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P., Karthick. "Some Classes on Product of Intuitionistic Fuzzy Graphs." Journal of Applied Mathematics and Statistical Analysis 1, no. 2 (2020): 1–8. https://doi.org/10.5281/zenodo.4067900.
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<em>Intuitionistic fuzzy graph is a generalization of a fuzzy graph. The purpose of this research paper is to calculate the degree of the vertices for an intuitionistic fuzzy graphs. This IFGs are derived from given intuitionistic fuzzy graphs based on the operators cartesian product, composition and tensor product</em>
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Yin, Songyi, Hongxu Li, and Yang Yang. "Product Operations on q-Rung Orthopair Fuzzy Graphs." Symmetry 11, no. 4 (2019): 588. http://dx.doi.org/10.3390/sym11040588.
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The q-rung orthopair fuzzy graph is an extension of intuitionistic fuzzy graph and Pythagorean fuzzy graph. In this paper, the degree and total degree of a vertex in q-rung orthopair fuzzy graphs are firstly defined. Then, some product operations on q-rung orthopair fuzzy graphs, including direct product, Cartesian product, semi-strong product, strong product, and lexicographic product, are defined. Furthermore, some theorems about the degree and total degree under these product operations are put forward and elaborated with several examples. In particular, these theorems improve the similar results in single-valued neutrosophic graphs and Pythagorean fuzzy graphs.
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Anwar, Abida, and Faryal Chaudhry. "On Certain Products of Complex Intuitionistic Fuzzy Graphs." Journal of Function Spaces 2021 (December 16, 2021): 1–9. http://dx.doi.org/10.1155/2021/6515646.
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A complex intuitionistic fuzzy set (CIFS) can be used to model problems that have both intuitionistic uncertainty and periodicity. A diagram composed of nodes connected by lines and labeled with specific information may be used to depict a wide range of real-life and physical events. Complex intuitionistic fuzzy graphs (CIFGs) are a broader type of diagram that may be used to manipulate data. In this paper, we define the key operations direct, semistrong, strong, and modular products for complex intuitionistic fuzzy graphs and look at some interesting findings. Further, the strong complex intuitionistic fuzzy graph is defined, and several significant findings are developed. Furthermore, we study the behavior of the degree of a vertex in the modular product of two complex intuitionistic fuzzy graphs.
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Parimala, Mani, Arwa Almunajam, Muthusamy Karthika, and Ibtesam Alshammari. "Pythagorean Fuzzy Digraphs and Its Application in Healthcare Center." Journal of Mathematics 2021 (September 6, 2021): 1–6. http://dx.doi.org/10.1155/2021/5459654.
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The notion of fuzzy set is introduced to deal with uncertainty, whereas the conventional sets are used for certainty. The extensions of fuzzy set theory such as intuitionistic fuzzy set (IFS) and Pythagorean fuzzy sets (PyFS) were introduced to overcome drawbacks in fuzzy theory. Fuzzy graph structure is used to deal with the uncertainty in a network and describe its relation on the nonempty vertex set. One of the extensions of intuitionistic fuzzy digraph (IFDG) is Pythagorean fuzzy digraph (PyFDG). IFDG cannot handle if the sum of degree of acceptance and degree of rejection for an arc weight exceeds 1. So, we introduced PyFDG to overcome the limitations in IFDG and it deals with the imprecise arc weight involving degree of acceptance and degree of rejection. Pythagorean fuzzy digraph (PyFDG) and its basic operations and score function of PyFDG are defined in this paper. Algorithm is proposed to solve application problem in healthcare center.
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Shoaib, Muhammad, Waqas Mahmood, Weded Albalawi, and Faria Ahmad Shami. "Notion of Complex Spherical Fuzzy Graph with Application." Journal of Function Spaces 2022 (May 18, 2022): 1–27. http://dx.doi.org/10.1155/2022/1795860.
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A complex spherical fuzzy set (CSFS) is a generalization of a spherical fuzzy set (CFS). CSFS handles vagueness more explicitly, and its range is expanded from the real subset to the complex with unit disc. The major goal of this research is to present the foundation of a complex spherical fuzzy graph (CSFG) due to the limitation of the complex neutral membership function in a complex Pythagorean fuzzy graph (CPFG). Complex spherical fuzzy models have more flexibility as compared to complex fuzzy models, complex intuitionistic fuzzy models, and complex Pythagorean fuzzy models due to their coverage in three directions: complex membership functions, neutral membership functions, and complex non-membership functions. Firstly, we present the motivation for CSFG. Furthermore, we define the order, degree of a vertex, size, and total degree of a vertex of CSFG. We elaborate on primary operations, including complement, join, and the union of CSFG. This research study introduces some operations, namely, strong product, composition, Cartesian product, and semi-strong product, on CSFG. Moreover, we present the application of CSFG, which ensures the ability to deal with problems in three directions.
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Tola, Keneni Abera, V. N. Srinivasa Rao Repalle, and Mamo Abebe Ashebo. "Theory and Application of Interval-Valued Neutrosophic Line Graphs." Journal of Mathematics 2024 (March 19, 2024): 1–17. http://dx.doi.org/10.1155/2024/5692756.
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Neutrosophic graphs are used to model inconsistent information and imprecise data about any real-life problem. It is regarded as a generalization of intuitionistic fuzzy graphs. Since interval-valued neutrosophic sets are more accurate, compatible, and flexible than single neutrosophic sets, interval-valued neutrosophic graphs (IVNGs) were defined. The interval-valued neutrosophic graph is a fundamental issue in graph theory that has wide applications in the real world. Also, problems may arise when partial ignorance exists in the datasets of membership [0, 1], and then, the concept of IVNG is crucial to represent the problems. Line graphs of neutrosophic graphs are significant due to their ability to represent and analyze uncertain or indeterminate information about edge relationships and complex networks in graphs. However, there is a research gap on the line graph of interval-valued neutrosophic graphs. In this paper, we introduce the theory of an interval-valued neutrosophic line graph (IVNLG) and its application. In line with that, some mathematical properties such as weak vertex isomorphism, weak edge isomorphism, effective edge, and other properties of IVNLGs are proposed. In addition, we defined the vertex degree of IVNLG with some properties, and by presenting several theorems and propositions, the relationship between fuzzy graph extensions and IVNLGs was explored. Finally, an overview of the algorithm used to solve the problems and the practical application of the introduced graphs were provided.
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Xiao, Wei, Arindam Dey, and Le Hoang Son. "A study on regular picture fuzzy graph with applications in communication networks." Journal of Intelligent & Fuzzy Systems 39, no. 3 (2020): 3633–45. http://dx.doi.org/10.3233/jifs-191913.
