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 num- bers in the real world, especially for optimization problems and via gray graphs.