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Some Root Level Modifications in Interval Valued Fuzzy Graphs and Their Generalizations Including Neutrosophic Graphs

1
Department of Mathematics and Statistics, International Islamic University Islamabad Pakistan, Islamabad Capital Territory 44000, Pakistan
2
School of Mathematics, Thapar Institute of Engineering & Technology, Deemed University, Patiala 147004, Punjab, India
3
Department of Mathematics, Yazd University, Yazd 89195-741, Iran
4
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 7616913439, Iran
5
Laboratory of Information Processing, University Hassan II, Casablanca 20000, Morocco
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(1), 72; https://doi.org/10.3390/math7010072
Received: 22 November 2018 / Revised: 26 December 2018 / Accepted: 3 January 2019 / Published: 10 January 2019
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Abstract

Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of some counterexamples. To achieve this, first, we aim to improve the existing definition for interval-valued FG, interval-valued intuitionistic FG and their complements, as these existing definitions are not well-defined; i.e., one can obtain some senseless intervals using the existing definitions. The limitations of the existing definitions and the validity of the new definitions are supported with some examples. It is also observed that the notion of a single-valued neutrosophic graph (SVNG) is not well-defined either. The consequences of the existing definition of SVNG are discussed with the help of examples. A new definition of SVNG is developed, and its improvement is demonstrated with some examples. The definition of an interval-valued neutrosophic graph is also modified due to the shortcomings in the current definition, and the validity of the new definition is proved. An application of proposed work is illustrated through a decision-making problem under the framework of SVNG, and its performance is compared with existing work. View Full-Text
Keywords: interval-valued fuzzy graph; intuitionistic fuzzy graph; interval-valued intuitionistic fuzzy graph; single-valued neutrosophic graph; interval-valued neutrosophic graph; complement interval-valued fuzzy graph; intuitionistic fuzzy graph; interval-valued intuitionistic fuzzy graph; single-valued neutrosophic graph; interval-valued neutrosophic graph; complement
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Jan, N.; Ullah, K.; Mahmood, T.; Garg, H.; Davvaz, B.; Saeid, A.B.; Broumi, S. Some Root Level Modifications in Interval Valued Fuzzy Graphs and Their Generalizations Including Neutrosophic Graphs. Mathematics 2019, 7, 72.

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