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Picture fuzzy graph (PFG) is an extended version of intuitionistic fuzzy graph (IFG) to model the uncertain real world problems, in which IFG may fail to model those problems properly. PFG is more precise, flexible and compatible than IFG to deal the real-life scenarios which consists of information these types: yes, abstain, no and refusal. The main focus of our study is to present the concept of isomorphic PFG, regular PFG (RPFG) and picture fuzzy multigraph. In this paper, we present the notation of RPFG. Many different types of RPFGs such as regular strong PFG, regular complete PFG, complete bipartite PFG and regular complement PFG are introduced. We also describe the concepts of dn and tdn-degree of a vertex in a RPFG. Based on those two types of degrees, we classify the regularity of PFG into 3 type’s namely, dn- RPFG, tdn-RPFG and n- highly irregular PFG. Several theorems of those RPFG are presented here. We define the busy vertex and free vertex in a RPFG. We present the notations of μ-complement, homomorphism, isomorphism, weak isomorphism and co weak isomorphism of RPFG. Some significant theorems on isomorphism and μ- complement of RPFG are derived here. We also introduce the notation of picture fuzzy multigraph. We present a mathematical model of communication network and transportation network by using picture fuzzy multigraph and real time data are collected so that the transportation network/communication network can work efficiently.
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Nazir, Nazia, Tanzeela Shaheen, LeSheng Jin, and Tapan Senapati. "An Improved Algorithm for Identification of Dominating Vertex Set in Intuitionistic Fuzzy Graphs." Axioms 12, no. 3 (2023): 289. http://dx.doi.org/10.3390/axioms12030289.
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In graph theory, a “dominating vertex set” is a subset of vertices in a graph such that every vertex in the graph is either a member of the subset or adjacent to a member of the subset. In other words, the vertices in the dominating set “dominate” the remaining vertices in the graph. Dominating vertex sets are important in graph theory because they can help us understand and analyze the behavior of a graph. For example, in network analysis, a set of dominant vertices may represent key nodes in a network that can influence the behavior of other nodes. Identifying dominant sets in a graph can also help in optimization problems, as it can help us find the minimum set of vertices that can control the entire graph. Now that there are theories about vagueness, it is important to define parallel ideas in vague structures, such as intuitionistic fuzzy graphs. This paper describes a better way to find dominating vertex sets (DVSs) in intuitive fuzzy graphs (IFGs). Even though there is already an algorithm for finding DVSs in IFGs, it has some problems. For example, it does not take into account the vertex volume, which has a direct effect on how DVSs are calculated. To address these limitations, we propose a new algorithm that can handle large-scale IFGs more efficiently. We show how effective and scalable the method is by comparing it to other methods and applying it to water flow. This work’s contributions can be used in many areas, such as social network analysis, transportation planning, and telecommunications.
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Myithili, K. K., and P. Nithyadevi. "Degrees and regularity of intuitionistic fuzzy semihypergraphs." Notes on Intuitionistic Fuzzy Sets 31, no. 1 (2025): 111–26. https://doi.org/10.7546/nifs.2025.31.1.111-126.
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This research work takes a new paradigm on the hypergraph concept which is a combination of a hypergraph and a semigraph. A semihypergraph is a connected hypergraph in which each hyperedge must have at least three vertices and any two hyperedges have at least one vertex in common. In a semihypergraph, vertices are classified as end, middle or middle-end vertices. This distinction, combined with membership and non-membership values, enables a more granular examination of vertices and their degrees in Intuitionistic Fuzzy Semihypergraphs (IFSHGs). This paper proposes four types of degrees: degree, end vertex degree, adjacent degree and consecutive adjacent degree on an IFSHG. Each degree reflects specific patterns within the intuitioistic fuzzy semihypergraphs. Additionally, three types of sizes are also defined: size, crisp size and pseudo size of IFSHGs. Concepts such as regular and totally regular IFSHGs with their properties are also defined.
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Isnaini Rosyida. "On Strong Fuzzy Resolving Set of Fuzzy Wheel Graphs." Advances in Nonlinear Variational Inequalities 28, no. 5s (2025): 132–39. https://doi.org/10.52783/anvi.v28.3655.
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Let Image be a fuzzy set on Image. A fuzzy labeling graph Image is a graph with bijective membership functions Image and Image so that each vertex’s membership degree and every edge’s membership value are different. Further, it satisfies Image for Image. “A graph formed from a single vertex connected to the vertices of a cycle of length Image is called a wheel graph with Image vertices". In this paper, we construct an algorithm for fuzzy labeling of wheel graphs. Under the algorithm, we have the degrees of vertices and edges are different and the monotone decreasing. The vertex Image gets the biggest degree among other vertices. Further, the edge Image is assigned to the biggest degree, and Image obtains the smallest degree. We also show the “strength of connectedness" between vertices in the fuzzy wheel graph. Finally, we investigate the “strong fuzzy resolving set" (SFRS) of the fuzzy labeling wheel graph with Image vertices and get the strong resolving number Image.
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S.Revathi, C. V. R. Harinarayanan, and R.Muthuraj. "Perfect Domination in Constant Intuitionistic Fuzzy Graph of Degree (ki ,kj)." International Journal of Fuzzy Mathematical Archive 14, no. 01 (2017): 91–99. http://dx.doi.org/10.22457/ijfma.v14n1a11.
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In this paper, the new kind of parameter perfect dominating set in constant intuitionistic fuzzy graph is defined and established the parametric conditions. Another new kind of parameter totally constant intuitionistic fuzzy graph is defined and established the parametric conditions. Some properties of Perfect dominating set in constant IFG and totally constant IFG with suitable examples are also discussed.
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Khan, Sami Ullah, Naeem Jan, Kifayat Ullah, and Lazim Abdullah. "Graphical Structures of Cubic Intuitionistic Fuzzy Information." Journal of Mathematics 2021 (May 11, 2021): 1–21. http://dx.doi.org/10.1155/2021/9994977.
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The theory developed in this article is based on graphs of cubic intuitionistic fuzzy sets (CIFS) called cubic intuitionistic fuzzy graphs (CIFGs). This graph generalizes the structures of fuzzy graph (FG), intuitionistic fuzzy graph (IFG), and interval-valued fuzzy graph (IVFG). Moreover, several associated concepts are established for CIFG, such as the idea subgraphs, degree of CIFG, order of CIFG, complement of CIFG, path in CIFG, strong CIFG, and the concept of bridges for CIFGs. Furthermore, the generalization of CIFG is proved with the help of some remarks. In addition, the comparison among the existing and the proposed ideas is carried out. Finally, an application of CIFG in decision-making problem is studied, and some future study is proposed.
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Talaee, B., and G. Nasiri. "On intersection graph of intuitionistic fuzzy submodules of a module." Lebanese Science Journal 20, no. 1 (2019): 104–21. http://dx.doi.org/10.22453/lsj-020.1.104-121.
