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2021 | Buch

Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs Kapitel lesen Erstes Kapitel lesen

Graphs for the Analysis of Bipolar Fuzzy Information

verfasst von: Prof. Muhammad Akram, Dr. Musavarah Sarwar, Prof. Wieslaw A. Dudek

Verlag: Springer Singapore

Buchreihe : Studies in Fuzziness and Soft Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

This monograph discusses decision making methods under bipolar fuzzy graphical models with the aim of overcoming the lack of mathematical approach towards bipolar information—positive and negative. It investigates the properties of bipolar fuzzy graphs, their distance functions, and concept of their isomorphism. It presents certain notions, including irregular bipolar fuzzy graphs, domination in bipolar fuzzy graphs, bipolar fuzzy circuits, energy in bipolar fuzzy graphs, bipolar single-valued neutrosophic competition graphs, and bipolar neutrosophic graph structures. This book also presents the applications of mentioned concepts to real-world problems in areas of product manufacturing, international relations, psychology, global terrorism and more, making it valuable for researchers, computer scientists, social scientists and alike.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Bipolar Fuzzy Sets and Bipolar Fuzzy Graphs
Abstract
In this chapter, we first review the notion of bipolar fuzzy sets and present several basic concepts concerning bipolar fuzzy graphs and bipolar fuzzy digraphs. We discuss different methods of construction of bipolar fuzzy graphs and their isomorphism properties. We describe certain types of bipolar fuzzy graphs, bipolar fuzzy walk, bipolar fuzzy bridge, strength of connectedness, weak and strong bipolar fuzzy edges. We establish the relations on bipolar fuzzy graphs, complement of bipolar fuzzy graphs, and crisp graphs with different operations, \(\alpha -\)cuts and \((\alpha ,\beta )-\)cuts. We also study certain operations and properties of complex bipolar fuzzy graphs. Moreover, with the help of composition of bipolar fuzzy relations, connectivity, and weighted matrices, we study the importance of bipolar fuzzy digraphs with a number of real-world problems. This chapter is basically adapted from [1‐3, 45, 46, 51].
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Chapter 2. Distance Measures in Bipolar Fuzzy Graphs
Abstract
In this chapter, we discuss the notion of distance in bipolar fuzzy graphs and present certain properties concerning distance functions in complete bipolar fuzzy graphs, complete bipartite bipolar fuzzy graphs, and products of bipolar fuzzy graphs.
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Chapter 3. Special Types of Bipolar Fuzzy Graphs
Abstract
In this chapter, we discuss the concept of irregularity in bipolar fuzzy graphs and present isomorphism properties of regular, m-totally regular, neighborly irregular, totally irregular, highly irregular, and neighborly totally irregular bipolar fuzzy graphs. We present certain characterizations under which, regular and totally regular bipolar fuzzy graphs, and highly irregular and neighborly irregular bipolar fuzzy graphs are equivalent. We discuss certain formulae of order and size of \(m-\)totally regular bipolar fuzzy graphs. We study the concept of bipolar fuzzy line graphs, and establish a necessary and sufficient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph.
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Chapter 4. Bipolar Fuzzy Competition Graphs
Abstract
In this chapter, we discuss the concept of bipolar fuzzy competition graphs and present several notions concerning bipolar fuzzy out neighborhoods, bipolar fuzzy in neighborhoods, bipolar fuzzy open neighborhood graphs, bipolar fuzzy closed neighborhood graphs, bipolar fuzzy \(\textit{\textbf{k}}-\)competition graphs, and underlying bipolar fuzzy graphs. We describe various methods for the construction of bipolar fuzzy competition graphs of certain products of bipolar fuzzy digraphs. Using various constraints, we study the relations of bipolar fuzzy \([\textit{\textbf{k}}]-\)competition graphs, bipolar fuzzy \((\textit{\textbf{k}})-\)competition graphs, and underlying bipolar fuzzy graphs. We elaborate certain algorithms to compute the strength of competition with a number of real-world applications in different fields including food webs, business marketing, politics, wireless communication networks, and social networking.
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Chapter 5. Bipolar Fuzzy Planar Graphs
Abstract
In this chapter, we study the concepts of bipolar fuzzy multisets, bipolar fuzzy multigraphs, strong and complete bipolar fuzzy multigraphs, bipolar fuzzy planar graphs, and bipolar fuzzy dual graphs. We discuss different types of bipolar fuzzy edges, intersection value and planarity value of bipolar fuzzy graphs, strong and weak bipolar fuzzy faces, and the relation of planarity and duality in bipolar fuzzy graphs. We elaborate various properties of bipolar fuzzy bridges, bipolar fuzzy cut vertices, bipolar fuzzy blocks, bipolar fuzzy cycles, and bipolar fuzzy trees in terms of level graphs. We describe the importance of bipolar fuzzy planar graphs with a number of real-world applications in road networks and electrical connections. The main results of this chapter are from [6, 7].
