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.
Highlights
The framework of the fuzzy set (FS) was introduced by Zadeh [1] in 1965 as a generalization of crisp sets, which describe the membership of an object to a set by assigning a membership grade from [0,1]
The results obtained using the current definition of interval-valued FG (IVFG) leads us to obtainobtain some some undefined intervals, and we propose a new definition for the undefined intervals, and we propose a new definition for the complement complement of
As the interval-valued neutrosophic graph (IVNG) is a generalization of IVFG and interval-valued IFG (IVIFG), its complement defined in [31] is not welldefined; we propose a new definition for the complement of IVNG as follows
Summary
The framework of the fuzzy set (FS) was introduced by Zadeh [1] in 1965 as a generalization of crisp sets, which describe the membership of an object to a set by assigning a membership grade from [0,1]. The concept of SVNS was further generalized to an interval-valued neutrosophic set (IVNS) in Wang et al [7] where the membership, abstinence and non-membership grades are defined in terms of closed sub-intervals of the unit interval. The concept of SVNG was further generalized to the interval-valued neutrosophic graph (IVNG) in [48], where the membership, abstinence and non-membership grades are described by closed subintervals of the unit interval. The existing definition of a complement does not satisfy the property (G ) = G These facts are described briefly, and based on these defects, a new definition for SVNG and its complement have been developed and their validity is proved;. Existingwith definition of the complement of IVNG is not well defined, which is described with the help of examples, and a new definition.
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