Fuzzy Graph Research Articles

Geodetic domination integrity in fuzzy graphs

Let N = (V, E) be a simple graph and let X be a subset of V (N). If every node not in X lies on a geodesic path between two nodes from X then it is called a geodetic set. The geodetic number g (N) is the minimum cardinality of such set X. The subset X is called a dominating set if every node not in X has at least one neighbour in X. The minimum number of nodes of a dominating set is known as domination number γ (N). If the subset X is a geodetic set as well as a dominating set then it is called a geodetic dominating set. The minimum cardinality of a geodetic dominating set is known as geodetic domination number γg (N). The geodetic domination integrity of N is defined to be DIg (N) = min {|X| + m (N - X) : X is a geodetic dominating set of N}, where m (N - X) denotes the order of the largest component of N - X. Uncertain networks can be modelled using fuzzy graphs. In a graph, each vertex and each edge are equally significant. However, in fuzzy graphs, each vertex and each edge is important in terms of fuzziness in their own right. In this study, the concepts of geodetic dominating sets in fuzzy graphs and geodetic domination number are defined and bounds are obtained. Moreover, the vulnerability parameter Geodetic domination integrity is introduced in fuzzy graphs. Further, the geodetic domination integrity for complete fuzzy graphs, complete bipartite fuzzy graphs, Cartesian product of two strong fuzzy graphs and bounds are also discussed. The applications of this parameter are applied to a telecommunication network system model to identify the key persons in the system and applied in a fuzzy social network to find the most influential group within the network.

A survey on cubic fuzzy graph structure with an application in the diagnosis of brain lesions

A cubic fuzzy graph is a fuzzy graph that simultaneously supports fuzzy membership and interval-valued fuzzy membership. This simultaneity leads to a better flexibility in modeling problems regarding uncertain variables. The cubic fuzzy graph structure, as a combination of cubic fuzzy graphs and graph structures, shows better capabilities in solving complex problems, especially where there are multiple relationships. Since many problems are a combination of different relationships, as well, applying some operations on them creates new problems; therefore, in this article, some of the most important product operations on cubic fuzzy graph structure have been investigated and some of their properties have been described. Studies have shown that the product of two strong cubic fuzzy graph structures is not always strong and sometimes special conditions are needed to be met. By calculating the vertex degree in each of the products, a clear image of the comparison between the vertex degrees in the products has been obtained. Also, the relationships between the products have been examined and the investigations have shown that the combination of some product operations with each other leads to other products. At the end, the cubic fuzzy graph structure application in the diagnosis of brain lesions is presented.

Defective vertex and stable connectivity of a fuzzy graph and their application to identify the chickenpox

A new concept of vertices in a fuzzy graph known as defective vertices is introduced here. A vertex in a fuzzy network is called defective if no edges incident with it are strong. Defective vertex cannot be ignored when determining dominance in a fuzzy network because they are a part of the network. Finding defective vertices in a network is not much difficult when the adjacency matrix is given. In this paper, the novel concept of defective vertices of a fuzzy graph is introduced. Based on this idea a stable domination set and a stable domination number are defined. This also optimised the network by establishing minimal connectivity. We have proposed three algorithms for finding the defective vertices, establishing stable connectivity, and determining the stable domination number for a given graph. An application of stable domination in the diagnosis of chickenpox disease is demonstrated to show the effectiveness of the proposed algorithms.

Dominations in bipolar picture fuzzy graphs and social networks

Dominations in bipolar picture fuzzy graphs and social networks

Signless Laplacian Energy of Interval-Valued Fuzzy Graph and its Applications

An interval-valued fuzzy graph (IVFG) emanates from a fuzzy graph (FG) where the membership is given in interval form. This framework give the user more flexibility in dealing with fuzzy information. In this paper, the signless Laplacian matrix of an interval-valued fuzzy-directed graph is defined. The eigenvalue, spectrum, spectral radius, and energy of an interval-valued fuzzy-directed graph associated with the signless Laplacian matrix are reported. In addition, the lower bound of the signless Laplacian energy in this graph is highlighted. Finally, these tools are employed to build an algorithm that helps in solving some real live problems.