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There are some interesting relations between submodules of a module and its intuitionistic fuzzy (IF) submodules. In this paper we investigate some relationshipsbetween submodules of a module and its IFsubmodules. Then we introduce a graph structure on IFsubmodules of a module and obtain some properties of it, that is the main goal of this paper. We define the intersection graph of submodules of a module M(G) and we show that a submodule Nof Mis a center in MGif and only if IFNis a center in IFG. We get some relationships between IFsubmodules of a module and their supports, as vertices of IFgraph and crisp graph of a module M, respectively. We show that an IFsubmodule Aof Mis center in IFgraph of Mif and only ifAis a center in crisp graph of M.In prime ring R, we show that every vertex of intersection graph of IFideals of Ris center. In general the nature of intersection graph of IFsubmodules of a module under intersection, homomorphic images, finite sum and other algebraic operations of its vertices, are investigated.
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Al-Masarwah, Anas, and Majdoleen Abu Qamar. "Certain Types of Fuzzy Soft Graphs." New Mathematics and Natural Computation 14, no. 02 (2018): 145–56. http://dx.doi.org/10.1142/s1793005718500102.
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In this paper, we introduce the concepts of uniform vertex fuzzy soft graphs, uniform edge fuzzy soft graphs, degree of a vertex, total degree of a vertex and complement fuzzy soft graphs with some examples. Also, we study regular and totally regular fuzzy soft graphs, and the conditions under which the complement of regular fuzzy soft graph becomes regular as well as totally regular are discussed. Also, we obtain some results related to regular, totally regular and complete fuzzy soft graphs.
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Sitara, Muzzamal, Muhammad Akram, and Muhammad Yousaf Bhatti. "Fuzzy Graph Structures with Application." Mathematics 7, no. 1 (2019): 63. http://dx.doi.org/10.3390/math7010063.
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In this article, we introduce the notions of maximal products of fuzzy graph structures, regular fuzzy graph structures, and describe these notions with examples and properties. Further, we present the degree and total degree of a vertex in maximal product of fuzzy graph structures and explain some of their properties. Furthermore, we develop a flowchart to show general procedure of application of fuzzy graph structure, regarding identification of most controversial issues among countries.
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Kang, Kyung Tae, Seok-Zun Song, and Young Bae Jun. "Multipolar Intuitionistic Fuzzy Set with Finite Degree and Its Application in BCK/BCI-Algebras." Mathematics 8, no. 2 (2020): 177. http://dx.doi.org/10.3390/math8020177.
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When events occur in everyday life, it is sometimes advantageous to approach them in two directions to find a solution for them. As a mathematical tool to handle these things, we can consider the intuitionistic fuzzy set. However, when events are complex and the key to a solution cannot be easily found, we feel the need to approach them for hours and from various directions. As mathematicians, we wish we had the mathematical tools that apply to these processes. If these mathematical tools were developed, we would be able to apply them to algebra, topology, graph theory, etc., from a close point of view, and we would be able to apply these research results to decision-making and/or coding theory, etc., from a distant point of view. In light of this view, the purpose of this study is to introduce the notion of a multipolar intuitionistic fuzzy set with finite degree (briefly, k-polar intuitionistic fuzzy set), and to apply it to algebraic structure, in particular, a BCK/BCI-algebra. The notions of a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra are introduced, and related properties are investigated. Relations between a k-polar intuitionistic fuzzy subalgebra and a k-polar intuitionistic fuzzy ideal are discussed. Characterizations of a k-polar intuitionistic fuzzy subalgebra/ideal are provided, and conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal are provided. In a BCI-algebra, relations between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal are discussed. A characterization of a closed k-polar intuitionistic fuzzy ideal is considered, and conditions for a k-polar intuitionistic fuzzy ideal to be closed are provided.
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Nisha., D., and B. Srividhya. "Characteristics of Fuzzy Wheel Graph and Hamilton Graph with Fuzzy Rule." International Journal of Trend in Scientific Research and Development 3, no. 6 (2019): 1061–64. https://doi.org/10.5281/zenodo.3589329.
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Graph theory is the concepts used to study and model various application in different areas. We proposed the wheel graph with n vertices can be defined as 1 skeleton of on n 1 gonal pyramid it is denoted by w nwith n 1 vertex n=3 . A wheel graph is hamiltonion, self dual and planar. In the mathematical field of graph theory, and a Hamilton path or traceable graph is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle is a hamiltonian path that is a cycle. In this paper, we consider the wheel graph and also the hamilton graph using if then rules fuzzy numbers. The results are related to the find the degree of odd vertices and even vertices are same by applying if then rules through the paths described by fuzzy numbers. Nisha. D | Srividhya. B "Characteristics of Fuzzy Wheel Graph and Hamilton Graph with Fuzzy Rule" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd29319.pdf
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Alanazi, Abdulaziz M., Ghulam Muhiuddin, Bashair M. Alenazi, Tanmoy Mahapatra, and Madhumangal Pal. "Utilizing m-Polar Fuzzy Saturation Graphs for Optimized Allocation Problem Solutions." Mathematics 11, no. 19 (2023): 4136. http://dx.doi.org/10.3390/math11194136.
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It is well known that crisp graph theory is saturated. However, saturation in a fuzzy environment has only lately been created and extensively researched. It is necessary to consider m components for each node and edge in an m-polar fuzzy graph. Since there is only one component for this idea, we are unable to manage this kind of circumstance using the fuzzy model since we take into account m components for each node as well as edges. Again, since each edge or node only has two components, we are unable to apply a bipolar or intuitionistic fuzzy graph model. In contrast to other fuzzy models, mPFG models produce outcomes of fuzziness that are more effective. Additionally, we develop and analyze these kinds of mPFGs using examples and related theorems. Considering all those things together, we define saturation for a m-polar fuzzy graph (mPFG) with multiple membership values for both vertices and edges; thus, a novel approach is required. In this context, we present a novel method for defining saturation in mPFG involving m saturations for each element in the membership value array of a vertex. This explains α-saturation and β-saturation. We investigate intriguing properties such as α-vertex count and β-vertex count and establish upper bounds for particular instances of mPFGs. Using the concept of α-saturation and α-saturation, block and bridge of mPFG are characterized. To identify the α-saturation and β-saturation mPFGs, two algorithms are designed and, using these algorithms, the saturated mPFG is determined. The time complexity of these algorithms is O(|V|3), where |V| is the number of vertices of the given graph. In addition, we demonstrate a practical application where the concept of saturation in mPFG is applicable. In this application, an appropriate location is determined for the allocation of a facility point.
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Radha, K., and S. R. Gayathri. "Distance Degree Sequence and Distance Neighborhood Degree Sequence of Vertices in Fuzzy Graphs and Some of Their Properties." Indian Journal Of Science And Technology 17, no. 41 (2024): 4349–57. http://dx.doi.org/10.17485/ijst/v17i41.1933.
