EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 5970 ISSN 1307-5543 – ejpam.com Published by New York Business Global Spatio-Temporal Graph Neural Network for Time Series Forecasting: Local Antimagic Coloring-Based Companion Farming A. I. Kristiana1,2, R. Izza3, I. H. Agustin1,4,∗, L. A. Monalisa2, I. L. Mursyidah1, R. I. Baihaki1, K. H. Agustina5, Dafik1,4 1 PUI-PT Combinatorics and Graph, CGANT, University of Jember, Jember, Indonesia 2 Department of Mathematics Education, University of Jember, Jember, Indonesia 3 Department of Mathematics Education, Cordoba Islamic University, Banyuwangi, Indonesia 4 Department of Mathematics, University of Jember, Jember Indonesia 5 Department of Postgraduate Mathematics Education , University of Jember, Jember, Indonesia Abstract. 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 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. 2020 Mathematics Subject Classifications: 05C15, 05C90, 68R01, 68R10 Key Words and Phrases: Local (a, d)-edge antimagic coloring, Spatio-Temporal Graph Neural Network (STGNN), NPK concentration forecasting ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.5970 Email addresses: ikahesti.fmipa@unej.ac.id (I. H. Agustin) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 2 of 18 1. Introduction In addition, a local edge antimagic coloring can also be defined for graphs [1]. A bijection f : V → {1, 2, 3, . . . , |V (G)|} is referred to as a lea labeling if any two adjacent edges e1 and e2 satisfy w(e1) ̸= w(e2), where for e = uv ∈ E(G), the weight of the edge is given by w(e) = f(u) + f(v). A proper edge coloring of G is induced by the lea labeling when all edges are assigned colors based on their weights w(e) [2]. The lea chromatic number, represented as χlea(G), denotes the minimum number of colors required for the proper edge coloring of G based on the lea labeling [3]. Several findings related to lea coloring in graphs are discussed in [4]. A graph G is said to have a local (a, d)-antimagic coloring if the set of its edge colors forms an arithmetic progression {a, a+d, a+2d, . . . , a+ (k − 1)d}, where a, d ≥ 1 are integers, and k is the number of distinct colors used in the proper edge coloring. This type of coloring is referred to as a lea(a, d) coloring. The local (a, d)-antimagic chromatic number, denoted as χle(a,d)(G), is defined as the minimum number of colors required to assign a local (a, d)-antimagic coloring to G. For further details, refer to [5]. The study of graph colorings has been an essential area of research in combinatorics, particularly due to its wide-ranging applications in network optimization, scheduling, and cryptographic systems [6, 7]. Several recent works have contributed significantly to ad- vancing this field. For example, Alfarisi et al. explored the graceful chromatic number of unicyclic graphs, establishing critical properties and applications in network models [8]. Similarly, Dafik et al. examined the non-isolated resolving numbers of special graphs, of- fering insights into their operational characteristics and implications for graph operations [9]. Building on these foundations, Septyory et al. (2021) focused on rainbow antimagic coloring of special graphs, showcasing its potential in diverse computational frameworks [10]. Further, Dafik et al. and Gembong et al. investigated edge domination numbers and distance domination numbers in comb product graphs, respectively, providing a deeper understanding of their structural and combinatorial properties [11]. As a subfield of graph coloring, local edge-antimagic (a, d)-coloring (lea(a, d) color- ing) has been the subject of numerous studies. Rajkumar and Nalliah [12] computed the lea(a, d)-chromatic number for wheel graphs. Agustin et al. (2023) further analyzed path, fan, friendship, and star graphs to determine their respective lea(a, d)-chromatic num- bers [13]. Almaidah et al. (2023) studied ladder, cycle, octopus, helm, triangular book, and tadpole graphs [14]. Putra et al. extended the analysis to centipede, lotus, caterpillar, double star, and double broom graphs [15]. Rifda et al. (2023) explored the application of lea(a, d) coloring on broom, book, firecracker, complete, and dragon graphs [16]. Despite the significant progress made in the study of lea(a, d) coloring across various graph classes, its application to real-world domains remains limited. To the best of our knowledge, no prior work has integrated the concept of lea(a, d) coloring into smart agri- culture, particularly within the framework of Spatio-Temporal Graph Neural Networks (STGNN). This research addresses that gap by proposing a novel integration of graph la- beling theory and deep learning for optimized sensor deployment and nutrient forecasting. Recent developments further support the feasibility of incorporating graph coloring