A Newfangled Interpretation on Fermatean Neutrosophic Dombi Fuzzy Graphs


 Neutrosophic Dombi fuzzy graph is an advancement of the Dombi fuzzy graph and intuitionistic Dombi fuzzy graph. In this paper, we have initiated a new concept of the Fermatean neutrosophic Dombi fuzzy graph. Further, we identified a few products of the direct, cartesian, composition of Fermatean neutrosophic Dombi fuzzy graphs. Also, we examined the related proposition with suitable illustrations with graphs.

Study of Various Dominations in Bipolar Intuitionistic Anti Fuzzy Graphs

Bipolar intuitionistic fuzzy graphs (BIFG) with perfectly connected domination are introduced in this study. The study also discusses the bipolar intuitionistic antifuzzy graphs (BIAFG) domination concepts. Additionally, the topic of 2-domination in BIAFG are explored. we developed the secure edge dominating set and its domination number in BIAFG also some results are derived with suitable examples.

Some indices of picture fuzzy graphs and their applications

Some indices of picture fuzzy graphs and their applications

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.

A Fuzzy Graph Theory Approach to the Facility Location Problem: A Case Study in the Indian Banking System

A fuzzy graph G is stated to have a set of trees as its tree cover if all the vertices of G are in their union. The maximum weight tree in the tree cover is assumed to be the cost of a tree cover for a fuzzy graph. For an integer β>0, finding a set of trees to cover all the vertices of a graph with minimum cost and at most β number of spanning trees is known as the β-tree cover problem. Combining the tree-covering concept and facility location problem in a fuzzy environment for solving critical real-life problems in the recent era is a more fruitful approach. This issue strongly inspires us to develop a model with a practical algorithm. This paper provides an algorithm and complexity analysis to determine the number of rooted trees s covering the given fuzzy graph. In addition, a model is constructed with three optimization programming problems in the facility location problem and a tree covering fuzzy graphs. The model includes two types of the facility location problem, simultaneously addressing a variable covering radius and a fixed covering radius. A numerical example is provided to further describe the model, then, in the application part of the paper, the proposed model is applied to solve the real-life problem of maximizing demand saturation by minimizing the number of small denominations in the Indian banking system. This problem involves the data input of different indicators in the banking system along with details of the denominations of banknotes.

Fuzzy topological invariants in uniform fuzzy graphs

In this paper, we have defined some fuzzy topological invariants for particular types of uniform fuzzy graph. Some particular useful types of uniform fuzzy graphs are Uniform Edge Fuzzy Graph, Uniform Vertex Fuzzy Graph, Uniform Vertex-Edge Fuzzy Graph and Totally Uniform Fuzzy Graph. For each particular type we have defined different kinds of degrees in a graph in accordance with the unique nature of it. In the end, we have applied all our output results to a cellular neural fuzzy graph as an example, to verify the predicting ability of topological invariants. The aim of this paper is to define more significant fuzzy topological invariants in fuzzy graphs. Our ideas will help to create a link between fuzzy graph theory and simple (crisp) graph theory.

Number of possible captures in a chess puzzle move using the fuzzy colouring technique

In this work, several operations on fuzzy graphs are introduced: u-product, strong edge product, and k th power. The relationship between the fuzzy chromatic number of resultant fuzzy graphs of operations union, join, and newly developed operations and the fuzzy chromatic number of associated fuzzy graphs is also investigated using fuzzy colouring techniques. The number of captures in a chess puzzle move is calculated using the fuzzy colouring approach.

Applications of Neutrosophic social network using max product networks

Neutrosophic graphs deals with more complex, uncertain problems in real-life applications which provides more flexibility and compatibility than intuitionistic fuzzy graphs. The aim of this paper is to enrich the efficiency of the maximized network in accordance with time management and quality. Here we maximize three neutrosophic graphs into a fully connected Neutrosophic network using the Max product of graphs. Such a type of network is formed from individuals with unique aspects in every field of work among them. This study proposes the max product of three graphs and forming a single-valued neutrosophic graph to find the efficient time management in the flow of information on a single source time-dependent network of single-valued neutrosophic network. The proposed approach is illustrated with applications. Also, a spanning tree algorithm comparative study is done with Said Broumi et al. [15] and enhanced the result by minimum score function.