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Objectives: The objective of this study is to explore new types of sequences in fuzzy graphs namely distance degree sequence and distance neighborhood degree sequence of fuzzy graphs which may have various applications in the real world. Methods: In this paper, two unique sequences for each vertex of fuzzy graphs are given by means of its degree and neighborhood degree which are called as the distance degree sequence and distance neighborhood degree sequence of vertices of fuzzy graphs respectively. The n-tuple of the distance degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance degree sequence of the fuzzy graph and the n-tuple of distance neighborhood degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance neighborhood degree sequence of the fuzzy graph. Findings: The distance degree sequence and distance neighborhood degree sequence of fuzzy graphs are introduced. The properties of these sequences and the general forms of these sequences of various fuzzy graphs are given. The relationship between the distance degree sequence and distance neighborhood degree sequences is discussed. Novelty: The distance degree sequence of graphs has already been introduced. In fuzzy graph theory, the distance degree sequence and distance neighborhood degree sequence are introduced in this paper. Mathematics Subject Classification (202 0): 03E72, 05C72, 05C12. Keywords: Fuzzy graphs, Distance degree sequence, Distance neighborhood-degree sequence, Regular fuzzy graphs
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Kumaran, Narayanan, Annamalai Meenakshi, Miroslav Mahdal, Jayavelu Udaya Prakash, and Radek Guras. "Application of Fuzzy Network Using Efficient Domination." Mathematics 11, no. 10 (2023): 2258. http://dx.doi.org/10.3390/math11102258.
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Let Heff (Veff, Eeff) be a finite simple connected graph of order m with vertex set Veff and edge set Eeff. A dominating set Sds⊆Veff is called an efficiently dominating set if, for every vertex ua∈VG, NGua∩Sds=1—where NG [ua] denotes the closed neighborhood of the vertex ua. Using efficient domination techniques and labelling, we constructed the fuzzy network. An algorithm has been framed to encrypt and decrypt the secret information present in the network, and furthermore, the algorithm has been given in pseudocode. The mathematical modelling of a strong fuzzy network is defined and constructed to elude the burgeoning intruder. Using the study of the efficient domination of fuzzy graphs, this domination parameter plays a nuanced role in encrypting and decrypting the framed network. One of the main purposes of fuzzy networks is encryption, so one of our contributions to this research is to build a novel combinatorial technique to encrypt and decode the built-in fuzzy network with a secret number utilizing effective domination. An illustration with an appropriate secret message is provided along with the encryption and decryption algorithms. Furthermore, we continued this study in intuitionistic fuzzy networks.
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Koam, Ali N. A., Muhammad Akram, and Peide Liu. "Decision-Making Analysis Based on Fuzzy Graph Structures." Mathematical Problems in Engineering 2020 (August 12, 2020): 1–30. http://dx.doi.org/10.1155/2020/6846257.
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A graph structure is a useful framework to solve the combinatorial problems in various fields of computational intelligence systems and computer science. In this research article, the concept of fuzzy sets is applied to the graph structure to define certain notions of fuzzy graph structures. Fuzzy graph structures can be very useful in the study of various structures, including fuzzy graphs, signed graphs, and the graphs having labeled or colored edges. The notions of the fuzzy graph structure, lexicographic-max product, and degree and total degree of a vertex in the lexicographic-max product are introduced. Further, the proposed concepts are explained through several numerical examples. In particular, applications of the fuzzy graph structures in decision-making process, regarding detection of marine crimes and detection of the road crimes, are presented. Finally, the general procedure of these applications is described by an algorithm.
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K, Radha, and R. Gayathri S. "Distance Degree Sequence and Distance Neighborhood Degree Sequence of Vertices in Fuzzy Graphs and Some of Their Properties." Indian Journal of Science and Technology 17, no. 41 (2024): 4349–57. https://doi.org/10.17485/IJST/v17i41.1933.
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Abstract <strong>Objectives:</strong> The objective of this study is to explore new types of sequences in fuzzy graphs namely distance degree sequence and distance neighborhood degree sequence of fuzzy graphs which may have various applications in the real world. <strong>Methods:</strong> In this paper, two unique sequences for each vertex of fuzzy graphs are given by means of its degree and neighborhood degree which are called as the distance degree sequence and distance neighborhood degree sequence of vertices of fuzzy graphs respectively. The n-tuple of the distance degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance degree sequence of the fuzzy graph and the n-tuple of distance neighborhood degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance neighborhood degree sequence of the fuzzy graph. <strong>Findings:</strong> The distance degree sequence and distance neighborhood degree sequence of fuzzy graphs are introduced. The properties of these sequences and the general forms of these sequences of various fuzzy graphs are given. The relationship between the distance degree sequence and distance neighborhood degree sequences is discussed. <strong>Novelty:</strong> The distance degree sequence of graphs has already been introduced. In fuzzy graph theory, the distance degree sequence and distance neighborhood degree sequence are introduced in this paper. Mathematics Subject Classification (202 0): 03E72, 05C72, 05C12. <strong>Keywords:</strong> Fuzzy graphs, Distance degree sequence, Distance neighborhood-degree sequence, Regular fuzzy graphs
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V., Ramadoss, and Kalpana D. "A SURVEY OF HYPO SOFT GRAPH STRUCTURES." International Journal of Applied and Advanced Scientific Research 3, no. 1 (2018): 159–64. https://doi.org/10.5281/zenodo.1193685.
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Akram [2] introduced the concept of bipolar fuzzy graphs and defined different operations on it. A. Nagoorgani and K. Radha [3, 4] introduced the concept of regular fuzzy graphs in 2008 and discussed about the degree of a vertex in some fuzzy graphs. K. Radha and N. Kumaravel [5] introduced the concept of edge degree, total edge degree and discussed about the degree of an edge in some fuzzy graphs. S. Arumugam and S. Velammal [6] discussed edge domination in fuzzy graphs. Soft set theory was introduced by Molodtsov [9] for modelling vagueness and uncertainty and it has been received much attention since Maji et al [10], Sezgin and Atagun [1] introduced and studied operations of soft sets. Soft set theory has also potential applications especially in decision making as in [10]. In this article, we have investigated the concept of hypo soft graph structures and its properties. Also we have discussed bell structures of hypo graphs with illustrative Examples.
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Ismayil, A. Mohamed, and S. Syed Asad Ahmed. "Degree, Neighbourhood, and Connectedness in Fuzzy Digraph." Indian Journal Of Science And Technology 17, no. 44 (2024): 4571–81. https://doi.org/10.17485/ijst/v17i44.3556.