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Chapter 6. Domination in Bipolar Fuzzy Graphs
Abstract
In this chapter, we discuss different concepts of dominating, total dominating, equitable dominating, total equitable dominating, and independent and equitable independent sets in bipolar fuzzy graphs. We present certain properties concerning the relationship of domination, total domination and independence numbers in complete bipolar fuzzy graphs, complete bipartite bipolar fuzzy graphs, and products of bipolar fuzzy graphs. We study the notions of private neighbor, lower and upper irredundant sets, and irredundance number in relation to independence and dominating numbers of bipolar fuzzy graphs. We describe the importance of domination in bipolar fuzzy graphs with a number of real-world applications in facility location problem, finding the set of representatives, and transmission tower location problem. The main discussion of this chapter is from [5, 14].
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Chapter 7. Bipolar Fuzzy Circuits
Abstract
In this chapter, we discuss the notions of bipolar fuzzy rank function, bipolar fuzzy vector spaces, bipolar fuzzy basis, bipolar fuzzy matroids, bipolar fuzzy circuits and, describe certain characterizations concerning the linear independence of linear bipolar fuzzy matroids, uniform bipolar fuzzy matroids, partition bipolar fuzzy matroids and cycle bipolar fuzzy matroids. We establish the relation of crisp matroids and bipolar fuzzy matroids using \((\alpha ,\beta )-\)cuts. We study the concepts of closure of bipolar fuzzy matroids, \(\mathcal {M}-\)induced matroid sequence, fundamental sequence, circuit rectangles, and put special emphasis on bipolar fuzzy circuits. We also introduce the concepts of matroids and circuits in a soft environment and a bipolar fuzzy soft environment. We present certain applications of bipolar fuzzy matroids and bipolar fuzzy soft matroids in decision support systems and network analysis. The main results of this chapter are from [25].
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Chapter 8. Energy of Bipolar Fuzzy Graphs
Abstract
In this chapter, we study the concept of energy of bipolar fuzzy graphs and present certain formulae, lower and upper bounds of Laplacian energy, signless Laplacian energy, dominating energy, out-dominating energy, double dominating energy, and double out-dominating energy of bipolar fuzzy graphs and bipolar fuzzy digraphs. We elaborate the presented concept of energy and its extensions with numerical and graphical examples. Using bipolar fuzzy preference relations, we present multi-criteria decision-making models based on the energy of bipolar fuzzy graphs in business partnerships and smooth communication problems. The main results of this chapter are from [3, 4, 14].
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Chapter 9. Bipolar Neutrosophic Competition Graphs
Abstract
In this chapter, we present a concise review of bipolar neutrosophic sets and apply this concept to graphs and digraphs. We discuss the notion of bipolar neutrosophic competition graphs and present certain characterizations of bipolar neutrosophic out-neighborhoods, bipolar neutrosophic in-neighborhoods, bipolar neutrosophic open neighborhood graphs, bipolar neutrosophic closed neighborhood graphs, bipolar neutrosophic p-competition graphs, m-step bipolar neutrosophic competition graphs, and strong preys and strong independent predator–prey relations.
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Chapter 10. Bipolar Neutrosophic Graph Structures
Abstract
In this chapter, we apply the powerful technique of bipolar neutrosophic set to graph structures and present a framework of bipolar neutrosophic graph structures with certain operations. We discuss the notions of \(B_k-\)edges, strong and complete bipolar fuzzy graph structures, and bipolar neutrosophic subgraph structures. Using \(\varphi \)-complement, we describe certain relations and isomorphism properties of self-complementary, totally self-complementary, and totally strong self-complementary bipolar neutrosophic graph structures. We study the importance of bipolar neutrosophic graph structures with a number of real-world applications in international relations, psychology, and global terrorism. This chapter is basically due to [4, 14].
Muhammad Akram, Musavarah Sarwar, Wieslaw A. Dudek
Backmatter
Metadaten
Titel
Graphs for the Analysis of Bipolar Fuzzy Information
verfasst von
Prof. Muhammad Akram
Dr. Musavarah Sarwar
Prof. Wieslaw A. Dudek
Copyright-Jahr
2021
Verlag
Springer Singapore
Electronic ISBN
978-981-15-8756-6
Print ISBN
978-981-15-8755-9
DOI
https://doi.org/10.1007/978-981-15-8756-6