(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let G(V,E) represent a simple, finite, and connected graph with |V| = p and |E| = q. A bijection f: V(G) {1,2,3, …, |V(G)|} is defined as an edge antimagic labeling of G if the edge weights w(uv) = f(u) + f(v), where uv E(G), are all distinct. This labeling induces a lea coloring of G, where each edge is assigned a color based on its weight w(e). The graph G is said to have a local (a,d)-edge antimagic coloring if the edge colors form an arithmetic sequence with an initial term a and a common difference d. The edge weights then belong to an arithmetic progression {a, a + d, a + 2d, …, a + (k - 1)d}, where a, d ≥ 1 are integers, and k represents the number of distinct colors used. The minimum number of colors required for G to admit a local (a,d)-edge antimagic coloring is called the local (a,d)-edge antimagic chromatic number, denoted by χ_le(a,d)(G). This coloring is referred to as lea(a,d) coloring. In this study, we establish both the lower and upper bounds of χ_le(a,d)(G) and identify the exact values for specific graph classes. Additionally, we demonstrate the practical application of local (a,d)-edge antimagic coloring by employing it in the analysis of the Spatio-Temporal Graph Neural Network (STGNN) model to support autonomous Controlled Environment Agriculture (CEA) in forecasting NPK concentration trends for companion plantations over multiple time steps.