Linear Diophantine Fuzzy Graph Structures: Theoretical Foundations and Applications in Road Crime Detection

Graph structures (GSs) have appeared as a robust mathematical framework for modelling and resolving complex combinatorial problems across diverse realms.At the same time, the linear Diophantine fuzzy set (LDFS) is a noteworthy expansion of the conventional concepts of the fuzzy set (FS), intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PFS), and q-Rung orthopair fuzzy set (q-ROFS).The LDFS framework introduces a flexible parameterization strategy that independently relaxes membership and non-membership restraints through reference parameters, thereby attaining enhanced expressiveness in apprehending ambiguous real-world phenomena.In this paper, a novel concept of linear Diophantine fuzzy graph structure (LDFGS) is introduced as a generalization of intuitionistic fuzzy graph structure (IFGS) and linear Diophantine fuzzy graph (LDFG) to GSs.Several cardinal fundamental notions in LDFGSs, including i -edge, i -path, strength of i -path, i -strength of connectedness, i -degree of a vertex, degree of a vertex, total i -degree of a vertex, and the total degree of a vertex in an LDFGS are discussed.Additionally, i -size of an LDFGS, the size of an LDFGS, and the order of an LDFGS are studied.Meanwhile, the ideas of the maximal product of two LDFGSs, strong LDFGS, degree, and i -degree of the maximal product are introduced with several concrete illustrations.To empirically validate the efficacy and practical utility of the proposed LDFGS framework, this study presents a case study analyzing road crime patterns across heterogeneous urban regions in Sindh province, Pakistan.