A Theoretical Development of Linear Diophantine Fuzzy Graph Structures

Saba Ayub ¹ ( Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. )
Muhammad Shabir ² ( Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. )

Submited date : 2024-07-19 Accepted date : 2024-09-27

Keywords: Linear diophantine fuzzy sets, Graph structure, Maximal product, Degree of a vertex, Total degree of a vertex.,

Abstract :

Graph structure (GS) is an advancement of the graph concept which effectively represents intricate situations with various connections, frequently used in computer science and mathematics to illustrate relationships among objects and extensively researched in fuzzy sets (FS), intuitionistic fuzzy set (IFS), pythagorean fuzzy set (PFS) and q-rung orthopair fuzzy set (q-ROFS). Meanwhile, a linear Diophantine fuzzy set (LDFS) is a remarkable extension of the existing notions of a FS, IFS, PFS and q-ROFS by comporting reference parameters that removed all the limitations related to membership degree (MD) and non-membership degree (NMD). According to the best of our knowledge, there is a lack of elegantly proposed GS extension for LDFSs in the current literature. As a result, this research focuses on introducing first linear Diophantine fuzzy graph structure (LDFGS) concept which extends the existing notions of GS in various contexts of FSs. Several key concepts in LDFGSs are presented, such as ˘ρi-edge, ˘ρi-path, strength of ˘ρi-path, ˘ρi-strength of connectedness, ˘ρi-degree of a vertex, vertex degree, total ρ˘i-degree of a vertex, and total vertex degree in an LDFGS. In addition, we introduce the ˘ρi-size, size, and order of an LDFGS. Moreover, this article presents the ideas of the maximal product of two LDFGSs, strong LDFGS, degree and ˘ρi-degree of the maximal product, ˘ρi-regular and regular LDFGSs, along with examples for clarification. Certain significant results related to the proposed concepts also demonstrated with explanatory examples such as the maximal product of two strong LDFGSs is also a strong LDFGS, the maximal product of two connected LDFGSs is also a connected LDFGS but the maximal product of two regular LDFGS may not be a regular LDGS. Moreover, many interesting and alternative formulas for calculating ˘ρi-degrees of an LDFGS in various situations are proved with examples. LDFGSs are highly beneficial for solving numerous combinatorial problems involving multiple relations, and they surpass existing concepts of GSs within the FS context due to their flexibility in selecting MD and NMD alongside their reference parameters.

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