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Objective: In the context of fuzzy digraphs, degree neighborhood, and connectedness play important roles in understanding the structure and relationships within the graph. Here are the objectives of degree, neighborhood, and connectedness in fuzzy digraphs: 1. Degree Neighborhood: The degree neighborhood of a vertex in a fuzzy digraph refers to the set of vertices that are directly connected to it. In fuzzy digraphs, the concept of degree neighborhood is extended to include the strength or degree of connection between vertices, which is represented by fuzzy values. 2. Connectedness: Connectedness in fuzzy digraphs refers to the existence of paths between any pair of vertices, considering the fuzzy nature of relationships. Methods: By using fuzzy vertex and fuzzy edges in fuzzy digraph to find degree (in-degree and out-degree), neighborhood (in-neighborhood and out-neighborhood), and to find connectedness in fuzzy digraph. Findings: In this study, some types of fuzzy digraphs are introduced, neighborhood (in-neighborhood and out-neighborhood) and degree (in-degree and out-degree) in fuzzy digraphs, and strong, weak, unilateral connectedness in fuzzy digraphs are introduced. Novelty: To show that let 𝐺𝐷 ∶ (𝜎, 𝜇) be a fuzzy digraph with fuzzy vertex set 𝜎 and fuzzy edge set 𝜇. In this paper, some types of fuzzy digraphs are introduced, neighborhood (inneighborhood and out-neighborhood) and degree (in-degree and out-degree) in fuzzy digraph and strong, weak, unilateral connectedness in fuzzy digraph are introduced. Furthermore, this paper establishes various results and theorems, and some properties of different fuzzy digraphs are discussed. Keywords: Fuzzy digraph; Degree in a fuzzy digraph; Neighborhood in a fuzzy digraph; Fuzzy diwalk; Fuzzy dicycle; Fuzzy diacyclic digraph
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Garai, Arindam, and Tapan Kumar Roy. "Intuitionistic Fuzzy Modeling to Travelling Salesman Problem." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 11, no. 9 (2013): 3015–24. http://dx.doi.org/10.24297/ijct.v11i9.3414.
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This paper presents solution technique for travelling salesman problem (TSP) under intuitionistic fuzzy environment. Travelling salesman problem is a non-deterministic polynomial-time (NP) hard problem in combinatorial optimization, studied in graph theory, operations research and theoretical computer science. It must be noted that a traveling sales man even face a situation in which he is not able to achieve his objectives completely. There must be a set of alternatives from which he can select one that best meets his aspiration level. For Multi-Objective Symmetric TSP, in fuzzy environment, it is converted into a Linear Program using Fuzzy Multi-Objective Linear Programming technique. A route cannot be simply chosen just as it will most minimize time or it will cover the least possible distance. Examples with requirements to consider the degree of rejection or hesitation (or both) are overflowing in our materialistic world. Here comes the need to consider TSP under intuitionistic fuzzy environment. The degree of rejection as well as the degree of hesitancy must be studied to find the solution in a truly optimum sense! Proposed technique is an extension as well as collaboration of ideas of fuzzy traveling salesperson problem and intuitionistic fuzzy (IF) optimization technique.
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A, Mohamed Ismayil, and Syed Asad Ahmed S. "Degree, Neighbourhood and Connectedness in Fuzzy Digraph." Indian Journal of Science and Technology 17, no. 44 (2024): 4571–81. https://doi.org/10.17485/IJST/v17i44.3556.
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Abstract <strong>Objective:</strong> In the context of fuzzy digraphs, degree neighborhood, and connectedness play important roles in understanding the structure and relationships within the graph. Here are the objectives of degree, neighborhood, and connectedness in fuzzy digraphs: 1. Degree Neighborhood: The degree neighborhood of a vertex in a fuzzy digraph refers to the set of vertices that are directly connected to it. In fuzzy digraphs, the concept of degree neighborhood is extended to include the strength or degree of connection between vertices, which is represented by fuzzy values. 2. Connectedness: Connectedness in fuzzy digraphs refers to the existence of paths between any pair of vertices, considering the fuzzy nature of relationships. <strong>Methods:</strong> By using fuzzy vertex and fuzzy edges in fuzzy digraph to find degree (in-degree and out-degree), neighborhood (in-neighborhood and out-neighborhood), and to find connectedness in fuzzy digraph. <strong>Findings:</strong> In this study, some types of fuzzy digraphs are introduced, neighborhood (in-neighborhood and out-neighborhood) and degree (in-degree and out-degree) in fuzzy digraphs, and strong, weak, unilateral connectedness in fuzzy digraphs are introduced. <strong>Novelty:</strong> To show that let 𝐺𝐷 ∶ (𝜎, 𝜇) be a fuzzy digraph with fuzzy vertex set 𝜎 and fuzzy edge set 𝜇. In this paper, some types of fuzzy digraphs are introduced, neighborhood (inneighborhood and out-neighborhood) and degree (in-degree and out-degree) in fuzzy digraph and strong, weak, unilateral connectedness in fuzzy digraph are introduced. Furthermore, this paper establishes various results and theorems, and some properties of different fuzzy digraphs are discussed. <strong>Keywords:</strong> Fuzzy digraph; Degree in a fuzzy digraph; Neighborhood in a fuzzy digraph; Fuzzy diwalk; Fuzzy dicycle; Fuzzy diacyclic digraph
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Indumathi, P., S. Santhosh Kumar, R. ,. Buvaneswari, and S. K. Mala. "Total degree of maximal product of two constant intuitionistic fuzzy graphs." Edelweiss Applied Science and Technology 8, no. 6 (2024): 5789–99. http://dx.doi.org/10.55214/25768484.v8i6.3258.
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This paper explains about the Total degree of Maximal product of Two constant IF graphs. Fuzzy graphs are derived from crisp graphs. Various properties of IF graphs are extended from Fuzzy graphs. Maximal product of Fuzzy graph structures with applications have been discussed in different papers and extended to IF graphs. Constant IF graphs are special type of IF graphs which have same degree for all its vertices. IF Graphs have many applications including the investigation of images by image segmentation, Brain mapping etc, Maximal product of fuzzy graphs is applied in various fields like effective logistic Management, Agricultural product mapping etc. Here in this paper, the Total degree of the vertices in maximal product of Constant IF graphs are studied in detail with definition and various examples and theorems.
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Shoaib, Muhammad, Waqas Mahmood, Ahmad N. Al-Kenani, and Sahidul Islam. "Notes on Upper and Lower Truncation of Picture Fuzzy Graphs." Discrete Dynamics in Nature and Society 2022 (April 26, 2022): 1–11. http://dx.doi.org/10.1155/2022/7646828.
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Due to the absence of a neutral function, there are drawbacks to the existing definition of an intuitionistic fuzzy graph (Int-FG). In that definition of Int-FG, membership function and nonmembership function are involved. In this study, lower truncation (Low-T) and upper truncation (Up-T) are applied to picture fuzzy graphs (Pic-FGs). Pic-FG has an additional neutral membership function. Furthermore, Low-T and Up-T of subdivision of the picture fuzzy line graph (Pic-FLnG) are also discussed. The degree of an edge in both Up-T and Low-T of Pic-FG is discussed. Some related theorems related to the degree of Up-T and Low-T of Pic-FG are discussed.
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Shoaib, Muhammad, Waqas Mahmood, Ahmad N. Al-Kenani, and Sahidul Islam. "Notes on Upper and Lower Truncation of Picture Fuzzy Graphs." Discrete Dynamics in Nature and Society 2022 (April 26, 2022): 1–11. http://dx.doi.org/10.1155/2022/7646828.
