A Study on Vague Graph Structures with an Application

Zheng Kou,Saber Omidi,Maryam Akhoundi,Xiang Chen,Meraj Ali Khan

doi:10.1155/2022/3182116

Abstract

Fuzzy graph (FG) models take on the presence being ubiquitous in environmental and fabricated structures by human, specifically the vibrant processes in physical, biological, and social systems. Owing to the unpredictable and indiscriminate data which are intrinsic in real-life, problems being often ambiguous, so it is very challenging for an expert to exemplify those problems through applying an FG. Vague graph structure (VGS), belonging to FGs family, has good capabilities when facing with problems that cannot be expressed by FGs. VGSs have a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. Therefore, in this paper, we apply the concept of vague sets (VSs) to GS. We define certain notions, VGS, strong vague graph structure (SVGS), and vague β i -cycle and describe these notions by several examples. Likewise, we introduce ψ -complement, self-complement (SC), strong self-complement (SSC), and totally strong self-complement (TSSC) in VGS and investigate some of their properties. Finally, an application of VGS is presented.

Highlights

Introduction eFuzzy graph (FG) concept serves as one of the most dominant and extensively employed tools for multiple real-word problem representations, modeling, and analysis
Many of the issues and phenomena around us are associated with complexities and ambiguities that make it difficult to express certainty. ese difficulties were alleviated by the introduction of fuzzy sets by Zadeh [1]
A GS facilitates studying the different relations and the equivalent edges simultaneously. e FS focuses on the membership degree of an object in a particular set. e existence of a single degree for a true membership could not resolve the ambiguity on uncertain issues, so the need for a degree of membership was felt

Summary

Preliminaries

A GS G∗ (V, R1, R2, . . . , Rk), consists of a nonempty set V together with relations R1, R2, . . . , Rk on V, which are mutually disjoint so that each Ri is irreflexive and symmetric. Rk), consists of a nonempty set V together with relations R1, R2, . A fuzzy subset σ on a set X is a map σ: X ⟶ [0, 1]. A fuzzy binary relation on X is a fuzzy subset σ on X × X. We mean a fuzzy binary relation given by σ: X × X ⟶ [0, 1]. Λk) is a λi-cycle if and only if (Supp(]), Supp(λ1), Supp(λ2), . A VS A is a pair (tA, fA) on set X where tA and fA are taken as real valued functions which can be defined on V ⟶ [0, 1] so that tA(a) + fA(b) ≤ 1, ∀ a ∈ X.

New Concepts in Vague Graph Structure

Application

Conclusion