EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6425 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the Edge-Gracefulness of Water Wheel-Type Graphs and Their Splitting Graphs Aaron D. C. Angel1,∗, John Rafael M. Antalan1, John Loureynz F. Gamurot2, Richard P. Tagle1 1 Faculty, Department of Mathematics and Physics, College of Science, Central Luzon State University (3120), Science City of Muñoz, Nueva Ecija, Philippines 2 Alumni, Department of Mathematics and Physics, College of Science, Central Luzon State University (3120), Science City of Muñoz, Nueva Ecija, Philippines Abstract. A graph G with p vertices and q edges is said to be edge-graceful if its edges can be labeled from 1 through q, in such a way that the labels induced on the vertices by adding over the labels of incident edges modulo p are distinct. Lo’s Theorem is a known result under this topic, which states that if a graph G with p vertices and q edges is edge-graceful, then p ∣∣∣(q2+q− p(p−1) 2 ) . Since the introduction of the concept of edge-graceful labeling, several works on the edge-graceful labeling of specific families of graphs have been conducted. We recall the definition of water wheel graphs (WWn) and introduces two new related graphs, the right water wheel graphs (RWWn) and left water wheel graphs (LWWn). Next, using Lo’s Theorem, and the concepts of divisibility and Diophantine equations, we proved the non-edge-gracefulness of these graphs. Then, using the same concepts, we determine all the edge-graceful splitting graphs of WWn, RWWn, and LWWn, denoted as S(WWn), S(RWWn), and S(LWWn), respectively. Finally, Python computer programs were created and used to conclude that among these graph families, only S(WW3), S(RWW3), and S(LWW3) are edge-graceful, by providing their corresponding edge-graceful labels. 2020 Mathematics Subject Classifications: 05C76, 05C78, 11D09, 11D45, 68R10 Key Words and Phrases: Graph labeling, edge-graceful labeling, Diophantine equation, water wheel graph 1. Introduction Graph Theory or the study of graphs is a vast field in mathematics which offers a plethora of unique and interesting topics. One of these particular topics in graph theory is graph labeling. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6425 Email addresses: aaronangel1123@gmail.com, aaron.angel@clsu2.edu.ph (A. D. C. Angel), jrantalan@clsu.edu.ph (J. R. M. Antalan), gamurotloureynz@gmail.com (J. L. F. Gamurot), richard_tagle@clsu.edu.ph (R. P. Tagle) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 2 of 33 In the mid-1960s, the concept of graph labeling was first introduced. Graph labeling is an assignment of integers to the vertices or edges, or both, of a graph under certain conditions [1]. In the last 60 years, over 200 types or variations of graph labeling have been studied and about 2500 articles have been published [2]. One type of graph labeling is graceful labeling. A graceful labeling or graceful numbering is a special graph labeling of a graph on m edges in which the vertices are labeled, using a subset of distinct nonnegative integers from 0 to m and each edge of the graph is uniquely labeled by the absolute difference between the labels of the vertices incident to it. If the resulting graph edge numbers run from 1 to m inclusive, it is a graceful labeling, and the graph is said to be a graceful graph [3]. On the other hand, the concepts of edge-gracefulness and edge-graceful graphs are defined and introduced by Lo in 1985 [4]. Edge-graceful labeling is considered as a reversal of graceful labeling because it labels the edges first, then the labels of the vertices would depend on the labels of the edges incident to them. That is, a graph G with p vertices and q edges is said to be edge-graceful if the edges can be labeled from 1 through q, in such a way that the labels induced on the vertices by adding over the labels of incident edges modulo p are distinct [3]. Since the introduction of the concept of edge-graceful labeling in 1985, several works on the edge-graceful labeling of specific families of graphs have been conducted. For instance, Kuan et al. [5] studied edge-graceful unicyclic graphs. Lee et al. [6] on the other hand, studied the edge-graceful labeling of complete graphs. The work of Small in [7] shows that regular (even) spider graphs are edge-graceful. In 1992, Cabannis et al. [8] studied the edge-gracefulness of regular graphs and trees. Fast-forward to 2017, Venkatesan and Sekar [3] attempted to prove that among all wheel graphs Wn, only W3 is edge-graceful while a complete proof of their result was provided recently in [9]. In this paper, we used essential graph theory concepts to study the basic properties of water wheel, right water wheel, and left water wheel graphs, and their splitting graphs. In particular, their vertex and edge sets were used to apply Lo’s Theorem, an initial condition that must be satisfied to conclude that a particular graph is edge-graceful. Then, using the definition of edge-graceful labeling, and some preliminary concepts in number theory, which include the concept of the Diophantine equation, this paper determined which among the said graphs satisfied Lo’s Theorem. If a particular graph does not satisfy Lo’s Theorem, then it automatically does not qualify to be an edge-graceful graph. Finally, by using Python computer programs that we created, this paper constructed actual examples of edge-graceful labeling of the graphs that satisfy these initial conditions, thereby showing that the graphs are indeed edge-graceful. 2. Preliminaries For completeness, we provide all the preliminary concepts and results necessary for the clear and organized presentation of the results of this study. We refer the readers who are unfamiliar with graph theory concepts that were briefly stated in this paper, to some standard graph theory references such as Gallian’s “A Dynamic Survey of Graph A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 3 of 33 Labeling” [1] and Gross’s “Graph theory and its applications”[10]. 2.1. Essential Graph Theory Concepts To begin this subsection, we briefly recall some key concepts in graph theory that are essential to this paper. Definition 1. A (p, q)-graph G(V,E) is called edge-graceful if there exist a bijection f : E → {1, 2, ...., q}, such that the induced mapping f+ : V → Zp, defined by f+(u) ≡ ( ∑ v∈N(u) f(uv) ) (mod p) is also a bijection, where Zp is the set of integers {0, 1, 2, .., (p − 1)}. Moreover, every f(uv) are the labels of the edges with endpoint u in G. Example 1. An example of an edge-graceful labeling of a graph is shown in Figure 1. First, the graph involved, has seven edges and seven vertices. By following Definition 1, the edges of the graph were labeled using distinct integers from 1 to 7, and the induced labels of the vertices are distinct integers from 0 to 6. Figure 1: An Edge-graceful Labeling of Cycle Graph C7. Definition 2 ([11]). Let n ≥ 3 be an integer. A water wheel graph denoted by WWn, is a graph with • V (WWn) = {u0} ∪ {ui, vi, wi : 1 ≤ i ≤ n}; and • E(WWn) = {u0ui, u0vi, uiwi, viwi : 1 ≤ i ≤ n} ∪ {uivi+1 : 1 ≤ i ≤ n− 1} ∪ {unv1}. Thus, |V (WWn)| = 3n+ 1 and |E(WWn)| = 5n. Example 2. An illustration of water wheel graph WWn is shown in Figure 2. A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 4 of 33 Figure 2: Water Wheel Graph WWn. Now, we introduce two new water wheel-related graphs, the right water wheel and left water wheel graphs. Definition 3. Let n ≥ 3 be an integer. A right water wheel graph denoted by RWWn, is obtained by adding edges wnv1 and wivi+1, for 1 ≤ i ≤ n− 1, to the water wheel graph WWn. Hence, • V (RWWn) = V (WWn); and • E(WWn) = E(WWn) ∪ {wivi+1 : 1 ≤ i ≤ n− 1} ∪ {wnv1}. Thus, |V (RWWn)| = 3n+ 1 and |E(RWWn)| = 6n. Example 3. An illustration of right water wheel graph RWWn is shown in Figure 3, in the next page. A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 5 of 33 Figure 3: Right Water Wheel Graph RWWn. Definition 4. Let n ≥ 3 be an integer. A left water wheel graph denoted by LWWn, is obtained by adding edges w1un and wiui−1, for 2 ≤ i ≤ n, to the water wheel graph WWn. Hence, • V (RWWn) = V (WWn); and • E(WWn) = E(WWn) ∪ {wiui−1 : 2 ≤ i ≤ n} ∪ {w1un}. Thus, |V (RWWn)| = 3n+ 1 and |E(RWWn)| = 6n. Example 4. An illustration of left water wheel graph LWWn is shown in Figure 4, in the next page. A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 6 of 33 Figure 4: Left Water Wheel Graph LWWn. Definition 5 ([12]). Suppose we have a graph G. The splitting graph of G denoted by S(G) can be constructed by introducing a new vertex v′ for each vertex v ∈ V (G), and connecting v′ to all the points in G that are adjacent to v. Example 5. The path graph P4 and its splitting graph S(P4) is shown in Figure 5 Figure 5: Splitting Graph of Path Graph P4 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 7 of 33 Remark 1. For convenience with the computer programs used in the study, we represent the new vertices in the splitting graph with capital letters. For example, a new vertex w′ 1 will be re-represented as simply vertex W1. Example 6. To illustrate Remark 1, the splitting graph S(WW3) is shown in Figure 6. The subgraph colored in black is the original water wheel graph WW3, while the newly introduced vertices and edges are colored in blue. Figure 6: Splitting Graph of S(WW3) We now introduce a proposition, used in this paper, regarding the order and size of splitting graphs. Proposition 1 ([12]). Suppose that G is a (p, q)- graph, then its splitting graph S(G) is a (2p, 3q)- graph. 2.2. Essential Number Theory Concepts In this subsection, we recall the important concepts and results in number theory that are used in this paper. For a more detailed discussion of basic concepts in number theory, readers may refer to Burton’s “Elementary Number Theory”[13]. Definition 6. Let a and b be two positive integers, where a ≤ b. We say that a divides b, written in symbols by a | b, if there is a positive integer c such that b = ac. Otherwise, we say that a does not divide b, denoted by a ∤ b. Now, if a | b, it also implies the following: (i) b is a multiple of a, (ii) a is a divisor of b , and (iii) a is a factor of b. A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 8 of 33 Definition 7. Let p be a positive integer. Also let a, b ∈ Z. We can say that a is congruent to b(modulo p) if p | (a− b). That is if a = b+ kp for some k ∈ Z. Definition 8. General Quadratic Diophantine Equation: Diophantine equations are equations wherein only integer solutions are allowed. A general quadratic Diophantine equation is an equation of the form ax2 + bxy + cy2 + dx+ ey + f = 0, where x and y are both integers. 2.3. Essential Results Used in this Paper This subsection discusses the Lo’s theorem, which is the main theorem that we will use to obtain the results of this paper. Furthermore, some essential transformations of quadratic Diophantine equations are also discussed in this section. Theorem 1. Lo’s Theorem [4]: If a graph G of p vertices and q edges is edge-graceful, then p ∣∣∣(q2 + q − p(p− 1) 2 ) . A complete proof of Theorem 1(Lo’s Theorem) was provided in [14]. Lemma 1. [15]The quadratic Diophantine equation ax2 + bxy + cy2 + dx+ ey + f = 0, (1) where a, b, c, d, e, and f are integer coefficients, and x and y are the unknown variables can be reduced to X2 −DY 2 = N, (2) where X = Dy + E, Y = 2ax+ by + d, and N = E2 −DF . Note that in the above transformation, D = b2 − 4ac,E = bd − 2ae, and F = d2 − 4af . Moreover, observe that if X and Y is a solution to equation (2), then there are integers x and y, such that x = Y − by − d 2a and y = X − E D , (3) where x and y in equation (3) are the integer solutions for equation (1). Remark 2. The Diophantine equations involved in this study have c = 0. Thus, equation (2) will just be written as X2 − (bY )2 = N (4) Observe that we can factor the left side of equation (4) as (X + bY )(X − bY ). Moreover, if the pair N1 and N2 is a factor pair of N , then we will have linear system of equation, (X + bY ) = N1 and (X − bY ) = N2. Now, solving for X and Y , we will have, X = N1 +N2 2 and Y = N1 −N2 2b A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 9 of 33 Having discussed all preliminary information, we are now ready to present the results of this study. 3. Results We begin the section by providing results on the non-edge-gracefulness of water wheel, right water wheel, and left water wheel graphs. 3.1. Non-edge-gracefulness of Water Wheel Graphs WWn, Right Water Wheel Graphs RWWn, and Left Water Wheel Graphs LWWn. Theorem 2. For n ≥ 3, water wheel graph WWn is not edge-graceful. Proof. First, recall from Definition 2 that water wheel graph WWn has 3n+1 vertices and 5n edges. Using Theorem 1(Lo’s Theorem), if a graph with p vertices and q edges is edge-graceful, then p ∣∣∣(q2 + q − p(p− 1) 2 ) . (5) Thus, we substitute p = 3n+ 1 and q = 5n in equation (5). That is, (3n+ 1) ∣∣∣((5n)2 + 5n− (3n+ 1)(3n) 2 ) , which further implies that, (41n2 + 7n 6n+ 2 ) ∈ Z. This means, 41n2 + 7n = (6n+ 2)k = 6nk + 2k, for some k ∈ Z. Thus, we solve for all the possible integer values of n and k in the Diophantine equation, 41n2 + 7n− 6nk − 2k = 0, (6) where n, k ∈ Z. This Diophantine equation is of the form, an2 + bnk + ck2 + dn+ ek + f = 0, where a = 41, b = −6, c = 0, d = 7, e = −2, and f = 0. Using the transformation stated in Lemma 1, equation (6) can be reduced to X2 − 36Y2 = 13120. (7) where Y = 2an+ bk + d = 82n− 6k + 7 and X = Dk + E = 36k + 122. By factoring the left side of the equation (7), we will have, (X + (−6)Y )(X − (−6)Y ) = 13120, A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 10 of 33 where b = −6. Therefore, by Lemma 1 and Remark 2, we have the following: X = N1 +N2 2 , Y = N1 −N2 −12 , k = X − 122 36 , n = Y + 6k − 7 82 , (8) where (N1, N2) are factor pairs of N = 13120. Using (8), we can now solve for the integer values of n and k, by considering all the factor pairs of 13120, as presented in Table 1 in the Appendices section. Table 1 shows that there are 4 possible solutions for the Diophantine equation 41n2 + 7n−6nk−2k = 0. These are: (n,k)= (0,0), (1,6), (-7,-49), and (-2, -15). However, since we are only concern with possible values of n for water wheel graph WWn, then by Definition 2, n must be a positive integer and n ≥ 3. Thus, none of our 4 solutions satisfies both of these conditions. Hence, we have shown that for all n ≥ 3, water wheel graph WWn does not satisfy the Lo’s Theorem. Therefore, we can conclude that for all n ≥ 3, water wheel graph WWn is not edge-graceful. Theorem 3. For n ≥ 3, right water wheel graph RWWn is not edge-graceful. Proof. Recall from Definition 3 that right water wheel graph RWWn has 3n + 1 vertices and 6n edges. Similar to the proof of Theorem 2, we substitute p = 3n + 1 and q = 6n in Theorem 1(Lo’s Theorem). That is, (3n+ 1) ∣∣∣((6n)2 + 6n− (3n+ 1)(3n) 2 ) , which further implies that, (63n2 + 9n 6n+ 2 ) ∈ Z. This means, 63n2 + 9n = (6n+ 2)k = 6nk + 2k, for some k ∈ Z. Thus, we solve for all the possible integer values of n and k in the Diophantine equation, 63n2 + 9n− 6nk − 2k = 0, (9) where n, k ∈ Z. This Diophantine equation is of the form, an2 + bnk + ck2 + dn+ ek + f = 0, where a = 63, b = −6, c = 0, d = 9, e = −2, and f = 0. Using the transformation stated in Lemma 1, equation (9) can be reduced to X2 − 36Y2 = 36288. (10) A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 11 of 33 where Y = 2an+ bk + d = 126n− 6k + 9 and X = Dk + E = 36k + 198. By factoring the left side of the equation (10), we will have, (X + (−6)Y )(X − (−6)Y ) = 36288, where b = −6. Therefore, by Lemma 1 and Remark 2, we have the following: X = N1 +N2 2 , Y = N1 −N2 −12 , k = X − 198 36 , n = Y + 6k − 9 126 , (11) where (N1, N2) are factor pairs of N = 36288. Using (11), we can now solve for the integer values of n and k, by considering all the factor pairs of 36288, as presented in Table 2 in the Appendices section. Table 2 shows that there are two possible solutions for the Diophantine equation 63n2 + 9n − 6nk − 2k = 0. These are: (n,k)= (0,0), and (1, 9). However, since we are only concern with possible values of n for right water wheel graph RWWn, then by Definition 3, n must be a positive integer and n ≥ 3. Thus, none of our two solutions satisfies both of these conditions. Hence, we have shown that for all n ≥ 3, right water wheel graph RWWn does not satisfy the Lo’s Theorem. Therefore, we conclude that for all n ≥ 3, right water wheel graph RWWn is not edge-graceful. Theorem 4. For n ≥ 3, left water wheel graph LWWn is not edge-graceful. Proof. Recall from Definition 4 that left water wheel graph LWWn, also has 3n+ 1 vertices and 6n edges, the same with right water wheel graph RWWn. Since LWWn has the same size and order as RWWn, then the proof of Theorem 4 will be the same as the proof of Theorem 3. Therefore, we already have enough evidence to conclude that for all n ≥ 3, no left water wheel graph LWWn is edge-graceful. We now present our results regarding the edge-gracefulness of the splitting graphs of water wheel, right water wheel, and left water wheel graphs. 