Application of neutrosophic optimal network using operations

Neutrosophic graphs deals with more complex, uncertain problems in real-life applications which provides more flexibility and compatibility than Intuitionistic fuzzy graphs. The aim of this paper is to enrich the efficiency of the network in accordance with productivity and quality. Here we develop two Neutrosophic graphs into a fully connected Neutrosophic network using the product of graphs. Such a type of network is formed from individuals with unique aspects in every field of work among them. This study proposes extending the other graph products and forming a single valued Neutrosophic graph to find the efficient productivity in the flow of information on a single source network of a single valued Neutrosophic network. An Optimal algorithm is proposed and illustrated with an application.

A Study on Different types of Product of Anti Fuzzy Graphs

A Study on Different types of Product of Anti Fuzzy Graphs

Interval-valued bipolar fuzzy line graphs

Objectives The notion of Bipolarity based on positive and negative outcomes. It is well known that bipolar models give more precision, flexibility, and compatibility to the system as compared to the classical models and fuzzy models. A bipolar fuzzy graph(BFG) provides more flexibility while modeling human thinking as compared with a fuzzy graph, and an interval valued bipolar fuzzy graph(IVBFG) has numerous applications where the real-life problem are time dependent and there is a network structure complexity. The aim of this paper is to introduce an interval-valued bipolar line fuzzy graph(IVBFLG).Result In this paper, we have proposed the notion of an IVBFLG and some of its characterizations. Also, some propositions and theorems related to an IVIFLGs are developed and proved. Furthermore, isomorphism between two IVIFLGs toward their IVIFGs was determined and verified. As a result, we derive a necessary and sufficient condition for an IVBFG to be isomorphic to its corresponding IVBFLG and some remarkable properties like degree, size, order, regularity, strength, and completeness of an IVBFLGs have been investigated, and the proposed concepts are illustrated with the examples.

DOMINATING SET ON CHAIN OF FUZZY GRAPHS

In this paper, we define fuzzy graph chains, which comprise vertex identification. These fuzzy graphs are isomorphic fuzzy graphs, provide that after applying various features to the chain of fuzzy graphs, which as special fuzzy graph chain of .

Fuzzy Domination Graphs in Decision Support Tasks

In decision support tasks, one often has to deal with uncertainty due to fuzzy judgments of the decision maker or the expert. This paper proposes methods that allow you to rank the alternatives based on fuzzy evaluations. This is achieved by using fuzzy weighted summation, fuzzy implication, a computation graph showing the criteria, and a fuzzy dominance graph showing the alternatives. If the criteria have equal importance, then fuzzy graphs corresponding to the dominance of each of the criteria are used. An algorithm that is used for both the transition from fuzzy dominance graphs and the ranking of alternatives is proposed. This algorithm is based on the idea of constructing Kemeny medians or other concordant rankings at a given confidence level in the existence of corresponding arcs. Computational experiments have shown the performance of these approaches. It is reasonable to apply them in problems that require complex expert evaluations with a large number of alternatives and criteria.

Characteristics and Operations of Complex Fuzzy Graphs

In this paper, Complex Fuzzy Graph (CFG) analyzed and introduced new concepts in CFG such as Spanning CFG, Complete CFG, path, arc, length, connected, strongest path and weakest path of CFG. We derived some properties of self complementary CFG and defined the operations on direct product, Semi strong product and strong product of CFG. Derived Isomorphic CFG with example is given in this paper. Moreover we introduced density of the graph and balanced complex fuzzy graph.

Bipolar intuitionistic fuzzy graph based decision-making model to identify flood vulnerable region.

Bipolar intuitionistic fuzzy graphs (BIFG) are an extension of fuzzy graphs that can effectively capture uncertain or imprecise information in various applications. In graph theory, the covering, matching, and domination problems are benchmark concepts applied to various domains. These concepts may not be defined precisely using a crisp graph when the vertices and edges are more uncertain. Therefore, this study defines the covering, matching and domination concepts in bipolar intuitionistic fuzzy graphs (BIFG) using effective edges with certain important results. To define these concepts when the effective edges are absent, some novel approaches are discussed. To illustrate the domination concepts, the applications in disaster management and location selection problems are discussed. Further, a BIFG-based decision-making model is designed to identify the flood-vulnerable zones in Chennai, where the city's most and least vulnerable zones are identified. From the proposed model, Kodambakkam ([Formula: see text]) is the most susceptible zone in Chennai. Finally, a comparative analysis is done with the existing techniques to show the efficiency of the model.