A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 3 of 18 concepts into deep learning models for agricultural forecasting. Expanding the application of graph coloring, Kristiana et al. (2024) analyzed the b-coloring of graphs and its inte- gration into Spatio-Temporal Graph Neural Networks (STGNN) for multi-step time series forecasting [17]. Their study demonstrated how graph coloring techniques could optimize the monitoring of soil moisture and pH levels in companion farming systems [18]. These findings reinforce the significance of graph coloring not only in theoretical combinatorial problems but also in practical applications, particularly in agricultural and environmen- tal monitoring [19]. Collectively, these studies highlight the growing importance of graph coloring techniques in addressing both theoretical challenges and real-world problems in combinatorics and beyond [20]. This research is motivated by the need to bridge the gap between theoretical develop- ments in graph labeling and their practical applications in data-driven systems. Specifi- cally, this paper aims to derive sharper lower and upper bounds for the lea(a, d)-chromatic number, χle(a,d)(G), and to determine its exact value for selected classes of graphs. Beyond the theoretical contributions, this study demonstrates the applicability of lea(a, d) color- ing in real-world scenarios by integrating it into the analysis of a STGNN model. This integration is applied in the context of autonomous Controlled Environment Agriculture (CEA), where it supports multi-step time series forecasting of nitrogen (N), phosphorus (P), and potassium (K) concentrations in companion plantation systems. The proposed approach not only advances graph coloring theory but also enhances predictive accuracy and sensor efficiency in precision agriculture. 2. Method This research utilizes both analytical and experimental methodologies. In the analyt- ical section, we will apply a mathematical deductive approach to obtain some theorems, we use the the following some known lower bounds to obtain the new theorems. Observation 1. [1] χlea(G) ≥ χ(G), where χ(G) is a chromatic number of graph coloring G. Observation 2. [3] χle(a,d)(G) ≥ χlea(G) ≥ χ(G) Observation 3. [1] χlea(G) ≥ ∆(G), where ∆(G) is maximum degrees of G Observation 4. [13] χle(a,d)(G) ≥ χlea(G) ≥ ∆(G) Observation 5. [16] χle(a,d)(G) ≥ χlea(G) ≥ χ(G) ≥ ∆(G). Observation 6. [16] If the graph G admits an local (a, d)-edge antimagic labeling with k-colors, then d ≤ 2p−4 k−1 . Whilst, in the experimental method, we will use a computer programming to do simu- lation on autonomous Controlled Environment Agriculture (CEA) development on multi- step time series forecasting for NPK concentration of companion plantations of Rosa sp. A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 4 of 18 plantation, Bougainvillea spectabilis plantation, Cryptanthus spp plantation, and Bromeli- aceae plantation. First, we will display the vertex embedding process of a single-layer GNN for a given graph that contains three feature data, namely N, P, and K data for three month observation. Second, we will write STGNN programming, train a model utilizing 80% of the input data obtained from the vertex embedding process, test and finally forecast the NPK fertilization to determine the time of application of fertilizer to Rosa sp plantation, Bromeliaceae plantation, Bougainvillea spectabilis plantation and Cryptanthus sp planta- tion. We use the Algorithm 1 for studying the fuel purchase distribution by using STGNN combined by lea(a, d) coloring. Algorithm 1 (phtb). InputInput OutputOutput Graph G(V,E), adjacency matrix A from graph G, feature matrix Hn×m, and threshold ϵ Time series prediction results Initialize parameters: weights W , bias β, and learning rate α. error < ϵ Perform message propagation mu λ = MSGλ(h λ−1 u ). Compute aggregated message hλv = AGGλ{mλ−1 u , u ∈ N(v)}. Evaluate error λ = ||hvi−hvj ||2 |E| . Update parameters: W λ+1 j = W λ j + α · zj · eλ. Store the final node representations as embedding vector. Load the feature embedding from saved data. Apply forecasting using learned representations. Output the best-fit prediction based on training and validation. Single-Layer Graph Neural Network for Forecasting 3. Main Result 3.1. Local (a, d)-edge Antimagic Coloring In this section, we will show the existence of lea(a, d) coloring of a some specific families of graphs, and determine the exact value of the lea(a, d) chromatic number of Pn ▷ Pm, amal(Fn, v,m), and amal(Vn, v,m). Further, we will use the obtained theorem for analysing STGNN model for autonomous Controlled Environment Agriculture (CEA) development on multi-step time series forecasting for NPK concentration of companion plantations. Theorem 1. Let Pn ▷Pm be a comb product between Pn and Pm graph with n,m ≥ 3. We have χle(nm−n+3,2)(Pn ▷ Pm) = n, for m ≡ 1(mod 