Spatio-Temporal Graph Neural Network for Time Series Forecasting: Local Antimagic Coloring Based Companion Farming

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 + 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 χle(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 χle(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.