Pełny tekst źródłaStreszczenie:
Due to the absence of a neutral function, there are drawbacks to the existing definition of an intuitionistic fuzzy graph (Int-FG). In that definition of Int-FG, membership function and nonmembership function are involved. In this study, lower truncation (Low-T) and upper truncation (Up-T) are applied to picture fuzzy graphs (Pic-FGs). Pic-FG has an additional neutral membership function. Furthermore, Low-T and Up-T of subdivision of the picture fuzzy line graph (Pic-FLnG) are also discussed. The degree of an edge in both Up-T and Low-T of Pic-FG is discussed. Some related theorems related to the degree of Up-T and Low-T of Pic-FG are discussed.
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Cai, Ruiqi, Buvaneswari Rangasamy, Senbaga Priya Karuppusamy, and Aysha Khan. "Characterization of Degree Energies and Bounds in Spectral Fuzzy Graphs." Symmetry 17, no. 5 (2025): 644. https://doi.org/10.3390/sym17050644.
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This study explores the degree energy of fuzzy graphs to establish fundamental spectral bounds and characterize adjacency structures. We derive upper bounds on the sum of squared degree eigenvalues based on vertex degree distributions and formulate constraints using the characteristic polynomial of the maximum degree matrix. Furthermore, we demonstrate that the average degree energy of a connected fuzzy graph remains strictly positive. The proposed framework is applied to protein–protein interaction networks, identifying critical proteins and enhancing network resilience analysis.
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Rahayu, H. S., I. M. Sulandra, and V. Kusumasari. "Product of bipolar anti fuzzy graph and their degree of vertex." Journal of Physics: Conference Series 1872, no. 1 (2021): 012013. http://dx.doi.org/10.1088/1742-6596/1872/1/012013.
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Kausar, Andleeb, Nabilah Abughazalah, and Naveed Yaqoob. "Multi Polar q-Rung Orthopair Fuzzy Graphs with Some Topological Indices." Symmetry 15, no. 12 (2023): 2131. http://dx.doi.org/10.3390/sym15122131.
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The importance of symmetry in graph theory has always been significant, but in recent years, it has become much more so in a number of subfields, including but not limited to domination theory, topological indices, Gromov hyperbolic graphs, and the metric dimension of graphs. The purpose of this monograph is to initiate the idea of a multi polar q-rung orthopair fuzzy graphs (m-PqROPFG) as a fusion of multi polar fuzzy graphs and q-rung orthopair fuzzy graphs. Moreover, for a vertex of multi polar q-rung orthopair fuzzy graphs, the degree and total degree of the vertex are defined. Then, some product operations, inclusive of direct, Cartesian, semi strong, strong lexicographic products, and the union of multi polar q-rung orthopair fuzzy graphs (m-PqROPFGs), are obtained. Also, at first we define some degree based fuzzy topological indices of m-PqROPFG. Then, we compute Zareb indices of the first and second kind, Randic indices, and harmonic index of a m-PqROPFG.
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Shoaib, Muhammad, Waqas Mahmood, Qin Xin, and Fairouz Tchier. "Maximal Product and Symmetric Difference of Complex Fuzzy Graph with Application." Symmetry 14, no. 6 (2022): 1126. http://dx.doi.org/10.3390/sym14061126.
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A complex fuzzy set (CFS) is described by a complex-valued truth membership function, which is a combination of a standard true membership function plus a phase term. In this paper, we extend the idea of a fuzzy graph (FG) to a complex fuzzy graph (CFG). The CFS complexity arises from the variety of values that its membership function can attain. In contrast to a standard fuzzy membership function, its range is expanded to the complex plane’s unit circle rather than [0,1]. As a result, the CFS provides a mathematical structure for representing membership in a set in terms of complex numbers. In recent times, a mathematical technique has been a popular way to combine several features. Using the preceding mathematical technique, we introduce strong approaches that are properties of CFG. We define the order and size of CFG. We discuss the degree of vertex and the total degree of vertex of CFG. We describe basic operations, including union, join, and the complement of CFG. We show new maximal product and symmetric difference operations on CFG, along with examples and theorems that go along with them. Lastly, at the base of a complex fuzzy graph, we show the application that would be important for measuring the symmetry or asymmetry of acquaintanceship levels of social disease: COVID-19.
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P., Sinthamani. "Microscopic Examinations of the Elements on the Behaviour of Group of Fellow Men using Fuzzy Graph." International Journal of Engineering and Advanced Technology (IJEAT) 9, no. 3 (2020): 220–26. https://doi.org/10.35940/ijeat.B4533.029320.
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In this work, the behavior of an individual of a group to his fellow men. In a group, if we consider the relationship between people, we note the following possibilities. Any two of them may like each other or dislikes each other or indifferent to each other. It can be also checked originally to describe the way psychological consistency is obtained. And signed degree vertex evolves interesting results in new parameters of Balanced signed fuzzy graph and unbalanced signed fuzzy graph also we obtained some properties over it.
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S.N, Suber Bathusha, and S. Angelin Kavitha Raj. "Climatic analysis based on interval-valued complex Pythagorean fuzzy graph structure." Annals of Mathematics and Computer Science 19 (December 18, 2023): 10–33. http://dx.doi.org/10.56947/amcs.v19.213.
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The Interval-valued Complex Pythagorean Fuzzy Set (IVCPFS), an extension of the Pythagorean Fuzzy Set (PFS), offers a more accurate description of uncertainty than traditional fuzzy sets. It has numerous uses in fuzzy control. In this research study, we show the concept of Interval-Valued Complex Pythagorean Fuzzy Graph Structure (IVCPFGS), as well as the results of some regular IVCPFGS and totally regular IVCPFGS, along with the degree of vertex present. We also discussed the subdivision of an IVCPFGS, the complement IVCPFGS, and the strength of a Path in an IVCPFGS. Finally, we discuss a realworld example based on temperature variation and climatic data analysis in an environment with an interval-valued complex Pythagorean fuzzy framework to demonstrate the applicability of the generated results.
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Renuka, Sahu, and Kumar Sharma Animesh. "Understanding the Unique Properties of Fuzzy Concept in Binary Trees." Indian Journal of Science and Technology 16, no. 33 (2023): 2631–36. https://doi.org/10.17485/IJST/v16i33.874.
