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A Novel Description of Perfectly Regular Fuzzy Graphs with Application in Psychological Sciences
Saeed Kosari, Xiaolong Shi, Janusz Kacprzyk, Zhihua Chen and Hossein Rashmanlou
Today, fuzzy graphs (FGs) have made significant progress in other sciences and help researchers to always make the best choice in complex problems. A type of FG that is widely used in medical and psychological sciences is vague graph (VG). VGs play an important role in various fields such as computer science, psychology, medicine, and political sciences and are used to find effective people in an organization or social institution. Regularity is very important in the VG theory because it can help a lot in identifying people with high ability in an institute or social group. In addition to regularity, the degree of nodes and edges also play an effective role in this relationship and can evaluate the ability and knowledge of people. Therefore, in this paper, first perfectly regular vague graph (PRVG) is introduced and investigated the regularity of vertices. The totally accurate communication between all connected vertices is described by defining completely open neighborhood degree (COND) and completely closed neighborhood degree (CCND) of vertices and edges in a VG. In the following, ((𝑠1, 𝑠2),𝑚, (𝑡1, 𝑡2))-regular vague graph and totally ((𝑠1, 𝑠2),𝑚, (𝑡1, 𝑡2))-regular vague graph are introduced and a necessary and sufficient condition under which they are equivalent is presented. Psychologists play a significant role in creating a healthy society in terms of physical and mental health. They help people in the society to better understand their emotions and behaviors and manage them to live a better life. So in the final part, an application of COND and CCND in psychological sciences has been introduced.
Keywords: Vague set, total degree, perfect regular, open neighborhood, close neighborhood