## Abstract

A complex fuzzy set (CFS) is described by a complex-valued truth membership function,

which is a combination of a standard true membership function plus a phase term. In this paper, we

extend the idea of a fuzzy graph (FG) to a complex fuzzy graph (CFG). The CFS complexity arises

from the variety of values that its membership function can attain. In contrast to a standard fuzzy

membership function, its range is expanded to the complex plane’s unit circle rather than [0,1]. As a

result, the CFS provides a mathematical structure for representing membership in a set in terms of

complex numbers. In recent times, a mathematical technique has been a popular way to combine

several features. Using the preceding mathematical technique, we introduce strong approaches

that are properties of CFG. We define the order and size of CFG. We discuss the degree of vertex

and the total degree of vertex of CFG. We describe basic operations, including union, join, and the

complement of CFG. We show new maximal product and symmetric difference operations on CFG,

along with examples and theorems that go along with them. Lastly, at the base of a complex fuzzy

graph, we show the application that would be important for measuring the symmetry or asymmetry

of acquaintanceship levels of social disease: COVID-19.

which is a combination of a standard true membership function plus a phase term. In this paper, we

extend the idea of a fuzzy graph (FG) to a complex fuzzy graph (CFG). The CFS complexity arises

from the variety of values that its membership function can attain. In contrast to a standard fuzzy

membership function, its range is expanded to the complex plane’s unit circle rather than [0,1]. As a

result, the CFS provides a mathematical structure for representing membership in a set in terms of

complex numbers. In recent times, a mathematical technique has been a popular way to combine

several features. Using the preceding mathematical technique, we introduce strong approaches

that are properties of CFG. We define the order and size of CFG. We discuss the degree of vertex

and the total degree of vertex of CFG. We describe basic operations, including union, join, and the

complement of CFG. We show new maximal product and symmetric difference operations on CFG,

along with examples and theorems that go along with them. Lastly, at the base of a complex fuzzy

graph, we show the application that would be important for measuring the symmetry or asymmetry

of acquaintanceship levels of social disease: COVID-19.

Original language | English |
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Article number | 1126 |

Number of pages | 24 |

Journal | Symmetry |

Volume | 14 |

Publication status | Published - 30 May 2022 |

## Keywords

- CFG
- vertex degree and total vertex degree
- maximal product
- symmetric difference
- complement
- application
- order
- size
- union
- join