# Maximal Product and Symmetric Difference of Complex Fuzzy Graph with Application

Muhammad Shoaib, Waqas Mahmood, Qin Xin, Fairouz Tchier

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

## 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.
Original language English 1126 24 Symmetry 14 Published - 30 May 2022

## Keywords

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

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