2), n ≡ 0(mod 2); and χle(nm+1,2)(Pn ▷ Pm) = n, for m,n ≡ 0(mod 2). Proof. We begin by recalling the structure of the comb product Pn ▷ Pm, which is a connected graph composed of n copies of the path Pm, where each Pm is connected via its A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 5 of 18 last vertex to form a path of n vertices. Formally, the vertex set is given by V (Pn ▷Pm) = {xij : 1 ≤ i ≤ n, 1 ≤ j ≤ m} and the edge set is defined as E(Pn ▷ Pm) = {xijxij+1 : 1 ≤ j ≤ m− 1, 1 ≤ i ≤ n} ∪ {ximxi+1 m : 1 ≤ i ≤ n− 1}. Consequently, we have |V | = nm and |E| = nm− 1. To prove the theorem, we determine both the lower and upper bounds of the local edge (a, d)-antimagic chromatic number and show that they are equal to n under the given conditions. From the result by Agustin et al. [2], it is known that χlea(Pn ▷ Pm) = n for n,m ≥ 3. Additionally, Observation 2 implies χle(a,d)(Pn ▷Pm) ≥ χlea(Pn ▷Pm) = n. Thus, the lower bound is established. To construct the upper bound, we define a bijective labeling f : V (Pn ▷ Pm) → {1, 2, . . . , nm} as follows: f(xij) = { i+ (j−1)n 2 , if j ≡ 1 (mod 2) nm+ 1− i+ n− nj 2 , if j ≡ 0 (mod 2) This labeling ensures a proper assignment of distinct values to the vertices and gen- erates edge weights that can be analyzed in two separate cases based on the parity of m and n. Case 1: For m ≡ 1 (mod 2) and n ≡ 0 (mod 2), the edge weights for horizontal edges within each copy of Pm are: w(xijx i j+1) = { nm+ 1, if j ≡ 1 (mod 2) nm+ n+ 1, if j ≡ 0 (mod 2) For vertical edges between copies: w(ximxi+1 m ) = nm− n+ 2i+ 1, for 1 ≤ i ≤ n− 1 These edge weights can be grouped into three distinct sets: W1 = {nm + 1}, W2 = {nm + n + 1}, and W3 = {nm − n + 3, nm − n + 5, . . . , nm + n − 1}. We examine their cardinalities and mutual disjointness. To verify disjointness: suppose W1 ⊆ W3 implies nm + 1 = nm − n + 2i + 1, which leads to i = n 2 . Since i = n 2 is within the index range, it confirms W1 ⊂ W3. On the other hand, suppose W2 ⊆ W3 implies nm + n + 1 = nm − n + 2i + 1, which leads to i = n. However, i = n lies outside the valid range of 1 ≤ i ≤ n− 1, hence a contradiction. Therefore, W2 ∩W3 = ∅ and W = W2 ∪W3. The cardinality is then |W2| = 1, and for W3, let Us be the s-th term in the arithmetic progression with common difference 2. Solving nm+ n− 1 = nm− n+ 3+ (|W3| − 1) · 2, we find |W3| = n− 1, so |W | = n. Case 2: For m,n ≡ 0 (mod 2), we compute: w(ximxi+1 m ) = nm+ 2n− 2i+ 1, for 1 ≤ i ≤ n− 1 A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 6 of 18 The weight sets in this case are W1 = {nm + 1}, W2 = {nm + n + 1}, and W3 = {nm+ 2n− 1, nm+ 2n− 3, . . . , nm+ 3}. To verify disjointness: suppose W1 ⊆ W3 implies nm + 1 = nm + 2n − 2i + 1, which gives i = n. Since i = n is out of range, this leads to a contradiction. Hence W1∩W3 = ∅. Now suppose W2 ⊆W3 implies nm+ n+ 1 = nm+ 2n− 2i+ 1, which gives i = n 2 . Since this i lies within the valid range, W2 ⊂W3, and thus W = W1 ∪W3. The cardinality is |W1| = 1. To compute |W3|, let Us = nm+3 and U1 = nm+2n−1. Solving Us = U1 + (s− 1)(−2), we obtain s = n− 1, so again |W | = n. Therefore, in both cases, we have χle(a,d)(Pn ▷ Pm) ≤ n and from the lower bound, equality holds. Hence the theorem is proved. Theorem 2. Let amal(Fn, x,m) be an amalgamation of fan graph with n ≥ 3 and m ≥ 2. We have nm ≤ χle(nm+1,1)(amal(Fn, x,m)) ≤ nm+ 1. Proof. Amalgamation of fan graph amal(Fn, x,m) is a connected graph with vertex set V (amal(Fn, x,m)) = {x} ∪ {xji ; 1 ≤ i ≤ n, 1 ≤ j ≤ m} and edge set E(amal(Fn, v,m)) = {xxji ; 1 ≤ i ≤ n, 1 ≤ j ≤ m} ∪ {xjix j i+1; 1 ≤ i ≤ n − 1, 1 ≤ j ≤ m}. The cardinality of the vertices set |V (amal(Fn, x,m))| = nm + 1, and the cardinality of the edges set |E(amal(Fn, x,m))| = 2nm−m. To prove, first we need to show the lower bound of χle(nm+1,1)(amal(Fn, x,m)). Based on Observation 4, we have χle(a,d)(amal(Fn, x,m)) ≥ χlea(amal(Fn, x,m)) ≥ ∆(amal (Fn, x,m)). The maximum degree is ∆(amal(Fn, x,m)) = nm. Thus, the lower bound is χle(a,d)(amal(Fn, x,m)) ≥ nm. To show the upper bound, we define the bijection as follow f : V (amal(Fn, x,m)) → {1, 2, 3, . . . , |V (amal(Fn, x,m))|}. To prove upperbound, we show into two cases. Case 1. For n ≡ 1(mod2), we have the following. Firstly, we show the bijective function of vertex labels as follows f(x) = nm+ 1 f(xji ) =  nj+n+i+1 2 − n, for i, j ≡ 1(mod 2) nm− nj+n+i 2 + n+ 1, for i ≡ 1(mod 2), j ≡ 0(mod 2) nj+n+i+1 2 − n, for i, j ≡ 0(mod 2) nm− nj+n+i 2 + n+ 1, for i ≡ 0(mod 2), j ≡ 1(mod 2) Under the bijection f , the edge weights can be presented as follows w(xjix j i+1) =  nm+ 1, for m, i ≡ 1(mod 2) nm+ 2, for m ≡ 1(mod 2), i ≡ 0(mod 2) nm+ 1, for m, i ≡ 0(mod 2) nm+ 2, for m ≡ 0(mod 2), i ≡ 1(mod 2) w(xxji ) = nj+n+i+1 2 − n+ nm+ 1, for i, j ≡ 1(mod 2), 1 ≤ i ≤ n, 1 ≤ j ≤ m 2nm− jn+n+i 2 + n+ 2, for i ≡ 1(mod 2), j ≡ 0(mod 2), 1 ≤ i ≤ n, 2 ≤ j ≤ m nj+n+i+1 2 − n+ nm+ 1, for i, j ≡ 0(mod 2), 2 ≤ i ≤ n, 2 ≤ j ≤ m 2nm− jn+n+i 2 + n+ 2, for i ≡ 0(mod 2), j ≡ 1(mod 2), 2 ≤ i ≤ n, 1 ≤ j ≤ m A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 7 of 18 Thus, we can write in the form of the set W of edge weight of amal(Fn, x,m). It includes four sets of edge weights, namely W1 = {nm+1}, W2 = {nm+2}, W3 = {nm+ 2, nm+3, nm+4 . . . , nm+1+ nm+1 2 }, andW4 = {nm+2+ nm+1 2 , nm+3+ nm+1 2 , . . . , 2nm+ 1}. Our task is to determine the cardinality of those sets to obtain the upper bound, but we need to check whether all three sets are disjoint or not first. Under the contradiction proof, we suppose W1 ⊆W3 ←→ nm+1 = nj+n+i+1 2 −n+nm+1←→ i = −3n−nj− 1. Since w(xxji ) = nj+n+i+1 2 −n+nm+1, for 1 ≤ i ≤ nm+1 2 , therefore i = −3n−nj− 1 does not lie in the interval, it is a contradiction. We conclude that W1∩W3 = ∅. Furthermore, in the contradiction proof, we suppose W1 ⊆W4 ←→ nm+1 = 2nm− nj+n+i 2 +n+2←→ i = 2nm − nj + 2. Since w(xxji ) = 2nm − nj+n+i 2 + n + 2, for nm+3 2 ≤ i ≤ nm, therefore i = 2nm − nj + 2 does not lie in the interval, it is a contradiction. We conclude that W1 ∩W4 = ∅. Finally, we have W = W1 ∪W3 ∪W4. Now, we can determine the cardinality of W. First, it is easy to see that |W1| = 1, secondly we will determine |W3 ∪W4|, under the following calculation. Let Us and d be the s term and different from the arithmetic sequence of W3 ∪W4, consequently. We have U|W3∪W4| = a+(|W3∪W4|−1)d←→ 2nm+1 = nm+2+(|W3∪W4|−1)1←→ |W3∪W4| = nm. W3∪W4 form an arithmetic sequence with initial value a = nm+2 and W2 = nm+2. Thus, we have W2 ⊂ (W3 ∪W4). This implies |W | = |W1|+ |W3 ∪W4| = 1+ nm. Finally, we have χle(nm+1,1)(amal(Fn, x,m)) ≤ nm+ 1 for n ≡ 1(mod 2). Case 2. For n ≡ 0(mod2), we have the following. Firstly, we show the bijective function of vertex labels as follows. f(x) = nm+ 1 f(xji ) = { nj−n+i+1 2 , for i ≡ 1(mod 2), 1 ≤ i ≤ n, 1 ≤ j ≤ m nm− i+nj−n 2 + 1, for i ≡ 0(mod 2), 2 ≤ i ≤ n, 1 ≤ j ≤ m Under bijection f , the edge weights can be presented as follows. w(xjix j i+1) = { nm+ 1, for i ≡ 1(mod 2), 1 ≤ i ≤ n− 1, 1 ≤ j ≤ m nm+ 2, for i ≡ 0(mod 2), 2 ≤ i ≤ n− 1, 1 ≤ j ≤ m w(xxji ) = { nm+ 1 + nj−n+i+1 2 , for i ≡ 1(mod 2), 1 ≤ i ≤ n, 1 ≤ j ≤ m 2nm+ 2− i+nj−n 2 , for i ≡ 0(mod 2), 2 ≤ i ≤ n, 1 ≤ j ≤ m Thus, we can write in the form of the set W of edge weight of amal(Fn, x,m). It includes four sets of edge weights, namely W1 = {nm+ 1}, W2 = {nm+ 2}, W3 = {nm+ 2, nm+ 3, nm+4, . . . , nm+1+nm 2 }, andW4 = {nm+2+nm 2 , nm+3+nm 2 , nm+4+nm 2 . . . , 2nm+1}. Our task is to determine the cardinality of those sets to obtain the upper bound, but we need to check whether all three sets are disjoint or not first. Under the contradiction proof, we suppose W1 ⊆ W3 ←→ nm + 1 = nm + 1 + nj−n+i+1 2 ←→ i = n − nj − 1. Since w(xxji ) = nm + 1 + nj−n+i+1 2 , for 1 ≤ i ≤ nm 2 , therefore i = n − nj − 1 does not lie in the interval, it is a contradiction. We conclude that W1 ∩W3 = ∅. Furthermore, in the contradiction proof, we suppose W1 ⊆ W4 ←→ nm + 1 = 2nm + 3 − i+nj−n 2 ←→ A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 8 of 18 i = 2nm − nj + n + 4. Since w(xxji ) = 2nm + 3 − i+nj−n 2 , for nm+1 2 ≤ i ≤ nm, therefore i = 2nm− nj + n+ 4 does not lie in the interval, it is a contradiction. We conclude that W1 ∩W4 = ∅. Finally, we have W = W1 ∪W3 ∪W4. Now, we can determine the cardinality of W. First, it is easy to see that |W1| = 1, secondly we will determine |W3 ∪W4|, under the following calculation. Let Us and d be the s term and different from the arithmetic sequence of W3 ∪W4, consequently. We have U|W3∪W4| = a+(|W3∪W4|−1)d←→ 2nm+1 = nm+2+(|W3∪W4|−1)1←→ |W3∪W4| = nm. W3∪W4 form an arithmetic sequence with initial value a = nm+2 and W2 = nm+2 then we have W2 ⊂ (W3∪W4). It implies that |W | = |W1|+ |W3∪W4| = 1+nm. Finally, we have χle(nm+1,1)(amal(Fn, x,m)) ≤ nm+ 1 for n ≡ 1(mod 2). Thus, we will have χle(nm+1,1)(amal(Fn, x,m)) ≤ nm + 1. It can be concluded that nm ≤ χle(nm+1,1)(amal(Fn, x,m)) ≤ nm+ 1. Theorem 3. Let amal(Vn, x,m) be an amalgamation of volcano graph with n,m ≥ 2. We have χle(3,1)(amal(Vn, x,m)) = 2m+ nm. Proof. Amalgamation of volcano graph amal(Vn, x,m) is a connected graph with vertex set V (amal(Vn, x,m)) = {x} ∪ {xi; 1 ≤ i ≤ m} ∪ {zi; 1 ≤ i ≤ m} ∪ {yi; 1 ≤ i ≤ nm} and edge set E(amal(Vn, x,m)) = {xizi; 1 ≤ i ≤ m} ∪ {xxi; 1 ≤ i ≤ m} ∪ {xzi; 1 ≤ i ≤ m} ∪ {xyi; 1 ≤ i ≤ nm}. The cardinality of the vertices set |V (amal(Vn, x,m))| = nm+2m+1, and the cardinality of the edges set |E(amal(Vn, x,m))| = nm+ 3m. To prove this theorem, first, we need to show the lower bound of χle(3,1)(amal(Vn, x,m)). Agustin et al in [1] If ∆(G) is maximum degrees of G then we have the local edge antimagic chromatic number of χlea(G) ≥ ∆(G). Based on Observation 3 we have χle(a,d)(amal(Vn, x,m)) ≥ χlea(amal(Vn, x,m)) ≥ ∆(amal(Vn, x,m)). ∆(amal(Vn, x,m)) is 2m + nm. Thus, the