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Abstract <strong>Objective:</strong> This study investigated the properties of fuzzy binaries, a type of rooted trees, and explored their implications in different domains. <strong>Method:</strong> This investigation included analyzing the structural properties of fuzzy binary tree, such as the number of vertices, the height, and the degree to which edges were assigned membership. Theorem: Combinatorial analysis and mathematical induction were used to derive the relation between the number of vertex and height of a fuzzy binary tree based on its internal vertex. The conditions that allow complete graphs to form within fuzzy binary tree were also determined by mathematical analysis and graph theory. <strong>Findings:</strong> The study found that the internal vertices of a fuzzy binary tree have a strong influence on the number of vertices and height. A theorem was developed that states a fuzzy binary tree full with ”i” terminal vertices will have (i+1), total vertices, and (i+1), terminal vertices. A fuzzy binary tree can form a complete graph under certain conditions. <strong>Novelty:</strong> This research provides novel insights into fuzzy binary trees’ structural properties. The derived theorem establishes a fundamental relation between the number and height of a fuzzy binary tree. This provides valuable insight into the complexity and organization of these trees. The identification of the conditions that lead to a complete graph in fuzzy binary tree further improves our understanding and appreciation of their comprehensiveness. These findings provide new perspectives on the application of fuzzy binary tree in different fields. <strong>Keywords:</strong> Fuzzy Binary Trees; Rooted Tree; Degrees of Membership; Structural Characteristics; Complete Graphs; Image Processing; Full Binary Tree; Terminal Vertices; Internal Vertices; Membership Function; Fuzzy Set Theory; Graph Theory; Data Mining
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T, Bharathi, and Leo S. "A Study on Plithogenic Product Fuzzy Graphs with Application in Social network." Indian Journal of Science and Technology 14, no. 40 (2021): 3051–63. https://doi.org/10.17485/IJST/v14i40.1298.
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Abstract <strong>Objectives:</strong> The main objective of this study is to define Plithogenic product fuzzy graphs (PPFGs), and introduce its properties.<strong> Method:</strong> PPFGs is newly introduced as a new graphical model where P-vertices are characterized by four or more attributes and the attribute values of P-edges are computed using the product operator. Findings: Theoretical discussions and results related to PPFGs, and subgraphs, paths, cycles, trees, bridge and cut vertex in PPFGs are demonstrated with examples. A social network model based on the notion of PPFGs has been presented and analyzed to show the utility and the advantage of Plithogenic product fuzzy graph model. <strong>Novelty:</strong> Strong and weak P-vertices, and highly strong, strong and weak P-edges are identified to analyze the strength of connectivity between different units. P-order, P-size, P-vertex range, P-edge range, degree, total degree, and average P-weight of P-vertices are computed to examine proximity, significance and centrality of units. <strong>Keywords:</strong> Plithogenic fuzzy sets; Fuzzy graphs; Plithogenic fuzzy graphs; Plithogenic product fuzzy graphs; Social networks
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Akram, Muhammad, Saba Siddique, and Majed G. Alharbi. "Clustering algorithm with strength of connectedness for $ m $-polar fuzzy network models." Mathematical Biosciences and Engineering 19, no. 1 (2021): 420–55. http://dx.doi.org/10.3934/mbe.2022021.
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<abstract><p>In this research study, we first define the strong degree of a vertex in an $ m $-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete $ m $-polar fuzzy graphs. Further, we introduce the concept of $ m $-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete $ m $-polar fuzzy graphs. We discuss connectivity parameters in $ m $-polar fuzzy graphs with precise examples, and we investigate the $ m $-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an $ m $-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as $ \epsilon_A $-reachable vertices in $ m $-polar fuzzy graphs, $ \epsilon_A $-connected $ m $-polar fuzzy graphs, or $ \epsilon_A $-connected $ m $-polar fuzzy subgraphs (in case the $ m $-polar fuzzy graph itself is not $ \epsilon_A $-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.</p></abstract>
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Rosyida, Isnaini, and Suryono Suryono. "Coloring picture fuzzy graphs through their cuts and its computation." International Journal of Advances in Intelligent Informatics 7, no. 1 (2021): 63. http://dx.doi.org/10.26555/ijain.v7i1.612.
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In a fuzzy set (FS), there is a concept of alpha-cuts of the FS for alpha in [0,1]. Further, this concept was extended into (alpha,delta)-cuts in an intuitionistic fuzzy set (IFS) for delta in [0,1]. One of the expansions of FS and IFS is the picture fuzzy set (PFS). Hence, the concept of (alpha,delta)-cuts was developed into (alpha,delta,beta)-cuts in a PFS where beta is an element of [0,1]. Since a picture fuzzy graph (PFG) consists of picture fuzzy vertex or edge sets or both of them, we have an idea to construct the notion of the (alpha,delta,beta)-cuts in a PFG. The steps used in this paper are developing theories and algorithms. The objectives in this research are to construct the concept of (alpha,delta,beta)-cuts in picture fuzzy graphs (PFGs), to construct the (alpha,delta,beta)-cuts coloring of PFGs, and to design an algorithm for finding the cut chromatic numbers of PFGs. The first result is a definition of the (alpha,delta,beta)-cut in picture fuzzy graphs (PFGs) where (alpha,delta,beta) are elements of a level set of the PFGs. Further, some properties of the cuts are proved. The second result is a concept of PFG coloring and the chromatic number of PFG based on the cuts. The third result is an algorithm to find the cuts and the chromatic numbers of PFGs. Finally, an evaluation of the algorithm is done through Matlab programming. This research could be used to solve some problems related to theories and applications of PFGs.
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Rao, Yongsheng, Saeed Kosari, Zehui Shao, Ruiqi Cai, and Liu Xinyue. "A Study on Domination in Vague Incidence Graph and Its Application in Medical Sciences." Symmetry 12, no. 11 (2020): 1885. http://dx.doi.org/10.3390/sym12111885.
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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.
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V., Ramadass, and Kalpana D. "REPRESENTATIONS OF EDGE REGULAR BIPOLAR FUZZY GRAPHS." International Journal of Applied and Advanced Scientific Research 2, no. 2 (2017): 267–72. https://doi.org/10.5281/zenodo.1069074.
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V. Ramadass & D. Kalpana, “Representations of Edge Regular Bipolar Fuzzy Graphs”, International Journal of Applied and Advanced Scientific Research, Volume 2, Issue 2, Page Number 267-272, 2017.
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"Degree of a Vertex in Max-Product of Intuitionistic Fuzzy Graph." International Journal of Recent Technology and Engineering 8, no. 4 (2019): 2902–5. http://dx.doi.org/10.35940/ijrte.c6411.118419.
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Graph theory has applications in many areas of computer science, including data mining, image segmentation, clustering and networking. Product on graphs has a wide range of application in networking system, automata theory, game theory and structural mechanics. In many cases, some aspects of a graph-theoretic problem may be uncertain. Intuitionistic fuzzy models provide more compatible to the system compared to the fuzzy models. An intuitionistic fuzzy graph can be derived from two given intuitionistic fuzzy graphs using max-product. In this paper, we studied the degree of vertex in intuitionistic fuzzy graph by the max-product of two given intuitionistic fuzzy graph. Also find the necessary and sufficient condition for max-product of two intuitionistic fuzzy graphs to be regular.
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Mohamed, S. Yahya, and A. Mohamed Ali. "Complement of max product of intuitionistic fuzzy graphs." Complex & Intelligent Systems, July 27, 2021. http://dx.doi.org/10.1007/s40747-021-00438-2.
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AbstractIn this paper, the complement of max product of two intuitionistic fuzzy graphs is defined. The degree of a vertex in the complement of max product of intuitionistic fuzzy graph is studied. Some results on complement of max product of two regular intuitionistic fuzzy graphs are stated and proved. Finally, we provide an application of intuitionistic fuzzy graphs in school determination using normalized Hamming distance.
