On Grey Graphs and their Applications in Optimization

In this research‎, ‎we use averages and relative measures of interval grey numbers to introduce grey vertices, ‎grey edges‎, ‎and grey graphs (graphs are based on interval grey numbers)‎. ‎To do so‎, ‎we design a grey graph based on a graph (as the underlying graph)‎. ‎Also‎, ‎we find a relation between grey vertices and grey edges of a grey graph‎. ‎The primary method used in this research is based on linear inequalities related to grey vertices and grey edges‎. ‎We find some necessary and sufficient conditions on the grey vertex (as (non-)discrete grey vertices) connectivity of grey graphs based on interval grey numbers and linear inequality systems}.‎The paper includes implications for the development of(non-)weighted graphs‎, ‎and the modeling of uncertainty problems by grey vertices‎, ‎grey edges‎, ‎and their relations in a grey model as a grey graph‎. ‎As a weighted graph‎, ‎a fuzzy graph is a vital graph that has some applications in the real world‎, ‎but with changes in conditions‎, ‎it loses its efficiency‎. ‎On the other hand‎, ‎the efficiency of a grey graph is stable under changes in the conditions‎. ‎So‎, ‎grey graphs cover the weaknesses of fuzzy graphs‎. ‎The new conception of grey graphs based on grey numbers is introduced in this study‎. ‎We propose an optimization method that can be applied for grey numbers in an extension of graphs‎, ‎and apply it for gray numbers in the real world, especially for optimization problems and via gray graphs.