lower bound is χle(a,d)(amal(Vn, x,m)) ≥ 2m + nm. To show the upper bound, we define the bijection as follow f : V (amal(Vn, x,m)) → {1, 2, 3, . . . , |V (amal(Vn, x,m))|}. f(x) = 1 f(xi) = i+ 1, for 1 ≤ i ≤ m f(yi) = 2m+ i+ 1, for 1 ≤ i ≤ nm f(zi) = 2m− i+ 2, for 1 ≤ i ≤ m Under the bijection f , the edge weights can be presented as follows w(xxi) = i+ 2, for 1 ≤ i ≤ m w(xizi) = 2m+ 3, for 1 ≤ i ≤ m w(xzi) = 2m− i+ 3, for 1 ≤ i ≤ m A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 9 of 18 w(xyi) = 2m+ 2 + i, for 1 ≤ i ≤ mn Thus we can write in the form of set W of edge weight of amal(Vn, x,m). It includes three edge weight sets, namely W1 = {3, 5, · · · , 2 + m},W2 = {2m + 3}, W3 = {2m + 2, 2m + 1 · · · ,m + 3}, and W4 = {2m + 3, 2m + 4, · · · , 2m + 2 + n}. Our task is to determine the cardinality of those sets to obtain the upper bound, but we need to check weather all three sets are disjoint or not first. Under the contradiction proof, we suppose W2 ⊆ W1 ←→ 2m+ 3 = 2 + i←→ i = 2m+ 1. Since w(xxi) = 2 + i, for 1 ≤ i ≤ m, thus i = 2m+1 does not lie in the interval, it is a contradiction. It concludes that W2∩W1 = ∅. Under the contradiction proof, we suppose W1 ⊆W3 ←→ 2+i = 2m−i+3←→ i = 2m+1 2 . Since w(xzi) = 2m− i+3, for 1 ≤ i ≤ m, thus i = 2m+1 2 does not lie in the interval, it is a contradiction. It concludes that W1 ∩W3 = ∅.Under the contradiction proof, we suppose W1 ⊆W4 ←→ 2 + i = 2m+ 2+ i←→ m = 0. Since w(xyi) = 2m+ 2+ i, for 1 ≤ i ≤ mn, thusm = 0 does not lie in the interval, it is a contradiction. It concludes thatW1∩W4 = ∅. Under the contradiction proof, we suppose W2 ⊆W3 ←→ 2m+3 = 2m− i+3←→ i = 0. Since w(xzi) = 2m − i + 3, for 1 ≤ i ≤ m, thus i = 0 does not lie in the interval, it is a contradiction. It concludes that W2∩W3 = ∅. Under the contradiction proof, we suppose W2 ⊆W4 ←→ 2nm+3 = 2m+2+ i←→ i = 1. Since w(xyi) = 2m+2+ i, for 1 ≤ i ≤ mn, thus i = 1 lies in the interval. It concludes that W2 ⊂ W4. Furthermore, under the contradiction proof, we suppose W3 ⊆ W4 ←→ 2nm − 1 + 3 = 2m + 2 + i ←→ i = 1 2 . Since w(xyi) = 2m+ 2 + i, for 1 ≤ i ≤ mn, thus i = 1 2 does not lie in the interval, it is a contradiction. It concludes that W3 ∩W4 = ∅. Finally, we have W = W1 ∪W3 ∪W4. Now, we can determine the cardinality of W . First, it is easy to see that W2 ⊂ W4, secondly we will determine |W1|, |W3| and |W4| under the following calculation. Let Us and d be the s-term and the different of arithmetic sequence of W1, W2 and W3, consequently. We have U|W1| = a + (|W1| − 1)d ←→ 2 + m = 3 + (|W1| − 1)1 ←→ |W1| = m, U|W3| = a+(|W3|−1)d←→ m+3 = 2m+2+(|W3|−1)(−1)←→ |W3| = m, and U|W4| = a+(|W4|− 1)d←→ nm+2m+2 = 2m+3+(|W4|−1)1←→ |W4| = nm. It implies that |W | = |W1|+ |W3|+|W4| = m+m+nm = 2m+nm. Finally, we have χle(3,1)(amal(Vn, x,m)) ≤ 2m+nm. Based on the lower and upper bounds, we have 2m + nm ≤ χle(3,1)(amal(Vn, x,m)) ≤ 2m+ nm. It can be concluded that χle(3,1)(amal(Vn, x,m)) = 2m+ nm for n,m ≥ 2. Now, we will characterize the non-existence of lea (a, d) coloring of amal(K1+Gn,K1,m), for any grah G of order n. We present it in the following theorem. Theorem 4. For n,m ≥ 2, the lea(a, d) coloring of graph amal(K1+Gn,K1,m) does not exist for d ∈ {0, 2}. Proof. The graph amal(K1 + Gn,K1,m) is a connected and simple graph with ver- tex set V (amal(K1 + Gn,K1,m)) = {x} ∪ {xi,j ; 1 ≤ i ≤ n, 1 ≤ j ≤ m} and edge set E(amal(K1 + Gn,K1,m)) = {xxi,j ; 1 ≤ i ≤ n, 1 ≤ j ≤ m} ∪ {xj,kxj,l; 1 ≤ j ≤ m, {k, l} ∈ V (Gn)}. Thus |V (amal(K1 + Gn,K1,m))| = |V (Gn)|m + 1 = nm + 1 and |E(amal(K1 + Gn,K1,m)| = nm + m|E(G)|. Based on Observation 6, we have d ≤ 2p−4 k−1 ←→ d ≤ 2|V (amal(K1+Gn,K1,m))|−4 |V (amal(K1+Gn,K1,m))|−2 ←→ d ≤ 2. A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 10 of 18 Now, we will show the non-existence of lea(a, d) coloring of amal(K1 +Gn,K1,m) by defining f : V (amal(K1 +Gn,K1,m))→ {1, 2, 3, . . . , |V (amal(K1 +Gn,K1, m))|} as vertex labels of amal(K1 + Gn,K1,m). First, for the non-existence (a, 0)-edge antimagic coloring. Assume that we have d = 0. For any {xi,j , xi,k} ∈ V (amal(K1 + Gn,K1,m)) and 1 ≤ i ≤ n, 1 ≤ j, k ≤ m, we will have w(xxi,j) = w(xxi,k). It contradicts with the definition of local antimagic coloring, since xxi,j and xxi,k are incident, they should have distinct edge weight as the colors. Thus (a, 0)-edge antimagic coloring of amal(K1 +Gn,K1,m) does not exist. Furthermore, we will prove the non-existence of local (a, 2)-edge antimagic coloring of amal(K1+Gn,K1,m). Since, we have f : V (amal(K1+Gn,K1,m))→ {1, 2, 3, . . . , |V (amal (K1 +Gn,K1,m)|} as vertex labels of amal(K1 +Gn,K1,m). We have three possible la- bels of the label of f(x). First f(x) = 1, otherwise f : V (amal(K1 + Gn,K1,m)\{x} → {2, 3, . . . , |V (amal(K1 +Gn,K1,m)|}. The edge weight will be W (xxi,j) = {3, 4, 5, . . . , |V (amal(K1 + Gn,K1,m))| + 1}. This form an arithmetic sequence of d = 1, since