The complex anti-fuzzy set (CAFS) is an extension of the traditional anti-fuzzy set with a wider range for membership function beyond real numbers to complex numbers with unit disc aims to address the uncertainty of data. The complex anti-fuzzy set is more significant because it provides two dimensional information and versatile representation of vagueness and ambiguity of data. In terms of the characteristics of complex anti-fuzzy sets, we proposed the concept of $(ϵ,\delta )$-CAFSs that offer a more comprehensive representation of the uncertainty of data than CAFSs by considering both the magnitude and phase of the membership functions and explain the $\left(ϵ,\delta \right)$-complex anti fuzzy subgroups (CAFS) in the context of CAFSs. Moreover, we showed that everyCAFSGis a $(ϵ,\delta )$-CAFSG. Also, we used this approach to define $(ϵ,\delta )$-complex anti-fuzzy(CAF) cosets and $(ϵ,\delta )$-CAF normal subgroups of a certain group as well as to investigate some of their algebraic properties. We elaborated the $(ϵ,\delta )$-CAFSG of the classical quotient group and demonstrated that the set of all $(ϵ,\delta )$-CAF cosets of such a particular CAFs normal subgroup formed a group. Furthermore, the index of $\left(ϵ,\delta \right)$-CAFSG was demonstrated and $(ϵ,\delta )$-complex anti fuzzification of Lagrange theorem corresponding to the Lagrange theorem of classical group theory was briefly examined.

Algebraic homomorphisms are essential mathematical structures that sustain operations across algebraic systems such as groups, rings, and fields. These mappings not only preserve the validity of algebraic operations but also make it easier to investigate structural similarities and equivalences across distinct algebraic entities. In this article, we establish the group isomorphism under the complex intuitionistic fuzzy set, an extended form of the complex fuzzy set that adds the complex degree of non-membership functions, which plays a significant role in the decision-making process. The complex algebraic structure provides effective tools for understanding complex phenomena. We discuss the more intricate features of homomorphism and isomorphism in the framework of a complex intuitionistic fuzzy set. In addition, we introduce the complex intuitionistic fuzzy normal subgroups. We establish the relationship between two complex intuitionistic fuzzy subgroups and analyze of complex intuitionistic fuzzy isomorphisms among these subgroups, proving the important theorems. Furthermore, we establish examples to explore the concept of complex intuitionistic fuzzy subgroups.

Graph labeling is an increasingly popular problem in graph theory. A mapping converts a collection of graph components into a set of integers known as labels. Graph labeling techniques typically label edges with positive integers, vertices with even numbers, and edge weights with consecutive numbers, known as edge irregular reflexive total labeling. This is achieved by utilizing the reflexive edge irregularity strength of the graphical structure. The edge calculates the exact values of the reflexive edge irregularity strength irregular reflexive labeling for the $m^t$-graph of cycle graph $m{C}_n$ on $t=1$ with $n\ge 3$ and $m\ge 4$. The maximum number of assignments assigned to each individual in a communication network, as well as providing a secure communication channel to ensure the unique identification of each employee, are potential applications for this problem.

A complex bipolar fuzzy set ( $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}$) is an extension of a complex fuzzy set and a $\mathbb{B}\mathbb{F}$ set with a wide range of values. A $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}$ is differentiated from a $\mathbb{B}\mathbb{F}$ set by the incorporation of negative and positive membership functions to the unit circle in the complex plane, which empowers one to handle the vagueness more effectively. We aim to generalize the notions of $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}s$ by proposing a general algebraic structure to dealing with complex bipolar ( $\mathbb{C}\mathbb{B}$) fuzzy data by integrating the idea of $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}s$ and subrings. The structure of $\mathbb{C}\mathbb{B}$ fuzzy subrings, such as the $\mathbb{C}\mathbb{B}$ fuzzy ring isomorphism, $\mathbb{C}\mathbb{B}$ fuzzy quotient ring, and $\mathbb{C}\mathbb{B}$ fuzzy ring homomorphism, are examined in this paper. We develop the ( $\delta ,\alpha ;\sigma ,\beta$)-cut of a $\mathbb{C}\mathbb{B}\mathbb{F}\mathbb{S}$ and explore its algebraic interpretations. Additionally, we describe the $\mathbb{C}\mathbb{B}$ fuzzy support set and demonstrate some significant characteristics associated with this idea. Furthermore, we use the concept of a naturally occurring complex ring homomorphism to describe a $\mathbb{C}\mathbb{B}$ fuzzy homomorphism. Additionally, we prove a $\mathbb{C}\mathbb{B}$ fuzzy homomorphism between the $\mathbb{C}\mathbb{B}$ fuzzy subrings of the ring and the $\mathbb{C}\mathbb{B}$ fuzzy subring of the complex quotient ring. We demonstrate a strong connection among two $\mathbb{C}\mathbb{B}$ fuzzy subrings of complex quotient rings under a specific $\mathbb{C}\mathbb{B}$ fuzzy surjective homomorphism. We construct a complex fuzzy isomorphism between both associated $\mathbb{C}\mathbb{B}$ fuzzy subrings. Furthermore, we introduce three basic results of the $\mathbb{C}\mathbb{B}$ fuzzy isomorphism to explain the relationship between two $\mathbb{C}\mathbb{B}$ fuzzy subrings. Finally, we use a complex bipolar fuzzy subring in decision making.