3.2. On Edge-gracefulness of the Splitting Graphs of Water Wheel Graphs S(WWn), Right Water Wheel Graphs S(RWWn), and Left Water Wheel Graphs S(LWWn). Theorem 5. Among all splitting graphs of water wheel graphs, only S(WW3) is edge- graceful. Proof. First, using Definiton 2 and Proposition 1, the splitting graph S(WWn) has 6n+ 2 vertices and 15n edges. Hence, we substitute p = 6n+ 2 and q = 15n in Theorem 1(Lo’s Theorem). That is, (6n+ 2) ∣∣∣((15n)2 + 15n− (6n+ 2)(6n+ 1) 2 ) , A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 12 of 33 which further implies that, (207n2 + 6n− 1 6n+ 2 ) ∈ Z. This means, 207n2 + 6n− 1 = (6n+ 2)k = 6nk + 2k, for some k ∈ Z. Thus, we solve for all the possible integer values of n and k in the Diophantine equation, 207n2 + 6n− 1− 6nk − 2k = 0, (12) where n, k ∈ Z. This Diophantine equation is of the form, an2 + bnk + ck2 + dn+ ek + f = 0, where a = 207, b = −6, c = 0, d = 6, e = −2, and f = −1. Using the transformation stated in Lemma 1, equation (12) can be reduced to X2 − 36Y2 = 596160. (13) where Y = 2an+ bk + d = 414n− 6k + 6 and X = Dk + E = 36k + 792. By factoring the left side of the equation (13), we will have, (X + (−6)Y )(X − (−6)Y ) = 596160, where b = −6. Therefore, by Lemma 1 and Remark 2, we have the following: X = N1 +N2 2 , Y = N1 −N2 −12 , k = X − 792 36 , n = Y + 6k − 6 414 , (14) where (N1, N2) are the factor pairs of N = 596160. Using (14), we can now solve for the integer values of n and k, by considering all the factor pairs of 596160, as presented in Table 3 in the Appendices section. Table 3 shows that there are 2 possible solutions for the Diophantine equation 207n2 + 6n− 1− 6nk− 2k = 0. These are: (n,k)= (3,94), and (-1,-50). However, since we are only concern with possible values of n for S(WWn), then only (3,94), where n = 3, is a valid solution under this conditions. Therefore, we have shown that S(WW3) is the only splitting graph of water wheel graph that satisfies the Lo’s Theorem. To complete this proof, an actual example of an edge-graceful labeling of S(WW3) must be provided. S(WW3) is a very large graph with 20 vertices and 45 edges. By permutation, there is a total of 45! possible labeling for its edges. Thus, the researchers created and used a Python computer program to generate an edge-graceful labeling for the said graph. One of the results from the computer program is the labeling shown in Figure 7. A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 13 of 33 Figure 7: An Edge-graceful Labeling of S(WW3) As shown in Figure 7, the 45 edges of S(WW3) are labeled using distinct integers from 1 to 45. Then, by Definition 1, the 20 vertices are successfully labeled using distinct integers from 0 to 19. Therefore, the labeled graph in Figure 7 is indeed an edge-graceful labeling of S(WW3). Since all the needed conditions are satisfied, then we can now conclude that S(WW3) is the only edge-graceful splitting graph of water wheel graphs. Theorem 6. Among all splitting graphs of right water wheel graphs, only S(RWW3) is edge-graceful. Proof. First, using Definiton 3 and Proposition 1, the splitting graph S(RWWn) has 6n+ 2 vertices and 18n edges. Hence, we substitute p = 6n+ 2 and q = 18n in Theorem 1(Lo’s Theorem). That is, (6n+ 2) ∣∣∣((18n)2 + 18n− (6n+ 2)(6n+ 1) 2 ) , A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 14 of 33 which further implies that, (306n2 + 9n− 1 6n+ 2 ) ∈ Z. This means, 306n2 + 9n− 1 = (6n+ 2)k = 6nk + 2k, for some k ∈ Z. Thus, we solve for all the possible integer values of n and k in the Diophantine equation, 306n2 + 9n− 1− 6nk − 2k = 0, (15) where n, k ∈ Z. This Diophantine equation is of the form, an2 + bnk + ck2 + dn+ ek + f = 0, where a = 306, b = −6, c = 0, d = 9, e = −2, and f = −1. Using the transformation stated in Lemma 1, equation (15) can be reduced to X2 − 36Y2 = 1321920. (16) where Y = 2an+ bk + d = 612n− 6k + 9 and X = Dk + E = 36k + 1170. By factoring the left side of the equation (16), we will have, (X + (−6)Y )(X − (−6)Y ) = 1321920, where b = −6. Therefore, by Lemma 1 and Remark 2, we have the following: X = N1 +N2 2 , Y = N1 −N2 −12 , k = X − 1170 36 , n = Y + 6k − 9 612 , (17) where (N1, N2) are the factor pairs of N = 1321920. Using (17), we can now solve for the integer values of n and k, by considering all the factor pairs of 1321920, as presented in Table 4 in the Appendices section. Table 4 shows that there are 2 possible solutions for the Diophantine equation 306n2 + 9n−1−6nk−2k = 0. These are: (n,k)= (3,139), and (-1,-74). However, since we are only concern with possible values of n for S(RWWn), then only (3,139), where n = 3, is a valid solution under this conditions. Therefore, we have shown that S(RWW3) is the only splitting graph of right water wheel graph that satisfies the Lo’s Theorem. Similar to Theorem 5, S(RWW3) is a very large graph with 20 vertices and 54 edges. By permutation, there is a total of 54! possible labeling for its edges. Thus, the researchers created and used a Python computer program to generate an edge-graceful labeling for the said graph. One of the results from the computer program is the labeling shown in Figure 8. A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 15 of 33 Figure 8: An Edge-graceful Labeling of S(RWW3) As shown in Figure 8, the 54 edges of S(RWW3) are labeled using distinct integers from 1 to 54. Then, by Definition 1, the 20 vertices are successfully labeled using distinct integers from 0 to 19. Therefore, the labeled graph in Figure 8 is indeed an edge-graceful labeling of S(RWW3). Since all the needed conditions are satisfied, then we can now conclude that S(RWW3) is the only edge-graceful splitting graph of right water wheel graphs. Theorem 7. Among all splitting graphs of left water wheel graphs, only S(LWW3) is edge-graceful. Proof. First, using Definiton 4 and Proposition 1, the splitting graph S(LWWn) also has 6n+ 2 vertices and 18n edges, the same as S(RWWn). Since the order and size of S(RWWn) and S(LWWn) are the same, then the proof of Theorem 7 will just be similar to the proof of Theorem 6. A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 16 of 33 With the proof of Theorem 6, we have also shown that S(LWW3) is the only splitting graph of left water wheel graph that satisfies the Lo’s Theorem. Similar to the preceding theorems, an actual edge-graceful labeling of S(LWW3) must be provided, in order to complete this proof. The same with S(RWW3), splitting graph S(LWW3) also has 20 vertices and 54 edges. Thus, a Python computer program was created and used by the researchers to generate an edge-graceful labeling of the said graph. One of the results from the computer program is the labeling shown in Figure 9. Figure 9: An Edge-graceful Labeling of S(LWW3) As shown in Figure 9, the 54 edges of S(LWW3) are labeled using distinct integers from 1 to 54. Then, by Definition 1, the 20 vertices are successfully labeled using distinct integers from 0 to 19. Therefore, the labeled graph in Figure 9 is indeed an edge-graceful labeling of S(LWW3). Since all the needed conditions are satisfied, then we can now conclude that S(LWW3) is the only edge-graceful splitting graph of left