Studying the applications of graph theory in functional materials and manufacturing through divisor 3-equitable labeling of graphs.

In the case of "modeling a manufacturing program represented through the product state concept, the nodes are state characteristics, and the lines are the (inter-) relations between the state characteristics. These can be within the product state, or between neighboring states, or system-wide. Once the graph of the model of a manufacturing program has been obtained, in this case, the visualization model, the resultant graph can be considered complex due to its size and the multiple relations among its nodes". Considering, that "the more nodes and lines a graph has, the more complex the graph is". "Divisor 3-equitable labeling (D3EL) is a bijection f: V(G) → { 1,2, ..., n; } that induces a function f′: E(G)→{0, 1, 2} defined by for each line e = xy, (i) f'(e) = 1 if f(x)|f(y) or f(y)|f(x), (ii) f'(e) = 2 if f(x)/f(y) = 2 or f(y)/f(e) = 2, and (iii) f'(e) = 0 otherwise such that |ef′(i)−ef′(j)| ≤ 1 for all 0 ≤ i, j ≤ 2, where ef′(i) is the number of lines having labels i under f'. A graph that admits D3EL is called a divisor 3-equitable graph (D3EG)". This paper shows the existence and non-existence of D3EL of certain graphs, besides highlighting the applications of graph theory in functional materials and manufacturing. [ABSTRACT FROM AUTHOR]