© 2020

Hybrid Soft Computing Models Applied to Graph Theory


  • Explains how construct and use rough fuzzy digraphs

  • Describes applications to different sets of data and complex problems

  • Describes relevant extensions, such as soft rough neutrosophic graphs


Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 380)

Table of contents

  1. Front Matter
    Pages i-xxv
  2. Muhammad Akram, Fariha Zafar
    Pages 1-77
  3. Muhammad Akram, Fariha Zafar
    Pages 79-128
  4. Muhammad Akram, Fariha Zafar
    Pages 129-209
  5. Muhammad Akram, Fariha Zafar
    Pages 211-259
  6. Muhammad Akram, Fariha Zafar
    Pages 261-322
  7. Muhammad Akram, Fariha Zafar
    Pages 323-352
  8. Muhammad Akram, Fariha Zafar
    Pages 353-370
  9. Muhammad Akram, Fariha Zafar
    Pages 371-419
  10. Back Matter
    Pages 421-434

About this book


This book describes a set of hybrid fuzzy models showing how to use them to deal with incomplete and/or vague information in different kind of decision-making problems. Based on the authors’ research, it offers a concise introduction to important models, ranging from rough fuzzy digraphs and intuitionistic fuzzy rough models to bipolar fuzzy soft graphs and neutrosophic graphs, explaining how to construct them. For each method, applications to different multi-attribute, multi-criteria decision-making problems, are presented and discussed. The book, which addresses computer scientists, mathematicians, and social scientists, is intended as concise yet complete guide to basic tools for constructing hybrid intelligent models for dealing with some interesting real-world problems. It is also expected to stimulate readers’ creativity thus offering a source of inspiration for future research.


Upper Approximate Fuzzy Digraphs Underlying Crisp Digraph Isomorphic Rough Fuzzy Digraphs Irregular Rough Fuzzy Digraphs Regular Fuzzy Soft Graphs Irregular Fuzzy Soft Graphs Strongest Directed Path Connected Fuzzy Subgraphs Fuzzy Directed Bridge Rough Fuzzy Directed Bridge Rough Fuzzy Directed Block Neutral Rough Fuzzy Digraphs Soft Rough Fuzzy Digraphs Bipolar Fuzzy Soft Graphs Traffic Densities of the Paths Multi-criteria Selection of Objects Suitable Career Selection Problem Detection of Bipolar Disorder Soft Rough Neutrosophic Graphs Soft Rough Neutrosophic Influence Graphs

Authors and affiliations

  1. 1.Department of MathematicsUniversity of the PunjabLahorePakistan
  2. 2.Department of MathematicsUniversity of the PunjabLahorePakistan

About the authors

Dr. Muhammad Akram received MSc degrees in Mathematics and Computer Science, MPhil in Computational Mathematics and PhD in Fuzzy Mathematics. He is currently a Professor in the Department of Mathematics at the University of the Punjab, Lahore, Pakistan, where he has been serving as a PhD supervisor of more than 10 students. Dr. Akram’s research interests include numerical solutions of parabolic PDEs, fuzzy graphs, fuzzy algebras, and fuzzy decision support systems. He has published 7 monographs and 300 research articles in international peer-reviewed journals. He has served as editorial board member of 10 international academic journals and as reviewer of 122 International journals, including Mathematical Reviews and Zentralblatt MATH.

Dr. Fariha Zafar received her PhD degree in Mathematics and MPhil degree in Mathematics from the University of the Punjab, Lahore. She has introduced the notions of Soft Trees and Fuzzy Soft Trees during her MPhil research work. She has published 10 research articles in top-ranked international journals. Her research interests include fuzzy graphs, soft set theory, rough set theory and decision-making.

Bibliographic information

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