water wheel graphs. A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 17 of 33 4. Conclusion In this paper, two new graphs are introduced, namely the right water wheel graph RWWn and the left water wheel graph LWWn. Using Lo’s Theorem, and the concepts of divisibility and Diophantine equations, we have proven the non-edge gracefulness of water wheel graphs WWn, right water wheel graphs RWWn, and left water wheel graphs LWWn. Using similar concepts and custom-developed Python computer programs, we have also successfully investigated the edge-gracefulness of the splitting graphs of water wheel, right water wheel, and left water wheel graphs. In Theorem 5, we proved that S(WW3) is the only edge-graceful splitting graph of water wheel graphs. In Theorem 6, we showed that S(RWW3) is the only edge-graceful splitting graph of right water wheel graphs. Finally, we proved in Theorem 7 that S(LWW3) is the only edge-graceful splitting graph of left water wheel graphs. Acknowledgements The completion of this paper would not be possible without the help and support of the following people and organizations: Central Luzon State University (CLSU), CLSU, College of Science, Department of Mathematics and Physics, Department of Science and Technology-Science Education Institute (DOST-SEI), so our heartfelt gratitude to them. We also express our gratitude to various reviewers for their comments and suggestions that helped to improve the overall content of this paper. References [1] Joseph A. Gallian. A dynamic survey of graph labeling. Electronic Journal of Com- binatorics, 6(25):4–623, 2022. [2] S. Ramachandran and T. Gnanaseelan. Prime labeling on some cycle related graphs. Advances and Applications in Mathematical Sciences, 21(12):6711–6719, 2022. [3] S. Venkatesan and P. Sekar. Not all wheel graphs were edge-graceful. Journal Home- page: http://www.ijesm.co.in, 6:5, 2017. [4] S.-P. Lo. On edge-graceful labelings of graphs. Congressus Numerantium, 50:231–241, 1985. [5] Q. Kuan, S. Lee, J. Mitchem, and A. Wang. On edge-graceful unicyclic graphs. Congressus Numerantium, 61:65–74, 1988. [6] L. Lee, S. Lee, and G. Muty. On edge-graceful labelings of complete graphs: solutions of lo’s conjecture. Congressus Numerantium, 62:225–233, 1988. [7] D. Small. Regular (even) spider graphs are edge-graceful. Congressus Numerantium, 74:247–254, 1990. [8] S. Cabaniss, R. Low, and J. Mitchem. On edge-graceful regular graphs and trees. Ars Combinatoria, 34:129–142, 1992. [9] A. Angel, J. L. Gamurot, J. R. Antalan, and R. Tagle. On edge-gracefulness of wheel A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 18 of 33 graphs. Communications in Combinatorics, Cryptography & Computer Science, pages 213–219, 2024. [10] Jonathan L. Gross, Jay Yellen, and Mark Anderson. Graph theory and its applications. Chapman and Hall/CRC, 2018. [11] Mohammed Aljohani and Salama Nagy Daoud. The edge odd graceful labeling of water wheel graphs. Axioms, 14(1):5, 2024. [12] E. Sampathkumar and H. B. Walikar. On the splitting graph of a graph. Karnatak University Science, 25(13):13–16, 1980. [13] David Burton. Ebook: Elementary number theory. McGraw Hill, 2010. [14] Aaron DC Angel, John Rafael M. Antalan, John Loureynz F. Gamurot, and Richard P. Tagle. Edge-graceful usual fan graphs, 2024. arXiv preprint arXiv:2412.08338. [15] Bal Bahadur Tamang. On the study of quadratic Diophantine equations. PhD thesis, Tribhuvan University Kathmandu, 2021. A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 19 of 33 Appendices This section shows all the numerical tables and computer programs that are used for this study. We begin by presenting the numerical tables used in this study. These tables show the complete set of solutions to the Diophantine equations that are involved in the study. Each table corresponds to a particular Diophantine equation associated with the main theorems, providing detailed numerical evidence that supports the theoretical results es- tablished in the earlier sections. Presented below, is the complete table of solutions for the Diophantine equation 41n2 + 7n− 6nk − 2k = 0, relevant to Theorem 2. Table 1: Solutions for 41n2 + 7n− 6nk − 2k = 0 N1 N2 X Y n k N1 N2 X Y n k 1 13120 6560.5 1093.25 26.33 178.85 -1 -13120 -6560.5 -1093.25 -27 -185.63 2 6560 3281 546.5 13 87.75 -2 -6560 -3281 -546.5 -13.67 -94.53 4 3280 1642 273 6.33 42.22 -4 -3280 -1642 -273 -7 -49 5 2624 1314.5 218.25 5 33.13 -5 -2624 -1314.5 -218.25 -5.67 -39.90 8 1640 824 136 3 19.5 -8 -1640 -824 -136 -3.67 -26.28 10 1312 661 108.5 2.33 14.97 -10 -1312 -661 -108.5 -3 -21.75 16 820 418 67 1.33 8.22 -16 -820 -418 -67 -2 -15 20 656 338 53 1 6 -20 -656 -338 -53 -1.67 -12.78 32 410 221 31.5 0.5 2.75 -32 -410 -221 -31.5 -1.17 -9.53 40 328 184 24 0.33 1.72 -40 -328 -184 -24 -1 -8.5 41 320 180.5 23.25 0.32 1.63 -41 -320 -180.5 -23.25 -0.98 -8.40 64 205 134.5 11.75 0.08 0.35 -64 -205 -134.5 -11.75 -0.75 -7.13 80 164 122 7 0 0 -80 -164 -122 -7 -0.67 -6.78 82 160 121 6.5 -0.01 -0.03 -82 -160 -121 -6.5 -0.66 -6.75 13120 1 6560.5 -1093.25 -0.33 178.85 -13120 -1 -6560.5 1093.25 -0.34 -185.63 6560 2 3281 -546.5 -0.33 87.75 -6560 -2 -3281 546.5 -0.34 -94.53 3280 4 1642 -273 -0.33 42.22 -3280 -4 -1642 273 -0.34 -49 2624 5 1314.5 -218.25 -0.32 33.13 -2624 -5 -1314.5 218.25 -0.34 -39.90 1640 8 824 -136 -0.32 19.5 -1640 -8 -824 136 -0.35 -26.28 1312 10 661 -108.5 -0.31 14.97 -1312 -10 -661 108.5 -0.35 -21.75 820 16 418 -67 -0.30 8.22 -820 -16 -418 67 -0.37 -15 656 20 338 -53 -0.29 6 -656 -20 -338 53 -0.37 -12.78 410 32 221 -31.5 -0.27 2.75 -410 -32 -221 31.5 -0.40 -9.53 328 40 184 -24 -0.25 1.72 -328 -40 -184 24 -0.41 -8.5 320 41 180.5 -23.25 -0.25 1.63 -320 -41 -180.5 23.25 -0.42 -8.40 205 64 134.5 -11.75 -0.20 0.35 -205 -64 -134.5 11.75 -0.46 -7.13 164 80 122 -7 -0.17 0 -164 -80 -122 7 -0.50 -6.78 160 82 121 -6.5 -0.17 -0.03 -160 -82 -121 6.5 -0.5 -6.75 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 20 of 33 Presented below, is the complete table of solutions for the Diophantine equation 63n2 + 9n− 6nk − 2k = 0, relevant to Theorem 3. Table 2: Solutions for 63n2 + 9n− 6nk − 2k = 0 N1 N2 X Y n k N1 N2 X Y n k 1 36288 18144.5 3023.92 47.67 498.51 -1 -36288 -18144.5 -3023.92 -48.33 -509.51 2 18144 9073 1511.83 23.67 246.53 -2 -18144 -9073 -1511.83 -24.33 -257.53 3 12096 6049.5 1007.75 15.67 162.54 -3 -12096 -6049.5 -1007.75 -16.33 -173.54 4 9072 4538 755.67 11.67 120.56 -4 -9072 -4538 -755.67 -12.33 -131.56 6 6048 3027 503.5 7.67 78.58 -6 -6048 -3027 -503.5 -8.33 -89.58 7 5184 2595.5 431.42 6.52 66.60 -7 -5184 -2595.5 -431.42 -7.19 -77.60 8 4536 2272 377.33 5.67 57.61 -8 -4536 -2272 -377.33 -6.33 -68.61 9 4032 2020.5 335.25 5 50.63 -9 -4032 -2020.5 -335.25 -5.67 -61.63 12 3024 1518 251 3.67 36.67 -12 -3024 -1518 -251 -4.33 -47.67 14 2592 1303 214.83 3.10 30.69 -14 -2592 -1303 -214.83 -3.76 -41.69 16 2268 1142 187.67 2.67 26.22 -16 -2268 -1142 -187.67 -3.33 -37.22 18 2016 1017 166.5 2.33 22.75 -18 -2016 -1017 -166.5 -3 -33.75 21 1728 874.5 142.25 1.95 18.79 -21 -1728 -874.5 -142.25 -2.62 -29.79 24 1512 768 124 1.67 15.83 -24 -1512 -768 -124 -2.33 -26.83 27 1344 685.5 109.75 1.44 13.54 -27 -1344 -685.5 -109.75 -2.11 -24.54 28 1296 662 105.67 1.38 12.89 -28 -1296 -662 -105.67 -2.05 -23.89 32 1134 583 91.83 1.17 10.69 -32 -1134 -583 -91.83 -1.83 -21.69 36 1008 522 81 1 9 -36 -1008 -522 -81 -1.67 -20 42 864 453 68.5 0.81 7.08 -42 -864 -453 -68.5 -1.48 -18.08 48 756 402 59 0.67 5.67 -48 -756 -402 -59 -1.33 -16.67 54 672 363 51.5 0.56 4.58 -54 -672 -363 -51.5 -1.22 -15.58 56 648 352 49.33 0.52 4.28 -56 -648 -352 -49.33 -1.19 -15.28 63 576 319.5 42.75 0.43 3.38 -63 -576 -319.5 -42.75 -1.10 -14.38 64 567 315.5 41.92 0.42 3.26 -64 -567 -315.5 -41.92 -1.08 -14.26 72 504 288 36 0.33 2.5 -72 -504 -288 -36 -1 -13.5 81 448 264.5 30.58 0.26 1.85 -81 -448 -264.5 -30.58 -0.93 -12.85 84 432 258 29 0.24 1.67 -84 -432 -258 -29 -0.90 -12.67 96 378 237 23.5 0.17 1.08 -96 -378 -237 -23.5 -0.83 -12.08 108 336 222 19 0.11 0.67 -108 -336 -222 -19 -0.78 -11.67 112 324 218 17.67 0.10 0.56 -112 -324 -218 -17.67 -0.76 -11.56 126 288 207 13.5 0.05 0.25 -126 -288 -207 -13.5 -0.71 -11.25 144 