Fuzzy Competition Graphs

Handbook of Research on Advanced Applications of Graph Theory in Modern Society

In this chapter, the authors introduce some basic definitions related to fuzzy graphs like directed and undirected fuzzy graph, walk, path and circuit of a fuzzy graph, complete and strong fuzzy graph, bipartite fuzzy graph, degree of a vertex in fuzzy graphs, fuzzy subgraph, etc. These concepts are illustrated with some examples. The recently developed concepts like fuzzy planar graphs are discussed where the crossing of two edges are considered. Finally, the concepts of fuzzy threshold graphs and fuzzy competitions graphs are also given as a generalization of threshold and competition graphs.

Mathematics

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

Symmetry, 2018

Hypergraph theory is the most developed tool for demonstrating various practical problems in different domains of science and technology. Sometimes, information in a network model is uncertain and vague in nature. In this paper, our main focus is to apply the powerful methodology of fuzziness to generalize the notion of competition hypergraphs and fuzzy competition graphs. We introduce various new concepts, including fuzzy column hypergraphs, fuzzy row hypergraphs, fuzzy competition hypergraphs, fuzzy k-competition hypergraphs and fuzzy neighbourhood hypergraphs, strong hyperedges, kth strength of competition and symmetric properties. We design certain algorithms for constructing different types of fuzzy competition hypergraphs. We also present applications of fuzzy competition hypergraphs in decision support systems, including predator–prey relations in ecological niche, social networks and business marketing.

In this study researchers propose the complete, regular and complement fuzzy graph. In these subjects we shall study some properties such that: self complementary, regular, total regular fuzzy graph and the relation between these subject. On the other hand we shall study a product fuzzy graph and what the relation between the product and the complete also the complement. We get some theorem, proposition and corollaries for these subjects.

1.1 Graph Graph theory is tremendously useful in modelling the essential features of systems with finite components. Wide applications of graphical models are in the field of railway network, telephone network, communication problems, traffic network etc. Graph theoretic models can sometimes provide a useful structure upon which analytic techniques can be used. A graph is also used to model a relationship between a given set of objects. Each object is represented by a vertex and the relationship between them is represented by an edge if the relationship is unordered and by means of a directed edge if the objects have an ordered relation between them. Relationship among the objects need not always be precisely defined criteria; when we think of an imprecise concept, the fuzziness arises. 1.2 Fuzzy graph A mathematical frame work to explain the concept of uncertainty in real life through the publication of a seminal paper is introduced by Zadeh [1]. A fuzzy set is defined mathematically by assigning to each possible individual in the universe of discourse a value, representing its grade of membership, which corresponds to the degree, to which that individual is similar or compatible with the concept represented by the fuzzy set. Rosenfeld [2] was introduced the fuzzy graph using fuzzy relation, represents the relationship between the objects by precisely indicating the level of the relationship between the objects of the given set. Also he coined many fuzzy analogous graph theoretic concepts like bridge, cut vertex and tree. Fuzzy graphs have many more applications in modelling real time systems where the level of information inherent in the system varies with different levels of precision.

Fuzzy Graphs are having numerous applications in problems like Network analysis, Clustering, Pattern Recognition and Neural Networks. The analysis of properties of fuzzy graphs has facilitated the study of many complicated networks like Internet. In this paper we study the structures of complement of many important fuzzy graphs such as Fuzzy cycles, Blocks etc. The complement of fuzzy graphs with a certain structural property is studied in this paper.

Complex & Intelligent Systems

Graph theory plays crucial role in structuring many real-world problems including, medical sciences, control theory, expert systems and network security. Product in graphs, an operation that consider two graphs and produce a new graph by simple or complex changes, has wide range of applications in games theory, automata theory, structural mechanics and networking system. An intuitionistic fuzzy model is used to handle the vagueness and uncertainty in network problems. A Pythagorean fuzzy model is a powerful tool for describing vagueness and uncertainty more accurately as compared to intuitionistic fuzzy model. The objective of this paper is to apply the concept of Pythagorean fuzzy sets to graphs and then combine two Pythagorean fuzzy graphs (PFGs) using two new graph products namely, maximal product and the residue product. This research paper investigates the regularity for these products. Moreover, it discusses some eminent properties such as strongness, connectedness and completeness. Further, it proposes some necessary and sufficient conditions for G 1 * G 2 and G 1 • G 2 to be regular. Finally, decision-making problems concerning evaluation of best company for investment and alliance partner selection of a software company are solved to better understand PFGs.

Journal of Uncertainty in Mathematics Science, 2014

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In this paper, we introduce some types of NF graphs and operations. Also, we define the partial NF subgraph, spanning NF subgraph, strong degree of the vertex, total strong degree of the vertex and its properties are included.

Mathematics, 2019

The picture fuzzy set is an efficient mathematical model to deal with uncertain real life problems, in which a intuitionistic fuzzy set may fail to reveal satisfactory results. Picture fuzzy set is an extension of the classical fuzzy set and intuitionistic fuzzy set. It can work very efficiently in uncertain scenarios which involve more answers to these type: yes, no, abstain and refusal. In this paper, we introduce the idea of the picture fuzzy graph based on the picture fuzzy relation. Some types of picture fuzzy graph such as a regular picture fuzzy graph, strong picture fuzzy graph, complete picture fuzzy graph, and complement picture fuzzy graph are introduced and some properties are also described. The idea of an isomorphic picture fuzzy graph is also introduced in this paper. We also define six operations such as Cartesian product, composition, join, direct product, lexicographic and strong product on picture fuzzy graph. Finally, we describe the utility of the picture fuzzy ...