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M., Aslam Malik, Rashmanlou Hossein, Shoaib Muhammad, A. Borzooei R., Taheri Morteza, and Broumi Said. "A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application." March 22, 2020. https://doi.org/10.5281/zenodo.3723630.
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Unipolar is less fundamental than bipolar cognition based on truth, and composure is a restraint for truth-based worlds. Bipolarity is the most powerful phenomenon that survives when truth disappeared in a black hole due to Hawking radiation or particular / anti particular emission. The purpose of this research study is to dene few four operations, including residue product, rejection, maximal product and symmetric difference of bipolar single-valued neutrosophic graph (BSVNG) and to explore some of their related properties with examples. Bipolar single-valued neutrosophic graph (BSVNG) is the generalization of the single-valued neutrosophic graph (SVNG), intuitionistic fuzzy graph, bipolar intuitionistic fuzzy graph, bipolar fuzzy graph and fuzzy graph. BSVNG plays a signicant role in the study of neural networks, daily energy issues, energy systems, and coding. Moreover, we will determine related properties like the degree of a vertex in a BSVNG or total degree of a vertex in a BSVNG. We provide examples of the vertex degree in BSVNG and the total vertex degree in BSVNG. In order to make this useful, we develop an algorithm for our useful method in steps.
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S.Ravi, Narayanan. "2-Pseudo Neighbourly Irregular Intuitionistic Fuzzy Graphs." November 30, 2016. https://doi.org/10.5281/zenodo.826796.
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Shaik, Noorjahan, and Sharief Basha Shaik. "Wiener index application in intuitionistic fuzzy rough graphs for transport network flow." Scientific Reports 15, no. 1 (2025). https://doi.org/10.1038/s41598-025-94488-y.
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Abstract The applicability of different topological indices is indispensable in fields such as chemistry, electronics, economics, business studies, medicine, and the social sciences. The most popular index in graph theory is the wiener index $$\left(\mathcal{W}\mathcal{I}\right)$$ , which is based on the geodesic distance between two vertices. It is assumed that the weight of the geodesic between vertex x and vertex y in intuitionistic fuzzy rough graphs (IFRG) is zero in the absence of a directed path. With regard to intuitionistic fuzzy rough graphs, the objective of this work is to investigate in detail the wiener index $$\left(\mathcal{W}\mathcal{I}\right)$$ and the average wiener index ( $$\mathcal{A}\mathcal{W}\mathcal{I})$$ . Also, the connectivity index $$\left(\mathcal{C}\mathcal{I}\right)$$ is one of the most significant indices, providing several examples and results. For intuitionistic fuzzy rough graphs, alternative distance and degree-based topological indices have also been developed. The research on intuitionistic fuzzy rough graphs that has been suggested is appropriate for representing imprecise data and uncertainty in practical situations. Additionally, examined is the connection between the wiener and connectivity indices. Finally, we proposed the use of wiener indices in transport network flow.
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Meenakshi, A., S. Dhanushiya, Hong Qin, and Muniyandy Elangovan. "Optimal Network Analysis through Vertex Order Coloring of Intuitionistic Fuzzy Graph Operations." Journal of Mathematics 2024, no. 1 (2024). http://dx.doi.org/10.1155/2024/8898813.
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Intuitionistic fuzzy graphs (IFGs) are a powerful tool for modeling uncertainty and complex relationships. They offer versatile frameworks for addressing real‐world challenges. In this research, we have introduced intuitionistic fuzzy vertex order coloring (IFVOC) and analyzed the alpha‐strong (), beta‐strong (), and gamma‐strong () vertices through their degree. We explored important theorems based on the types of strong vertices, broadening the scope of our study. We analyzed multiple IFG products to determine the most optimal network based on some important metrics, including the weight and total number of vertices, the chromatic number, and the weight of the graph’s minimum spanning tree. We systematically evaluate different types of IFG products, such as conormal, modular, residue, and maximal, and consider their implications for fuzzy graph structure and connectivity. We have investigated new metrics to measure the presence and importance of vertices in the graph, assessing both their frequency and overall impact. Additionally, we have investigated the impact of product operations on the significance of nodes and the resultant minimum spanning tree, providing insights into the overall robustness and efficiency of each product. We have explored how variations in product operations impact the distribution of vertices and the characteristics of the minimum spanning tree and chromatic number, elucidating the trade‐offs between different product options. We have modeled the product graph as a network to represent complex systems composed of interconnected entities. The outcomes of this research contribute to the advancement of IFG theory and its use in various fields, such as network design, strategic planning, and system analysis.
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Gong, Zengtai, and Lele He. "Connectivity Status of Intuitionistic Fuzzy Graphs and Its Applications." International Journal of Foundations of Computer Science, October 19, 2023, 1–17. http://dx.doi.org/10.1142/s0129054123500223.
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The connectivity is one of the crucial parameters of network used to transport network flow, routing problems and bandwidth allocation problems. The increased connectivity makes a network more stable. In this paper, a new parameter called connectivity status of a vertex is introduced in the intuitionistic fuzzy graph. The related definitions and propositions of connectivity status of a vertex are proposed and investigated in an intuitionistic fuzzy graph. Specifically, connectivity status and status sequence are defined and analysed with various examples. After deleting a vertex, we classify the vertices of an intuitionistic fuzzy graph as connectivity status enhancing vertices, connectivity status neutral vertices and connectivity status reducing vertices because of the change of connectivity status. Finally, we establish two algorithms for these concepts and give an application to illustrate feasibility of algorithms.
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"Degree of intuitionistic L-fuzzy graph." Journal of Mathematical and Computational Science, 2021. http://dx.doi.org/10.28919/jmcs/5221.
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Li, Li, Saeed Kosari, Seyed Hossein Sadati, and Ali Asghar Talebi. "Concepts of vertex regularity in cubic fuzzy graph structures with an application." Frontiers in Physics 10 (April 20, 2023). http://dx.doi.org/10.3389/fphy.2022.1087225.
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The cubic fuzzy graph structure, as a combination of cubic fuzzy graphs and fuzzy graph structures, shows better capabilities in solving complex problems, especially in cases where there are multiple relationships. The quality and method of determining the degree of vertices in this type of fuzzy graphs simultaneously supports fuzzy membership and interval-valued fuzzy membership, in addition to the multiplicity of relations, motivated us to conduct a study on the regularity of cubic fuzzy graph structures. In this context, the concepts of vertex regularity and total vertex regularity have been informed and some of its properties have been studied. In this regard, a comparative study between vertex regular and total vertex regular cubic fuzzy graph structure has been carried out and the necessary and sufficient conditions have been provided. These degrees can be easily compared in the form of a cubic number expressed. It has been found that the condition of the membership function is effective in the quality of degree calculation. In the end, an application of the degree of vertices in the cubic fuzzy graph structure is presented.