252 198 9 0 0 -144 -252 -198 -9 -0.67 -11 162 224 193 5.17 -0.04 -0.14 -162 -224 -193 -5.17 -0.63 -10.86 168 216 192 4 -0.05 -0.17 -168 -216 -192 -4 -0.62 -10.83 189 192 190.5 0.25 -0.08 -0.21 -189 -192 -190.5 -0.25 -0.59 -10.79 36288 1 18144.5 -3023.92 -0.33 498.51 -36288 -1 -18144.5 3023.92 -0.33 -509.51 18144 2 9073 -1511.83 -0.33 246.53 -18144 -2 -9073 1511.83 -0.34 -257.53 12096 3 6049.5 -1007.75 -0.33 162.54 -12096 -3 -6049.5 1007.75 -0.34 -173.54 9072 4 4538 -755.67 -0.33 120.56 -9072 -4 -4538 755.67 -0.34 -131.56 6048 6 3027 -503.5 -0.33 78.58 -6048 -6 -3027 503.5 -0.34 -89.58 5184 7 2595.5 -431.42 -0.32 66.60 -5184 -7 -2595.5 431.42 -0.34 -77.60 4536 8 2272 -377.33 -0.32 57.61 -4536 -8 -2272 377.33 -0.34 -68.61 4032 9 2020.5 -335.25 -0.32 50.63 -4032 -9 -2020.5 335.25 -0.35 -61.63 3024 12 1518 -251 -0.32 36.67 -3024 -12 -1518 251 -0.35 -47.67 2592 14 1303 -214.83 -0.31 30.69 -2592 -14 -1303 214.83 -0.35 -41.69 2268 16 1142 -187.67 -0.31 26.22 -2268 -16 -1142 187.67 -0.35 -37.22 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 21 of 33 2016 18 1017 -166.5 -0.31 22.75 -2016 -18 -1017 166.5 -0.36 -33.75 1728 21 874.5 -142.25 -0.31 18.79 -1728 -21 -874.5 142.25 -0.36 -29.79 1512 24 768 -124 -0.30 15.83 -1512 -24 -768 124 -0.37 -26.83 1344 27 685.5 -109.75 -0.30 13.54 -1344 -27 -685.5 109.75 -0.37 -24.54 1296 28 662 -105.67 -0.30 12.89 -1296 -28 -662 105.67 -0.37 -23.89 1134 32 583 -91.83 -0.29 10.69 -1134 -32 -583 91.83 -0.38 -21.69 1008 36 522 -81 -0.29 9 -1008 -36 -522 81 -0.38 -20 864 42 453 -68.5 -0.28 7.08 -864 -42 -453 68.5 -0.39 -18.08 756 48 402 -59 -0.27 5.67 -756 -48 -402 59 -0.40 -16.67 672 54 363 -51.5 -0.26 4.58 -672 -54 -363 51.5 -0.40 -15.58 648 56 352 -49.33 -0.26 4.28 -648 -56 -352 49.33 -0.41 -15.28 576 63 319.5 -42.75 -0.25 3.38 -576 -63 -319.5 42.75 -0.42 -14.38 567 64 315.5 -41.92 -0.25 3.26 -567 -64 -315.5 41.92 -0.42 -14.26 504 72 288 -36 -0.24 2.5 -504 -72 -288 36 -0.43 -13.5 448 81 264.5 -30.58 -0.23 1.85 -448 -81 -264.5 30.58 -0.44 -12.85 432 84 258 -29 -0.22 1.67 -432 -84 -258 29 -0.44 -12.67 378 96 237 -23.5 -0.21 1.08 -378 -96 -237 23.5 -0.46 -12.08 336 108 222 -19 -0.19 0.67 -336 -108 -222 19 -0.48 -11.67 324 112 218 -17.67 -0.19 0.56 -324 -112 -218 17.67 -0.48 -11.56 288 126 207 -13.5 -0.17 0.25 -288 -126 -207 13.5 -0.5 -11.25 252 144 198 -9 -0.14 0 -252 -144 -198 9 -0.52 -11 224 162 193 -5.17 -0.12 -0.14 -224 -162 -193 5.17 -0.55 -10.86 216 168 192 -4 -0.11 -0.17 -216 -168 -192 4 -0.56 -10.83 192 189 190.5 -0.25 -0.08 -0.21 -192 -189 -190.5 0.25 -0.58 -10.79 Presented below, is the complete table of solutions for the Diophantine equation 207n2 + 6n− 1− 6nk − 2k = 0, relevant to Theorem 5. Table 3: Solutions for 207n2 + 6n− 1− 6nk − 2k = 0 N1 N2 X Y n k N1 N2 X Y n k 1 596160 298080.5 49679.92 239.67 8258.01 -1 -596160 -298080.5 -49679.92 -240.33 -8302.01 2 298080 149041 24839.83 119.67 4118.03 -2 -298080 -149041 -24839.83 -120.33 -4162.03 3 198720 99361.5 16559.75 79.67 2738.04 -3 -198720 -99361.5 -16559.75 -80.33 -2782.04 4 149040 74522 12419.67 59.67 2048.06 -4 -149040 -74522 -12419.67 -60.33 -2092.06 5 119232 59618.5 9935.58 47.67 1634.07 -5 -119232 -59618.5 -9935.58 -48.33 -1678.07 6 99360 49683 8279.5 39.67 1358.08 -6 -99360 -49683 -8279.5 -40.33 -1402.08 8 74520 37264 6209.33 29.67 1013.11 -8 -74520 -37264 -6209.33 -30.33 -1057.11 9 66240 33124.5 5519.25 26.33 898.13 -9 -66240 -33124.5 -5519.25 -27 -942.13 10 59616 29813 4967.17 23.67 806.14 -10 -59616 -29813 -4967.17 -24.33 -850.14 12 49680 24846 4139 19.67 668.17 -12 -49680 -24846 -4139 -20.33 -712.17 15 39744 19879.5 3310.75 15.67 530.21 -15 -39744 -19879.5 -3310.75 -16.33 -574.21 16 37260 18638 3103.67 14.67 495.72 -16 -37260 -18638 -3103.67 -15.33 -539.72 18 33120 16569 2758.5 13 438.25 -18 -33120 -16569 -2758.5 -13.67 -482.25 20 29808 14914 2482.33 11.67 392.28 -20 -29808 -14914 -2482.33 -12.33 -436.28 23 25920 12971.5 2158.08 10.10 338.32 -23 -25920 -12971.5 -2158.08 -10.77 -382.32 24 24840 12432 2068 9.67 323.33 -24 -24840 -12432 -2068 -10.33 -367.33 27 22080 11053.5 1837.75 8.56 285.04 -27 -22080 -11053.5 -1837.75 -9.22 -329.04 30 19872 9951 1653.5 7.67 254.42 -30 -19872 -9951 -1653.5 -8.33 -298.42 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 22 of 33 32 18630 9331 1549.83 7.17 237.19 -32 -18630 -9331 -1549.83 -7.83 -281.19 36 16560 8298 1377 6.33 208.5 -36 -16560 -8298 -1377 -7 -252.5 40 14904 7472 1238.67 5.67 185.56 -40 -14904 -7472 -1238.67 -6.33 -229.56 45 13248 6646.5 1100.25 5 162.63 -45 -13248 -6646.5 -1100.25 -5.67 -206.63 46 12960 6503 1076.17 4.88 158.64 -46 -12960 -6503 -1076.17 -5.55 -202.64 48 12420 6234 1031 4.67 151.17 -48 -12420 -6234 -1031 -5.33 -195.17 54 11040 5547 915.5 4.11 132.08 -54 -11040 -5547 -915.5 -4.78 -176.08 60 9936 4998 823 3.67 116.83 -60 -9936 -4998 -823 -4.33 -160.83 64 9315 4689.5 770.92 3.42 108.26 -64 -9315 -4689.5 -770.92 -4.08 -152.26 69 8640 4354.5 714.25 3.14 98.96 -69 -8640 -4354.5 -714.25 -3.81 -142.96 72 8280 4176 684 3 94 -72 -8280 -4176 -684 -3.67 -138 80 7452 3766 614.33 2.67 82.61 -80 -7452 -3766 -614.33 -3.33 -126.61 81 7360 3720.5 606.58 2.63 81.35 -81 -7360 -3720.5 -606.58 -3.30 -125.35 90 6624 3357 544.5 2.33 71.25 -90 -6624 -3357 -544.5 -3 -115.25 92 6480 3286 532.33 2.28 69.28 -92 -6480 -3286 -532.33 -2.94 -113.28 96 6210 3153 509.5 2.17 65.58 -96 -6210 -3153 -509.5 -2.83 -109.58 108 5520 2814 451 1.89 56.17 -108 -5520 -2814 -451 -2.56 -100.17 115 5184 2649.5 422.42 1.75 51.60 -115 -5184 -2649.5 -422.42 -2.42 -95.60 120 4968 2544 404 1.67 48.67 -120 -4968 -2544 -404 -2.33 -92.67 135 4416 2275.5 356.75 1.44 41.21 -135 -4416 -2275.5 -356.75 -2.11 -85.21 138 4320 2229 348.5 1.41 39.92 -138 -4320 -2229 -348.5 -2.07 -83.92 144 4140 2142 333 1.33 37.5 -144 -4140 -2142 -333 -2 -81.5 160 3726 1943 297.17 1.17 31.97 -160 -3726 -1943 -297.17 -1.83 -75.97 162 3680 1921 293.17 1.15 31.36 -162 -3680 -1921 -293.17 -1.81 -75.36 180 3312 1746 261 1 26.5 -180 -3312 -1746 -261 -1.67 -70.5 184 3240 1712 254.67 0.97 25.56 -184 -3240 -1712 -254.67 -1.64 -69.56 192 3105 1648.5 242.75 0.92 23.79 -192 -3105 -1648.5 -242.75 -1.58 -67.79 207 2880 1543.5 222.75 0.83 20.88 -207 -2880 -1543.5 -222.75 -1.49 -64.88 216 2760 1488 212 0.78 19.33 -216 -2760 -1488 -212 -1.44 -63.33 230 2592 1411 196.83 0.71 17.19 -230 -2592 -1411 -196.83 -1.38 -61.19 240 2484 1362 187 0.67 15.83 -240 -2484 -1362 -187 -1.33 -59.83 270 2208 1239 161.5 0.56 12.42 -270 -2208 -1239 -161.5 -1.22 -56.42 276 2160 1218 157 0.54 11.83 -276 -2160 -1218 -157 -1.20 -55.83 288 2070 1179 148.5 0.5 10.75 -288 -2070 -1179 -148.5 -1.17 -54.75 320 1863 1091.5 128.58 0.42 8.32 -320 -1863 -1091.5 -128.58 -1.08 -52.32 324 1840 1082 126.33 0.41 8.06 -324 -1840 -1082 -126.33 -1.07 -52.06 345 1728 1036.5 115.25 0.36 6.79 -345 -1728 -1036.5 -115.25 -1.03 -50.79 360 1656 1008 108 0.33 6 -360 -1656 -1008 -108 -1 -50 368 1620 994 104.33 0.32 5.61 -368 -1620 -994 -104.33 -0.99 -49.61 405 1472 938.5 88.92 0.26 4.07 -405 -1472 -938.5 -88.92 -0.93 -48.07 414 1440 927 85.5 0.25 3.75 -414 -1440 -927 -85.5 -0.91 -47.75 432 1380 906 79 0.22 3.17 -432 -1380 -906 -79 -0.89 -47.17 460 1296 878 69.67 0.19 2.39 -460 -1296 -878 -69.67 -0.86 -46.39 480 1242 861 63.5 0.17 1.92 -480 -1242 -861 -63.5 -0.83 -45.92 540 1104 822 47 0.11 0.83 -540 -1104 -822 -47 -0.78 -44.83 552 1080 816 44 0.10 0.67 -552 -1080 -816 -44 -0.77 -44.67 576 1035 805.5 38.25 0.08 0.38 -576 -1035 -805.5 -38.25 -0.75 -44.38 621 960 790.5 28.25 0.05 -0.04 -621 -960 -790.5 -28.25 -0.72 -43.96 648 920 784 22.67 0.04 -0.22 -648 -920 -784 -22.67 -0.70 -43.78 690 864 777 14.5 0.01 -0.42 -690 -864 -777 -14.5 -0.68 -43.58 720 828 774 9 0 -0.5 -720 -828 -774 -9 -0.67 -43.5 736 810 773 6.17 -0.01 -0.53 -736 -810 -773 -6.17 -0.66 -43.47 596160 1 298080.5 -49679.92 -0.33 8258.01 -596160 -1 -298080.5 49679.92 -0.33 -8302.01 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 23 of 33 298080 2 149041 -24839.83 -0.33 4118.03 -298080 -2 -149041 24839.83 -0.33 -4162.03 198720 3 99361.5 -16559.75 -0.33 2738.04 -198720 -3 -99361.5 16559.75 -0.33 -2782.04 149040 4 74522 -12419.67 -0.33 2048.06 -149040 -4 -74522 12419.67 -0.33 -2092.06 119232 5 59618.5 -9935.58 -0.33 1634.07 -119232 -5 -59618.5 9935.58 -0.34 -1678.07 99360 6 49683 -8279.5 -0.33 1358.08 -99360 -6 -49683 8279.5 -0.34 -1402.08 74520 8 37264 -6209.33 -0.33 1013.11 -74520 -8 -37264 6209.33 -0.34 -1057.11 66240 9 33124.5 -5519.25 -0.33 898.13 -66240 -9 -33124.5 5519.25 -0.34 -942.13 59616 10 29813 -4967.17 -0.33 806.14 -59616 -10 -29813 4967.17 -0.34 -850.14 49680 12 24846 -4139 -0.33 668.17 -49680 -12 -24846 4139 -0.34 -712.17 39744 15 19879.5 -3310.75 -0.33 530.21 -39744 -15 -19879.5 3310.75 -0.34 -574.21 37260 16 18638 -3103.67 -0.33 495.72 -37260 -16 -18638 3103.67 -0.34 -539.72 33120 18 16569 -2758.5 -0.33 438.25 -33120 -18 -16569 2758.5 -0.34 -482.25 29808 20 14914 -2482.33 -0.33 392.28 -29808 -20 -14914 2482.33 -0.34 -436.28 25920 23 12971.5 -2158.08 -0.32 338.32 -25920 -23 -12971.5 2158.08 -0.34 -382.32 24840 24 12432 -2068 -0.32 323.33 -24840 -24 -12432 2068 -0.34 -367.33 22080 27 11053.5 -1837.75 -0.32 285.04 -22080 -27 -11053.5 1837.75 -0.34 -329.04 19872 30 9951 -1653.5 -0.32 254.42 -19872 -30 -9951 1653.5 -0.35 -298.42 18630 32 9331 -1549.83 -0.32 237.19 -18630 -32 -9331 1549.83 -0.35 -281.19 16560 36 8298 -1377 -0.32 208.5 -16560 -36 -8298 1377 -0.35 -252.5 14904 40 7472 -1238.67 -0.32 185.56 -14904 -40 -7472 1238.67 -0.35 -229.56 13248 45 6646.5 -1100.25 -0.32 162.63 -13248 -45 -6646.5 1100.25 -0.35 -206.63 12960 46 6503 -1076.17 -0.31 158.64 -12960 -46 -6503 1076.17 -0.35 -202.64 12420 48 6234 -1031 -0.31 151.17 -12420 -48 -6234 1031 -0.35 -195.17 11040 54 5547 -915.5 -0.31 132.08 -11040 -54 -5547 915.5 -0.36 -176.08 9936 60 4998 -823 -0.31 116.83 -9936 -60 -4998 823 -0.36 -160.83 9315 64 4689.5 -770.92 -0.31 108.26 -9315 -64 -4689.5 770.92 -0.36 -152.26 8640 69 4354.5 -714.25 -0.31 98.96 -8640 -69 -4354.5 714.25 -0.36 -142.96 8280 72 4176 -684 -0.30 94 -8280 -72 -4176 684 -0.36 -138 7452 80 3766 -614.33 -0.30 82.61 -7452 -80 -3766 614.33 -0.37 -126.61 7360 81 3720.5 -606.58 -0.30 81.35 -7360 -81 -3720.5 606.58 -0.37 -125.35 6624 90 3357 -544.5 -0.30 71.25 -6624 -90 -3357 544.5 -0.37 -115.25 6480 92 3286 -532.33 -0.30 69.28 -6480 -92 -3286 532.33 -0.37 -113.28 6210 96 3153 -509.5 -0.29 65.58 -6210 -96 -3153 509.5 -0.37 -109.58 5520 108 2814 -451 -0.29 56.17 -5520 -108 -2814 451 -0.38 -100.17 5184 115 2649.5 -422.42 -0.29 51.60 -5184 -115 -2649.5 422.42 -0.38 -95.60 4968 120 2544 -404 -0.29 48.67 -4968 -120 -2544 404 -0.38 -92.67 4416 135 2275.5 -356.75 -0.28 41.21 -4416 -135 -2275.5 356.75 -0.39 -85.21 4320 138 2229 -348.5 -0.28 39.92 -4320 -138 -2229 348.5 -0.39 -83.92 4140 144 2142 -333 -0.28 37.5 -4140 -144 -2142 333 -0.39 -81.5 3726 160 1943 -297.17 -0.27 31.97 -3726 -160 -1943 297.17 -0.40 -75.97 3680 162 1921 -293.17 -0.27 31.36 -3680 -162 -1921 293.17 -0.40 -75.36 3312 180 1746 -261 -0.26 26.5 -3312 -180 -1746 261 -0.41 -70.5 3240 184 1712 -254.67 -0.26 25.56 -3240 -184 -1712 254.67 -0.41 -69.56 3105 192 1648.5 -242.75 -0.26 23.79 -3105 -192 -1648.5 242.75 -0.41 -67.79 2880 207 1543.5 -222.75 -0.25 20.88 -2880 -207 -1543.5 222.75 -0.42 -64.88 2760 216 1488 -212 -0.25 19.33 -2760 -216 -1488 212 -0.42 -63.33 2592 230 1411 -196.83 -0.24 17.19 -2592 -230 -1411 196.83 -0.43 -61.19 2484 240 1362 -187 -0.24 15.83 -2484 -240 -1362 187 -0.43 -59.83 2208 270 1239 -161.5 -0.22 12.42 -2208 -270 -1239 161.5 -0.44 -56.42 2160 276 1218 -157 -0.22 11.83 -2160 -276 -1218 157 -0.44 -55.83 2070 288 1179 -148.5 -0.22 10.75 -2070 -288 -1179 148.5 -0.45 -54.75 1863 320 1091.5 -128.58 -0.20 8.32 -1863 -320 -1091.5 128.58 -0.46 -52.32 1840 324 1082 -126.33 -0.20 8.06 -1840 -324 -1082 126.33 -0.46 -52.06 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 24 of 33 1728 345 1036.5 -115.25 -0.19 6.79 -1728 -345 -1036.5 115.25 -0.47 -50.79 1656 360 1008 -108 -0.19 6 -1656 -360 -1008 108 -0.48 -50 1620 368 994 -104.33 -0.19 5.61 -1620 -368 -994 104.33 -0.48 -49.61 1472 405 938.5 -88.92 -0.17 4.07 -1472 -405 -938.5 88.92 -0.50 -48.07 1440 414 927 -85.5 -0.17 3.75 -1440 -414 -927 85.5 -0.5 -47.75 1380 432 906 -79 -0.16 3.17 -1380 -432 -906 79 -0.51 -47.17 1296 460 878 -69.67 -0.15 2.39 -1296 -460 -878 69.67 -0.52 -46.39 1242 480 861 -63.5 -0.14 1.92 -1242 -480 -861 63.5 -0.53 -45.92 1104 540 822 -47 -0.12 0.83 -1104 -540 -822 47 -0.55 -44.83 1080 552 816 -44 -0.11 0.67 -1080 -552 -816 44 -0.56 -44.67 1035 576 805.5 -38.25 -0.10 0.38 -1035 -576 -805.5 38.25 -0.57 -44.38 960 621 790.5 -28.25 -0.08 -0.04 -960 -621 -790.5 28.25 -0.58 -43.96 920 648 784 -22.67 -0.07 -0.22 -920 -648 -784 22.67 -0.59 -43.78 864 690 777 -14.5 -0.06 -0.42 -864 -690 -777 14.5 -0.61 -43.58 828 720 774 -9 -0.04 -0.5 -828 -720 -774 9 -0.62 -43.5 810 736 773 -6.17 -0.04 -0.53 -810 -736 -773 6.17 -0.63 -43.47 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 25 of 33 Presented below, is the complete table of solutions for the Diophantine equation 306n2 + 9n− 1− 6nk − 2k = 0, relevant to Theorem 6. Table 4: Solutions for 306n2 + 9n− 1− 6nk − 2k = 0 N1 N2 X Y n k N1 N2 X Y n k 1 1321920 660960.5 110159.9 359.7 18327.5 -1 -1321920 -660960.5 -110159.9 -360.3 -18392.5 2 660960 330481 55079.8 179.7 9147.5 -2 -660960 -330481 -55079.8 -180.3 -9212.5 3 440640 220321.5 36719.8 119.7 6087.54 -3 -440640 -220321.5 -36719.8 -120.3 -6152.5 4 330480 165242 27539.7 89.7 4557.56 -4 -330480 -165242 -27539.7 -90.3 -4622.6 5 264384 132194.5 22031.6 71.7 3639.57 -5 -264384 -132194.5 -22031.6 -72.3 -3704.6 6 220320 110163 18359.5 59.7 3027.58 -6 -220320 -110163 -18359.5 -60.3 -3092.6 8 165240 82624 13769.3 44.7 2262.61 -8 -165240 -82624 -13769.3 -45.3 -2327.6 9 146880 73444.5 12239.3 39.7 2007.63 -9 -146880 -73444.5 -12239.3 -40.3 -2072.6 10 132192 66101 11015.2 35.7 1803.64 -10 -132192 -66101 -11015.2 -36.3 -1868.6 12 110160 55086 9179 29.7 1497.67 -12 -110160 -55086 -9179 -30.3 -1562.7 15 88128 44071.5 7342.8 23.7 1191.71 -15 -88128 -44071.5 -7342.8 -24.3 -1256.7 16 82620 41318 6883.7 22.2 1115.22 -16 -82620 -41318 -6883.7 -22.8 -1180.2 17 77760 38888.5 6478.6 20.8 1047.74 -17 -77760 -38888.5 -6478.6 -21.5 -1112.7 18 73440 36729 6118.5 19.7 987.75 -18 -73440 -36729 -6118.5 -20.3 -1052.75 20 66096 33058 5506.3 17.7 885.78 -20 -66096 -33058 -5506.3 -18.3 -950.8 24 55080 27552 4588 14.7 732.83 -24 -55080 -27552 -4588 -15.3 -797.8 27 48960 24493.5 4077.8 13 647.88 -27 -48960 -24493.5 -4077.75 -13.7 -712.9 30 44064 22047 3669.5 11.7 579.92 -30 -44064 -22047 -3669.5 -12.3 -644.9 32 41310 20671 3439.8 10.9 541.69 -32 -41310 -20671 -3439.8 -11.6 -606.7 34 38880 19457 3237.2 10.3 507.97 -34 -38880 -19457 -3237.2 -10.9 -572.97 36 36720 18378 3057 9.7 478 -36 -36720 -18378 -3057 -10.3 -543 40 33048 16544 2750.7 8.7 427.06 -40 -33048 -16544 -2750.7 -9.3 -492.06 45 29376 14710.5 2444.3 7.7 376.13 -45 -29376 -14710.5 -2444.3 -8.3 -441.13 48 27540 13794 2291 7.2 350.67 -48 -27540 -13794 -2291 -7.8 -415.67 51 25920 12985.5 2155.8 6.7 328.21 -51 -25920 -12985.5 -2155.8 -7.4 -393.21 54 24480 12267 2035.5 6.3 308.25 -54 -24480 -12267 -2035.5 -7 -373.25 60 22032 11046 1831 5.7 274.33 -60 -22032 -11046 -1831 -6.3 -339.33 64 20655 10359.5 1715.9 5.3 255.26 -64 -20655 -10359.5 -1715.9 -5.96 -320.26 68 19440 9754 1614.3 5.0 238.44 -68 -19440 -9754 -1614.3 -5.6 -303.44 72 18360 9216 1524 4.7 223.5 -72 -18360 -9216 -1524 -5.3 -288.5 80 16524 8302 1370.3 4.2 198.11 -80 -16524 -8302 -1370.3 -4.8 -263.11 81 16320 8200.5 1353.3 4.1 195.29 -81 -16320 -8200.5 -1353.3 -4.8 -260.29 85 15552 7818.5 1288.9 3.9 184.68 -85 -15552 -7818.5 -1288.9 -4.6 -249.68 90 14688 7389 1216.5 3.7 172.75 -90 -14688 -7389 -1216.5 -4.3 -237.75 96 13770 6933 1139.5 3.4 160.08 -96 -13770 -6933 -1139.5 -4.1 -225.08 102 12960 6531 1071.5 3.2 148.92 -102 -12960 -6531 -1071.5 -3.9 -213.92 108 12240 6174 1011 3 139 -108 -12240 -6174 -1011 -3.7 -204 120 11016 5568 908 2.7 122.17 -120 -11016 -5568 -908 -3.3 -187.17 135 9792 4963.5 804.8 2.3 105.375 -135 -9792 -4963.5 -804.75 -3 -170.38 136 9720 4928 798.7 2.3 104.39 -136 -9720 -4928 -798.7 -2.98 -169.39 144 9180 4662 753 2.2 97 -144 -9180 -4662 -753 -2.83 -162 153 8640 4396.5 707.3 2.02 89.63 -153 -8640 -4396.5 -707.25 -2.69 -154.63 160 8262 4211 675.2 1.9 84.47 -160 -8262 -4211 -675.2 -2.58 -149.47 162 8160 4161 666.5 1.9 83.08 -162 -8160 -4161 -666.5 -2.56 -148.08 170 7776 3973 633.8 1.8 77.86 -170 -7776 -3973 -633.8 -2.45 -142.86 180 7344 3762 597 1.7 72 -180 -7344 -3762 -597 -2.33 -137 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 26 of 33 192 6885 3538.5 557.8 1.5 65.79 -192 -6885 -3538.5 -557.75 -2.21 -130.79 204 6480 3342 523 1.4 60.33 -204 -6480 -3342 -523 -2.10 -125.33 216 6120 3168 492 1.3 55.5 -216 -6120 -3168 -492 -2 -120.5 240 5508 2874 439 1.2 47.33 -240 -5508 -2874 -439 -1.83 -112.33 243 5440 2841.5 433.1 1.1 46.43 -243 -5440 -2841.5 -433.1 -1.81 -111.43 255 5184 2719.5 410.8 1.1 43.04 -255 -5184 -2719.5 -410.75 -1.75 -108.04 270 4896 2583 385.5 1 39.25 -270 -4896 -2583 -385.5 -1.67 -104.25 272 4860 2566 382.3 0.99 38.78 -272 -4860 -2566 -382.3 -1.66 -103.78 288 4590 2439 358.5 0.9 35.25 -288 -4590 -2439 -358.5 -1.58 -100.25 306 4320 2313 334.5 0.8 31.75 -306 -4320 -2313 -334.5 -1.51 -96.75 320 4131 2225.5 317.6 0.8 29.32 -320 -4131 -2225.5 -317.6 -1.46 -94.32 324 4080 2202 313 0.8 28.67 -324 -4080 -2202 -313 -1.44 -93.67 340 3888 2114 295.7 0.7 26.22 -340 -3888 -2114 -295.7 -1.39 -91.22 360 3672 2016 276 0.7 23.5 -360 -3672 -2016 -276 -1.33 -88.5 405 3264 1834.5 238.3 0.6 18.46 -405 -3264 -1834.5 -238.25 -1.22 -83.46 408 3240 1824 236 0.5 18.17 -408 -3240 -1824 -236 -1.22 -83.17 432 3060 1746 219 0.5 16 -432 -3060 -1746 -219 -1.17 -81 459 2880 1669.5 201.8 0.5 13.88 -459 -2880 -1669.5 -201.75 -1.12 -78.88 480 2754 1617 189.5 0.4 12.42 -480 -2754 -1617 -189.5 -1.08 -77.42 486 2720 1603 186.2 0.4 12.03 -486 -2720 -1603 -186.2 -1.07 -77.03 510 2592 1551 173.5 0.4 10.58 -510 -2592 -1551 -173.5 -1.04 -75.58 540 2448 1494 159 0.3 9 -540 -2448 -1494 -159 -1 -74 544 2430 1487 157.2 0.3 8.81 -544 -2430 -1487 -157.2 -0.995 -73.81 576 2295 1435.5 143.3 0.3 7.38 -576 -2295 -1435.5 -143.25 -0.96 -72.38 612 2160 1386 129 0.3 6 -612 -2160 -1386 -129 -0.92 -71 648 2040 1344 116 0.2 4.83 -648 -2040 -1344 -116 -0.89 -69.83 680 1944 1312 105.3 0.2 3.94 -680 -1944 -1312 -105.3 -0.86 -68.94 720 1836 1278 93 0.2 3 -720 -1836 -1278 -93 -0.83 -68 765 1728 1246.5 80.3 0.1 2.13 -765 -1728 -1246.5 -80.25 -0.80 -67.13 810 1632 1221 68.5 0.1 1.42 -810 -1632 -1221 -68.5 -0.78 -66.42 816 1620 1218 67 0.1 1.33 -816 -1620 -1218 -67 -0.77 -66.33 864 1530 1197 55.5 0.1 0.75 -864 -1530 -1197 -55.5 -0.75 -65.75 918 1440 1179 43.5 0.1 0.25 -918 -1440 -1179 -43.5 -0.73 -65.25 960 1377 1168.5 34.8 0.04 -0.04 -960 -1377 -1168.5 -34.75 -0.71 -64.96 972 1360 1166 32.3 0.04 -0.11 -972 -1360 -1166 -32.3 -0.70 -64.89 1020 1296 1158 23 0.02 -0.33 -1020 -1296 -1158 -23 -0.69 -64.67 1080 1224 1152 12 0 -0.5 -1080 -1224 -1152 -12 -0.67 -64.5 1088 1215 1151.5 10.6 -0.002 -0.51 -1088 -1215 -1151.5 -10.6 -0.66 -64.49 1321920 1 660960.5 -110159.9 -0.33 18327.5 -1321920 -1 -660960.5 110159.9 -0.33 -18392.51 660960 2 330481 -55079.8 -0.33 9147.53 -660960 -2 -330481 55079.8 -0.33 -9212.53 440640 3 220321.5 -36719.8 -0.33 6087.54 -440640 -3 -220321.5 36719.8 -0.33 -6152.54 330480 4 165242 -27539.7 -0.33 4557.56 -330480 -4 -165242 27539.7 -0.33 -4622.56 264384 5 132194.5 -22031.6 -0.33 3639.57 -264384 -5 -132194.5 22031.6 -0.33 -3704.57 220320 6 110163 -18359.5 -0.33 3027.58 -220320 -6 -110163 18359.5 -0.33 -3092.58 165240 8 82624 -13769.3 -0.33 2262.61 -165240 -8 -82624 13769.3 -0.34 -2327.61 146880 9 73444.5 -12239.3 -0.33 2007.63 -146880 -9 -73444.5 12239.3 -0.34 -2072.63 132192 10 66101 -11015.2 -0.33 1803.64 -132192 -10 -66101 11015.2 -0.34 -1868.64 110160 12 55086 -9179 -0.33 1497.67 -110160 -12 -55086 9179 -0.34 -1562.67 88128 15 44071.5 -7342.8 -0.33 1191.71 -88128 -15 -44071.5 7342.75 -0.34 -1256.71 82620 16 41318 -6883.7 -0.33 1115.22 -82620 -16 -41318 6883.7 -0.34 -1180.22 77760 17 38888.5 -6478.6 -0.33 1047.74 -77760 -17 -38888.5 6478.6 -0.34 -1112.74 73440 18 36729 -6118.5 -0.33 987.75 -73440 -18 -36729 6118.5 -0.34 -1052.75 66096 20 33058 -5506.3 -0.33 885.78 -66096 -20 -33058 5506.3 -0.34 -950.78 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 27 of 33 55080 24 27552 -4588 -0.33 732.83 -55080 -24 -27552 4588 -0.34 -797.83 48960 27 24493.5 -4077.8 -0.33 647.88 -48960 -27 -24493.5 4077.8 -0.34 -712.88 44064 30 22047 -3669.5 -0.33 579.92 -44064 -30 -22047 3669.5 -0.34 -644.92 41310 32 20671 -3439.8 -0.32 541.69 -41310 -32 -20671 3439.8 -0.34 -606.69 38880 34 19457 -3237.2 -0.32 507.97 -38880 -34 -19457 3237.2 -0.34 -572.97 36720 36 18378 -3057 -0.32 478 -36720 -36 -18378 3057 -0.34 -543 33048 40 16544 -2750.7 -0.32 427.06 -33048 -40 -16544 2750.7 -0.34 -492.06 29376 45 14710.5 -2444.3 -0.32 376.13 -29376 -45 -14710.5 2444.3 -0.35 -441.13 27540 48 13794 -2291 -0.32 350.67 -27540 -48 -13794 2291 -0.35 -415.67 25920 51 12985.5 -2155.8 -0.32 328.21 -25920 -51 -12985.5 2155.75 -0.35 -393.21 24480 54 12267 -2035.5 -0.32 308.25 -24480 -54 -12267 2035.5 -0.35 -373.25 22032 60 11046 -1831 -0.32 274.33 -22032 -60 -11046 1831 -0.35 -339.33 20655 64 10359.5 -1715.9 -0.32 255.26 -20655 -64 -10359.5 1715.9 -0.35 -320.26 19440 68 9754 -1614.3 -0.31 238.44 -19440 -68 -9754 1614.3 -0.35 -303.44 18360 72 9216 -1524 -0.31 223.5 -18360 -72 -9216 1524 -0.35 -288.5 16524 80 8302 -1370.3 -0.31 198.11 -16524 -80 -8302 1370.3 -0.36 -263.11 16320 81 8200.5 -1353.3 -0.31 195.29 -16320 -81 -8200.5 1353.25 -0.36 -260.29 15552 85 7818.5 -1288.9 -0.31 184.68 -15552 -85 -7818.5 1288.9 -0.36 -249.68 14688 90 7389 -1216.5 -0.31 172.75 -14688 -90 -7389 1216.5 -0.36 -237.75 13770 96 6933 -1139.5 -0.31 160.08 -13770 -96 -6933 1139.5 -0.36 -225.08 12960 102 6531 -1071.5 -0.31 148.92 -12960 -102 -6531 1071.5 -0.36 -213.92 12240 108 6174 -1011 -0.30 139 -12240 -108 -6174 1011 -0.36 -204 11016 120 5568 -908 -0.30 122.17 -11016 -120 -5568 908 -0.37 -187.17 9792 135 4963.5 -804.8 -0.30 105.38 -9792 -135 -4963.5 804.75 -0.37 -170.38 9720 136 4928 -798.7 -0.30 104.39 -9720 -136 -4928 798.7 -0.37 -169.39 9180 144 4662 -753 -0.29 97 -9180 -144 -4662 753 -0.37 -162 8640 153 4396.5 -707.3 -0.29 89.63 -8640 -153 -4396.5 707.25 -0.38 -154.63 8262 160 4211 -675.2 -0.29 84.47 -8262 -160 -4211 675.2 -0.38 -149.47 8160 162 4161 -666.5 -0.29 83.08 -8160 -162 -4161 666.5 -0.38 -148.08 7776 170 3973 -633.8 -0.29 77.86 -7776 -170 -3973 633.8 -0.38 -142.86 7344 180 3762 -597 -0.28 72 -7344 -180 -3762 597 -0.38 -137 6885 192 3538.5 -557.8 -0.28 65.79 -6885 -192 -3538.5 557.75 -0.39 -130.79 6480 204 3342 -523 -0.28 60.33 -6480 -204 -3342 523 -0.39 -125.33 6120 216 3168 -492 -0.27 55.5 -6120 -216 -3168 492 -0.39 -120.5 5508 240 2874 -439 -0.27 47.33 -5508 -240 -2874 439 -0.40 -112.33 5440 243 2841.5 -433.1 -0.27 46.43 -5440 -243 -2841.5 433.1 -0.40 -111.43 5184 255 2719.5 -410.8 -0.26 43.04 -5184 -255 -2719.5 410.75 -0.40 -108.04 4896 270 2583 -385.5 -0.26 39.25 -4896 -270 -2583 385.5 -0.41 -104.25 4860 272 2566 -382.3 -0.26 38.78 -4860 -272 -2566 382.3 -0.41 -103.78 4590 288 2439 -358.5 -0.25 35.25 -4590 -288 -2439 358.5 -0.41 -100.25 4320 306 2313 -334.5 -0.25 31.75 -4320 -306 -2313 334.5 -0.42 -96.75 4131 320 2225.5 -317.6 -0.25 29.32 -4131 -320 -2225.5 317.6 -0.42 -94.32 4080 324 2202 -313 -0.25 28.67 -4080 -324 -2202 313 -0.42 -93.67 3888 340 2114 -295.7 -0.24 26.22 -3888 -340 -2114 295.7 -0.43 -91.22 3672 360 2016 -276 -0.24 23.5 -3672 -360 -2016 276 -0.43 -88.5 3264 405 1834.5 -238.3 -0.22 18.46 -3264 -405 -1834.5 238.25 -0.44 -83.46 3240 408 1824 -236 -0.22 18.17 -3240 -408 -1824 236 -0.44 -83.17 3060 432 1746 -219 -0.22 16 -3060 -432 -1746 219 -0.45 -81 2880 459 1669.5 -201.8 -0.21 13.88 -2880 -459 -1669.5 201.75 -0.46 -78.88 2754 480 1617 -189.5 -0.20 12.42 -2754 -480 -1617 189.5 -0.46 -77.42 2720 486 1603 -186.2 -0.20 12.03 -2720 -486 -1603 186.2 -0.47 -77.03 2592 510 1551 -173.5 -0.19 10.58 -2592 -510 -1551 173.5 -0.47 -75.58 2448 540 1494 -159 -0.19 9 -2448 -540 -1494 159 -0.48 -74 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 28 of 33 2430 544 1487 -157.2 -0.19 8.81 -2430 -544 -1487 157.2 -0.48 -73.81 2295 576 1435.5 -143.3 -0.18 7.375 -2295 -576 -1435.5 143.25 -0.49 -72.38 2160 612 1386 -129 -0.17 6 -2160 -612 -1386 129 -0.5 -71 2040 648 1344 -116 -0.16 4.83 -2040 -648 -1344 116 -0.51 -69.83 1944 680 1312 -105.3 -0.15 3.94 -1944 -680 -1312 105.3 -0.52 -68.94 1836 720 1278 -93 -0.14 3 -1836 -720 -1278 93 -0.53 -68 1728 765 1246.5 -80.3 -0.13 2.13 -1728 -765 -1246.5 80.25 -0.54 -67.13 1632 810 1221 -68.5 -0.11 1.42 -1632 -810 -1221 68.5 -0.55 -66.42 1620 816 1218 -67 -0.11 1.33 -1620 -816 -1218 67 -0.56 -66.33 1530 864 1197 -55.5 -0.10 0.75 -1530 -864 -1197 55.5 -0.57 -65.75 1440 918 1179 -43.5 -0.08 0.25 -1440 -918 -1179 43.5 -0.58 -65.25 1377 960 1168.5 -34.8 -0.07 -0.04 -1377 -960 -1168.5 34.75 -0.59 -64.96 1360 972 1166 -32.33 -0.07 -0.11 -1360 -972 -1166 32.3 -0.60 -64.89 1296 1020 1158 -23 -0.06 -0.33 -1296 -1020 -1158 23 -0.61 -64.67 1224 1080 1152 -12 -0.04 -0.5 -1224 -1080 -1152 12 -0.63 -64.5 1215 1088 1151.5 -10.6 -0.04 -0.51 -1215 -1088 -1151.5 10.6 -0.63 -64.49 Next, we present the computer programs that are used in the study. These are Python computer programs that were used to generate actual edge-graceful labeling of needed par- ticular graphs, in order to complete the proof of the main results of this study. Provided below, is the Python program used to generate an actual edge-graceful labeling of S(WW3). 