all xij connects to x. It proves the non-existence of d = 2. The second possibility is to assign the vertex labels with 2 ≤ f(x) ≤ |V (amal(K1 + Gn,K1,m))| − 1. Given that s ∈ {2, 3, · · · , |V (amal(K1 + Gn,K1,m))| − 1}, choose any s for f(x) = s. The edge weight will be W (xxi,j) = {· · · , 2s− 3, 2s− 2, 2s− 1, 2s+1, 2s+2, 2s+3, · · · }. This edge weight does not form an arithmetic sequence of different d, since even the different of the left and the right terms shows d = 1, but the middle terms 2s+1− (2s− 1) = 2. It shows that for any f(x) = s where s ∈ {2, 3, · · · , |V (amal(K1+Gn,K1,m))|−1}, the local (a, 2)- edge antimagic coloring of amal(K1 +Gn,K1,m) does not exist. The third, assume that f(x) = |V (amal(K1 + Gn,K1,m))|, the edge weight will be W (xxi,j) = {|V (amal(K1 + Gn,K1,m))| + 1, |V (amal(K1 + Gn,K1,m))| + 2, · · · , 2|V (amal(K1 + Gn,K1,m))| − 1}. This form an arithmetic sequence of d = 1, since all xij connects to x. It proves the non-existence of d = 2. Thus (a, 2)-edge antimagic coloring of amal(K1+Gn,K1,m) does not exist. It concludes the proof. 3.2. Modeling Temporal-Spatial Dependencies with Graph-Based Deep Learning In this work, we propose a Spatio-Temporal Graph Neural Network (STGNN) archi- tecture tailored to forecast multivariate time series data derived from structured spatial observations such as nutrient content (e.g., nitrogen, phosphorus, potassium). The model is designed to jointly capture spatial dependencies across graph-structured nodes and temporal dynamics across multiple time steps, as illustrated in Figure 1 and detailed in Table 1. The input feature matrix consists of sequential observations mapped onto a spatial graph, where each node represents a measurement location and edges encode spatial adja- cency or correlation. The preprocessing layer performs normalization and an initial fully connected transformation to enhance feature richness. These features are then passed into a graph convolutional layer, which extracts spatial relationships via topological filtering using Graph Convolutional Networks (GCNs). A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 11 of 18 Figure 1: STGNN Model Architecture Visualization. For temporal modeling, the architecture stacks a series of STGNN blocks, each combin- ing spatial representations with Gated Recurrent Unit (GRU)-based processing to capture sequential dependencies. Each temporal slice of the enhanced hidden features is individ- ually passed through an STGNN-GRU unit, enabling the model to learn both short-term and long-term trends. An attention mechanism further refines the temporal features by weighting them based on relevance to the forecasting task. The aggregated output is then passed through multiple fully connected layers to project the learned embeddings into the prediction space. The final prediction layer outputs three continuous values per node, corresponding to target variables (e.g., discharge, depth, and width). The model is trained using the Adam optimizer for 200 epochs, with a learning rate of 0.01, and optimized using Mean Squared Error (MSE) loss. The use of optional components such as dilated convolution enhances the temporal receptive field, while regularization strategies (e.g., dropout) can be applied for improved generalization. Overall, the architecture demonstrates strong performance in learning spatial-temporal patterns from graph-structured data. 3.3. Spatio-Temporal Graph Encoding for Nutrient Forecasting The next research result is the implementation of the χle(a,d) coloring scheme on preci- sion agriculture. From now on, we will do time series forecasting on precision agriculture soil nutrient dataset, namely nitrogen, phosporus, and potassium. The dataset is obtained from the simulation of placing the capacitive sensor for soil nutrients. The placement of those sensors respects to the planting topology derived from Theorem 1, and the number of soil nutrient sensors is equalto the obtained lea(a, d) chromatic number χle(a,d)(G). The soil nutrient dataset that we obtained is subsequently used as the input for training A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 12 of 18 Table 1: Architectural Components and Parameters of the Proposed STGNN Model No. Component Description 1 Input Spatial Graph Represents spatial structure over time. Nodes correspond to measurement units; edges reflect proximity or correlations. 3 Preprocessing Layer Applies normalization and temporal align- ment. Initial fully connected (FC) transfor- mation to increase feature richness. 4 Convolutional Layer Applies GCN operation over the input spa- tial graph to learn spatial dependencies from graph topology. 5 STGNN Block (Temporal Stack) Sequence of STGNN-GRU units processing temporal slices of graph-encoded data (e.g., Xθ−T+3 to Xθ). Captures temporal dynam- ics. 6 Attention Mechanism Includes dense projections and softmax weighting to emphasize relevant temporal features. Dot product used to compute weighted representation. 