attribute based concepts. The analyses obtained from both of the proposed methods are compared with recently available approaches on handling the m-polar fuzzy attributes.

Morphism of m-Polar Intuitionistic Fuzzy Graphs

Morphism of <i>m</i>-Polar Intuitionistic Fuzzy Graphs

Genus value of m-polar fuzzy graphs

Genus value of m-polar fuzzy graphs

M-polar fuzzy graphs and m-polar fuzzy line graphs

We first introduce the concepts of regular m-polar fuzzy graphs, totally regular m-polar fuzzy graphs and complete m-polar fuzzy graphs. We investigate some of their related properties. Next, we introduce the notion of an m-polar fuzzy line graph and investigate some of its properties. Finally, we present an application of m-polar fuzzy graphs.

Concept lattice visualization of data with m-polar fuzzy attribute

In the current, year’s most of the attention has been paid towards analysis of uncertainty and vagueness in fuzzy attributes. In this case, a major problem arises when the uncertainty exists in the attributes beyond the $$\left\{ \in , \notin \right\}$$ boundaries. The most of the time researchers faced problem in precise representation of given multi-polar or multi-decision attributes. To deal with this type of attributes, current paper introduces a method based on the properties of m-polar fuzzy set and its concept lattice diagram. The generation of line diagram is established using the vertices and nodes of a defined m-polar fuzzy graph and its properties. A method is also proposed to generate some of the pattern from the given m-polar fuzzy context at user required subset of attributes. In addition, the analysis derived from the proposed method is compared with Akram and Younas (J Appl Math Comput 53(1–2):365–382, 2017) as well as Mukherjee and Das (in: Proceedings of Third International Conference on Advanced Computing, Networking and Informatics, pp 21–28, 2016) based on various parameters.