Government policymakers face a problem about illustrating and evaluating strategies and plan to design active communities which need to be assessed for studying the collaboration between ministries. This work aims at proposing a complex hesitant fuzzy graph that represents the collaboration and its factors between ministries. The present graph can transform complex information using two variables, which modify the consistency and performance of hesitant fuzzy graphs in representing information and the phases. The additional variable represents the phase/level/factors of information that may affect the meaning of the information. In addition, three types of products are defined between two complex hesitant fuzzy graphs. The vertex degrees of these new graphs are investigated and proved. Finally, the proposed graphs are illustrated by an application problem of evaluating the cooperation between ministries and inside one ministry to enhance the tool in representing two variables.

Correctly determining a company's market worth during an entire year or a certain period presents a difficulty to decision-makers. In the case of the merger of companies, the need performs heavier when both the companies' owners are attracted to establishing a fair price at the optimal time to merge. The effectiveness of representing, connecting and manipulating both uncertainty and periodicity information becomes highly required. Hence, study and nhance some properties and conditions of the algebraic structure of complex hesitant fuzzy graphs. Therefore, the degree of composition between two complex hesitant fuzzy graphs is proposed. Also, the formal definitions of union, joint and complement are presented to be covered in the realm of complex hesitant fuzzy graphs. A real-life application is illustrated to show the relation between vertices and edges in the form of complex hesitant fuzzy graphs.

Graph invariants are the properties of graphs that do not change under graph isomorphisms, the independent set decision problem, vertex covering problem, and matching number problem are known to be NP-Hard, and hence it is not believed that there are efficient algorithms for solving them. In this paper, the graph invariants matching number, vertex covering number, and independence number for the zero-divisor graph over the rings $Z_{p^k}$ and $Z_{p^k{q}^r}$ are determined in terms of the sets $S_{p^i}$ and $S_{p^i{q}^j}$ respectively. Accordingly, a formula in terms of $p,q,k,$ and r, with $n=p^k$ , $n=p^k{q}^r$ is provided.

A complex Fermatean fuzzy set provides a detailed framework for representing a specific type of information and has been effectively applied to decision-making problems. This study introduces complex Fermatean fuzzy subgroups (CFFSGs), an extension of Fermatean fuzzy subgroups, and complex Pythagorean fuzzy subgroups. The key innovation of CFFSGs lies in their capacity to represent two variables within their algebraic structure, surpassing the capabilities of traditional Fermatean fuzzy subgroups. The research establishes the formal definition and properties of CFFSGs, adapting them to a complex framework that incorporates amplitude and phase components. Additionally, the concepts of complex Fermatean fuzzy cosets and complex Fer-matean fuzzy normal subgroups are introduced. The study also investigates and examines the characteristics of homomorphisms between complex Fermatean fuzzy subgroups.

In this paper, we continue to discuss the concept of harmonized fuzzy subgroups. We present the harmonized fuzzy coset and harmonized fuzzy normal subgroup and their properties. We also study the effect of group homomorphism on harmonized fuzzy subgroups. Finally, we define and study the cartesian product of two harmonized fuzzy subgroups.