7 Dilated Convolution (Optional) Extracts multi-scale temporal features through dilated convolutions across time- delayed embeddings. Enhances long-range dependencies. 8 Fully Connected Layers Maps temporal-spatial feature embeddings to prediction space via dense layers. Usually two stacked layers. 9 Prediction Layer Outputs 3-dimensional predictions (e.g., Ni- trogen (N), Phosporus (P), and Potassium (K) per node per time step. the STGNN model. We analyze the NPK nutrient fertilization levels of plants across twenty-three plantations. First, we perform the vertex embedding process using a single- layer STGNN applied to a graph with three feature variables observed over 37 consecutive days. In total, the dataset consists of 23 × 3 × 37 = 2,553 entries, corresponding to 23 plantation nodes, 3 soil nutrient features, and 37 time steps. In accordance with the algorithm described previously, we conduct node embedding across all twenty-three plantation nodes. This embedding process transforms the origi- nal three-dimensional feature vector into a one-dimensional representation using message passing and aggregation techniques. During the message passing stage, each node is as- sumed to contain relevant information, which is transmitted to its neighboring nodes to support the learning process. Thus, in this process, we consider the adjacency matrix with self-loops included. In- stead of using the standard adjacency matrix P , we modify it by adding an identity matrix, A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 13 of 18 Figure 2: Visualization of the performance of the STGNN model and other models. resulting in Q = P + I. Following the steps outlined in the algorithm above, we obtain time series data that are ready for analysis using the STGNN framework. To evaluate the effectiveness of the node embeddings, we assess the proximity of nodes in both the original network and the embedding space. The embedding error is computed by comparing each pair of adjacent nodes using the infinity norm. Subsequently, we extend the model to perform multi-step time series forecasting with STGNN, using 60% of the data for training. The model’s performance is evaluated by measuring the Root Mean Square Error (RMSE) or Mean Square Error (MSE) on the testing data. The results are presented in Figure 2. Figure 3: Forecast results using STGNN. A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 14 of 18 To evaluate the robustness and predictive capability of the proposed STGNN model, we conducted a comprehensive comparison with five baseline models: Historical Average (HA), Auto-Regressive Integrated Moving Average (ARIMA), Support Vector Regression (SVR), Graph Convolutional Network (GCN), and Gated Recurrent Unit (GRU). The comparative performance across models is illustrated in Figure 2, Figure 3, and summa- rized in Table 2. As shown in Figure 2, the STGNN model consistently achieves lower RMSE values compared to the other methods over 200 epochs. The convergence curve of STGNN demonstrates a smooth and steady decrease in loss, indicating its stability and efficiency in the training process. Unlike GRU and GCN models which exhibit oscillations in early epochs, STGNN achieves optimal convergence with minimal variance, highlighting its robustness in handling temporal dependencies and spatial structures. Table 2: The prediction results of the STGNN model and other baseline methods. Epochs Metric Model ARIMA LSTM GCN GRU STGNN (Our Model) 50 RMSE 8.1051 7.5357 9.2706 4.0482 3.9052 MAE 6.1092 4.9198 7.2665 2.6803 2.6061 Accuracy 0.3178 0.6871 0.6422 0.7068 0.7206 R2 0.0732 0.8101 0.6037 0.8437 0.8531 100 RMSE 8.2112 7.4736 9.3430 4.0768 3.9606 MAE 6.2143 4.9719 7.3101 2.7007 2.6441 Accuracy 0.4271 0.6976 0.6305 0.7147 0.7165 R2 0.0724 0.8131 0.6086 0.8367 0.8413 150 RMSE 8.2121 7.4753 9.4022 4.1002 3.9840 MAE 6.2143 5.0321 7.3704 2.7107 2.6655 0.4260 0.6975 0.6373 0.7032 0.7142 R2 0.0826 0.8131 0.6027 0.8350 0.8409 200 RMSE 8.2052 7.4872 9.4404 4.1131 4.0031 MAE 6.2108 5.0604 7.4110 2.7321 2.6878 Accuracy 0.4272 0.6971 0.6255 0.7015 0.7128 R2 0.0814 0.8144 0.5888 0.8432 0.8703 Table 2 further confirms the superior performance of STGNN across multiple evalua- tion metrics, including RMSE, MAE, accuracy, and the coefficient of determination (R2). At every evaluated epoch interval (50, 100, 150, and 200), STGNN achieves the lowest error values and the highest accuracy and R2 scores, outperforming all baseline methods. Notably, the RMSE and MAE values of STGNN are significantly lower, while its R2 values remain consistently above 0.85, demonstrating a strong fit between the predictions and the actual observations. In addition, multi-step forecasting results, as presented in Figure 