A Degree-Based Exponential Fuzzy Graph for Pollution Impact Analysis

Abstract

Exponential Fuzzy Graphs (EFGs) are a new family of fuzzy graphs in which the vertices and edges’ membership functions exhibit exponential decay. An EFG, which is characterised as a pair G = (V; E); captures the uncertainty and degradation of influence in complex systems by giving each edge (r; w) 2 E a membership value #E(r; w) = αE(r; w) · e-λαE(r;w); and each vertex v 2 V a membership value #V (w) = αV (w) · e-λαV (w); where λ > 0 is a decay parameter. To maintain consistency inside the fuzzy structure, the edge membership values are limited by #E(r; w) ≤ minf#V (r); #V (w)g. Some fundamental aspects such as vertex degree, order, and size are explored in relation to the idea of exponential fuzzy graphs (EFGs). EFGs are characterised as complete and complement. Some basic operations like semi-strong product, union, join, composition, and cartesian product are defined with graphical representing examples. The vertex degree of the generated vertices is examined for each operation, and associated theorems are demonstrated. The theoretical findings are shown using examples. The use of EFGs in modelling real-life imprecise and uncertain data is explained in an application related to environmental contamination