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

Abstract

Let $G(V,E)$ represent a simple, finite, and connected graph with $|V| = p$ and $|E| = q$. A bijection $f: V(G) \longrightarrow \{1,2,3, \dots, |V(G)|\}$ is defined as an edge antimagic labeling of $G$ if the edge weights $w(uv) = f(u) + f(v)$, where $uv \in E(G)$, 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 $\{a, a + d, a + 2d, \dots, a + (k - 1)d\}$, where $a, d \geq 1$ 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 $\chi_{le(a,d)}(G)$. This coloring is referred to as lea$(a,d)$ coloring. In this study, we establish both the lower and upper bounds of $\chi_{le(a,d)}(G)$ 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.