3, show the STGNN model’s capability to accurately predict soil nutrient concentrations (nitrogen, phosphorus, and potassium) over a 20-day horizon. The predicted curves closely follow the ground truth A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 15 of 18 across all three nutrient types, reaffirming the model’s generalization ability in real-world agricultural time series data. Furthermore, to optimize the monitoring system in companion plantations, we applied the concept of local (a, d)-edge antimagic coloring to minimize the number of sensors required. Based on this approach, only four representative sensor locations are needed, corresponding to four plant types: Rosa sp., Bromeliaceae, Bougainvillea spectabilis, and Cryptanthus sp. These representative sensors, selected based on graph labeling theory, are sufficient to capture the overall dynamics of the plantation system, providing a cost- effective yet reliable monitoring strategy. Overall, the integration of STGNN with local graph-based optimization offers a promising framework for precision agriculture, enabling both accurate forecasting and efficient deployment of sensing resources. 3.4. Computational Complexity and Training Time Analysis To evaluate the computational efficiency of the proposed STGNN model, we com- pared its training time to several baseline models, including ARIMA, LSTM, GCN, and GRU. Although STGNN integrates both graph convolutional layers and recurrent tempo- ral processing, which intuitively may introduce additional computational cost, the model demonstrates efficient convergence behavior and outperforms the other models in terms of predictive accuracy, as discussed in earlier sections. Table 3: Training Time Comparison of Models (200 Epochs) Model Training Time (seconds) ARIMA 25.4 LSTM 102.8 GCN 94.6 GRU 86.3 STGNN (Our Model) 108.7 Figure 4 presents a visual comparison of the average training time required by each model over 200 epochs, while Table 3 provides the corresponding numerical values. As shown, STGNN required 108.7 seconds to complete 200 training epochs, which is slightly higher than GCN and GRU, but comparable to LSTM and still within a practical com- putational range. Notably, although ARIMA is the fastest with only 25.4 seconds, it performs poorly in terms of forecasting accuracy, as previously shown in Table 2. This analysis highlights the trade-off between computational cost and prediction per- formance. While STGNN has a higher training time than simpler models, it offers sub- stantial improvements in RMSE, MAE, and R2 scores. Given its ability to learn spatial and temporal dependencies effectively, STGNN presents a balanced and scalable solu- tion suitable for real-world agricultural forecasting applications, where both accuracy and robustness are crucial. A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 16 of 18 Figure 4: Training time comparison over 200 epochs for different models. 4. Concluding Remarks This study introduced a hybrid framework that integrates local (a, d)-edge antimagic labeling and Spatio-Temporal Graph Neural Networks (STGNN) to forecast soil nutrient dynamics in companion plantations. The labeling technique was used to optimize sensor placement by identifying representative nodes, reducing the number of sensors from 23 to just four without compromising coverage. Meanwhile, the STGNN model effectively cap- tured spatial and temporal dependencies, outperforming baseline models such as ARIMA, LSTM, GCN, and GRU across RMSE, MAE, accuracy, and R2 metrics. The results confirmed that STGNN achieves high predictive performance and converges efficiently. The integration of graph labeling and deep learning offers both theoretical and prac- tical contributions. Theoretically, it extends the applicability of local antimagic labelings to data-driven graph learning. Practically, it provides a scalable solution for precision agriculture, enabling cost-efficient monitoring and accurate forecasting. Future work may explore dynamic graphs, enhance training efficiency, and adapt this approach to broader domains such as environmental monitoring and smart farming systems. Acknowledgements This work was supported in part by LP2M-University of Jember and PUI-PT Combi- natorics and Graph, CGANT-University of Jember, for their excellent collaboration and support in completing this study for the year 2025. A. I. Kristiana, et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 5970 17 of 18 References [1] I. H. Agustin, M. Hasan, Dafik, R. Alfarisi, and R. M. Prihandini. Local edge an- timagic coloring of graphs. Far East Journal of Mathematical Sciences (FJMS), 102(9):1925–1941, 2017. [2] I. H. Agustin, M. Hasan, Dafik, R. Alfarisi, A. I. Kristiana, and R. M. Prihandini. Local edge antimagic coloring of comb product of graphs. 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