1 #Python Program for S(WW_3) 2 from itertools import permutations 3 4 numbers = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] 5 6 # Function to compute u0 to V1 7 def compute_v_values(w1u1, u1v2, v2w2, w2u2, u2v3, v3w3, w3u3, u3v1, v1w1, u0u1, u0v2, u0u2, u0v3, u0u3, u0v1, W1v1, W1u1, U1w1, U1u0, U1v2, V2u1, V2u0, V2w2, W2v2, W2u2, U2w2, U2u0, U2v3, V3u2, V3u0, V3w3, W3v3, W3u3, U3w3, U3u0, U3v1, V1u3, V1u0, V1w1, U0v1, U0u3, U0v3, U0u2, U0v2, U0u1): 8 u0 = (u0v1 + u0u1 + u0v2 + u0u2 + u0v3 + u0u3 + V1u0 + U1u0 + V2u0 + U2u0 + V3u0 + U3u0) % 20 9 w1 = (v1w1 + w1u1 + V1w1 + U1w1) % 20 10 u1 = (w1u1 + u1v2 + u0u1 + W1u1 + V2u1 + U0u1) % 20 11 v2 = (u1v2 + v2w2 + u0v2 + U1v2 + W2v2 + U0v2) % 20 12 w2 = (v2w2 + w2u2 + V2w2 + U2w2) % 20 13 u2 = (w2u2 + u2v3 + u0u2 + W2u2 + V3u2 + U0u2) % 20 14 v3 = (u2v3 + v3w3 + u0v3 + U2v3 + W3v3 + U0v3) % 20 15 w3 = (v3w3 + w3u3 + V3w3 + U3w3) % 20 16 u3 = (w3u3 + u3v1 + u0u3 + W3u3 + V1u3 + U0u3) % 20 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 29 of 33 17 v1 = (u3v1 + v1w1 + u0v1 + U3v1 + W1v1 + U0v1) % 20 18 U0 = (U0v1 + U0u3 + U0v3 + U0u2 + U0v2 +U0u1) % 20 19 W1 = (W1v1 + W1u1) % 20 20 U1 = (U1w1 + U1u0 + U1v2) % 20 21 V2 = (V2u1 + V2u0 + V2w2) % 20 22 W2 = (W2v2 + W2u2) % 20 23 U2 = (U2w2 + U2u0 + U2v3) % 20 24 V3 = (V3u2 + V3u0 + V3w3) % 20 25 W3 = (W3v3 + W3u3) % 20 26 U3 = (U3w3 + U3u0 + U3v1) % 20 27 V1 = (V1u3 + V1u0 + V1w1) % 20 28 return u0, w1, u1, v2, w2, u2, v3, w3, u3, v1, U0, W1, U1, V2, W2, U2, V3, W3, U3, V1 29 30 # Backtracking function to find valid permutations 31 def backtrack(perm, used): 32 # If we have a complete permutation , verify u0 to V1 33 if len(perm) == 45: 34 v_values = compute_v_values(*perm) 35 # Verify if u0 to V1 are distinct 36 if len(set(v_values)) == 20: 37 print(f"Permutation: {perm}, u0 to V1: {v_values}") 38 return 39 40 # Try to add each unused number to the permutation 41 for num in numbers: 42 if not used[num - 1]: # Check if number is not used 43 # Mark the number as used 44 used[num - 1] = True 45 perm.append(num) 46 47 # Recursively try the next number 48 backtrack(perm, used) 49 50 # Undo the choice (backtrack) 51 perm.pop() 52 used[num - 1] = False 53 54 # Start the backtracking process 55 def find_valid_permutations(): 56 used = [False] * 45 # Keeps track of used numbers 57 backtrack([], used) 58 59 # Start finding the valid permutations 60 find_valid_permutations() Provided below, is the Python program used to generate an actual edge-graceful labeling of S(RWW3). A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 30 of 33 1 #Python Program for S(RWW_3) 2 from itertools import permutations 3 4 numbers = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] 5 6 # Function to compute u0 to V1 7 def compute_v_values(w1u1, u1v2, v2w2, w2u2, u2v3, v3w3, w3u3, u3v1, v1w1, u0u1, u0v2, u0u2, u0v3, u0u3, u0v1, W1v1, W1v2, U1w1, U1v2, V2w1, V2u1, V2w2, W2v2, W2v3, U2w2, U2v3, V3w2, V3u2, V3w3, W3v3, W3v1, U3w3, U3v1, V1w3, V1u3, V1w1, U0v1, U0u3, U0v3, U0u2, U0v2, U0u1, w3v1, w2v3, w1v2, V1u0, U3u0, W3u3, V3u0, U2u0, W2u2, V2u0, U1u0, W1u1): 8 u0 = (u0v1 + u0u1 + u0v2 + u0u2 + u0v3 + u0u3 + V1u0 + U1u0 + V2u0 + U2u0 + V3u0 + U3u0) % 20 9 w1 = (v1w1 + w1u1 + w1v2 + V1w1 + U1w1 + V2w1) % 20 10 u1 = (w1u1 + u0u1 + u1v2 + W1u1 + U0u1 + V2u1) % 20 11 v2 = (w1v2 + u0v2 + u1v2 + v2w2 + W1v2 + U0v2 + U1v2 + W2v2) % 20 12 w2 = (v2w2 + w2u2 + w2v3 + V2w2 + U2w2 + V3w2) % 20 13 u2 = (w2u2 + u0u2 + u2v3 + W2u2 + U0u2 + V3u2) % 20 14 v3 = (w2v3 + u2v3 + u0v3 + v3w3 + W2v3 + U2v3 + U0v3 + W3v3) % 20 15 w3 = (v3w3 + w3u3 + w3v1 + V3w3 + U3w3 + V1w3) % 20 16 u3 = (w3u3 + u0u3 + u3v1 + W3u3 + U0u3 + V1u3) % 20 17 v1 = (u3v1 + u0v1 + v1w1 + w3v1 + U3v1 + U0v1 + W1v1 + W3v1) % 20 18 U0 = (U0v1 + U0u1 + U0v2 + U0u2 + U0v3 + U0u3) % 20 19 W1 = (W1v1 + W1u1 + W1v2) % 20 20 U1 = (U1w1 + U1u0 + U1v2) % 20 21 V2 = (V2w1 + V2u1 + V2u0 + V2w2) % 20 22 W2 = (W2v2 + W2u2 + W2v3) % 20 23 U2 = (U2w2 + U2u0 + U2v3) % 20 24 V3 = (V3w2 + V3u2 + V3u0 + V3w3) % 20 25 W3 = (W3v3 + W3u3 + W3v1) % 20 26 U3 = (U3w3 + U3u0 + U3v1) % 20 27 V1 = (V1w3 + V1u3 + V1u0 + V1w1) % 20 28 return u0, w1, u1, v2, w2, u2, v3, w3, u3, v1, U0, W1, U1, V2, W2, U2, V3, W3, U3, V1 29 30 # Backtracking function to find valid permutations 31 def backtrack(perm, used): 32 # If we have a complete permutation , verify u0 to V1 33 if len(perm) == 54: 34 v_values = compute_v_values(*perm) 35 # Verify if u0 to V1 are distinct 36 if len(set(v_values)) == 20: 37 print(f"Permutation: {perm}, u0 to V1: {v_values}") A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 31 of 33 38 return 39 40 # Try to add each unused number to the permutation 41 for num in numbers: 42 if not used[num - 1]: # Check if number is not used 43 # Mark the number as used 44 used[num - 1] = True 45 perm.append(num) 46 47 # Recursively try the next number 48 backtrack(perm, used) 49 50 # Undo the choice (backtrack) 51 perm.pop() 52 used[num - 1] = False 53 54 # Start the backtracking process 55 def find_valid_permutations(): 56 used = [False] * 54 # Keeps track of used numbers 57 backtrack([], used) 58 59 # Start finding the valid permutations 60 find_valid_permutations() Provided below, is the Python program used to generate an actual edge-graceful labeling of S(LWW3). 1 #Python Program for S(LWW_3) 2 from itertools import permutations 3 4 numbers = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] 5 6 # Function to compute u0 to V1 7 def compute_v_values(v1w1, u3v1, w3u3, v3w3, u2v3, w2u2, v2w2, u1v2, w1u1, u0v1, u0u3, u0v3, u0u2, u0v2, u0u1, W1u1, W1u3, V1w1, V1u3, U3w1, U3v1, U3w3, W3u3, W3u2, V3w3, V3u2, U2w3, U2v3, U2w2, W2u2, W2u1, V2w2, V2u1, U1w2, U1v2, U1w1, U0u1, U0v2, U0u2, U0v3, U0u3, U0v1, w2u1, w3u2, w1u3, U1u0, V2u0, W2v2, U2u0, V3u0, W3v3, U3u0, V1u0, W1v1): 8 u0 = (u0v1 + u0u1 + u0v2 + u0u2 + u0v3 + u0u3 + V1u0 + U1u0 + V2u0 + U2u0 + V3u0 + U3u0) % 20 9 w1 = (v1w1 + w1u1 + w1u3 + V1w1 + U1w1 + U3w1) % 20 10 u1 = (w1u1 + u1v2 + u0u1 + w2u1 + W1u1 + V2u1 + U0u1 + W2u1) % 20 11 v2 = (u1v2 + v2w2 + u0v2 + U1v2 + W2v2 + U0v2) % 20 12 w2 = (v2w2 + w2u2 + w2u1 + V2w2 + U2w2 + U1w2) % 20 A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 32 of 33 13 u2 = (w2u2 + u2v3 + u0u2 + w3u2 + W2u2 + V3u2 + U0u2 + W3u2) % 20 14 v3 = (u2v3 + v3w3 + u0v3 + U2v3 + W3v3 + U0v3) % 20 15 w3 = (v3w3 + w3u3 + w3u2 + V3w3 + U3w3 + U2w3) % 20 16 u3 = (w3u3 + u3v1 + u0u3 + w1u3 + W3u3 + V1u3 + U0u3 + W1u3) % 20 17 v1 = (u3v1 + v1w1 + u0v1 + U3v1 + W1v1 + U0v1) % 20 18 U0 = (U0v1 + U0u3 + U0v3 + U0u2 + U0v2 + U0u1) % 20 19 W1 = (W1u1 + W1v1 + W1u3) % 20 20 U1 = (U1w1 + U1u0 + U1v2 + U1w2) % 20 21 V2 = (V2u1 + V2u0 + V2w2) % 20 22 W2 = (W2u1 + W2v2 + W2u2) % 20 23 U2 = (U2w2 + U2u0 + U2v3 + U2w3) % 20 24 V3 = (V3u2 + V3u0 + V3w3) % 20 25 W3 = (W3u2 + W3v3 + W3u3) % 20 26 U3 = (U3w3 + U3u0 + U3v1 + U3w1) % 20 27 V1 = (V1u3 + V1u0 + V1w1) % 20 28 return u0, w1, u1, v2, w2, u2, v3, w3, u3, v1, U0, W1, U1, V2, W2, U2, V3, W3, U3, V1 29 30 # Backtracking function to find valid permutations 31 def backtrack(perm, used): 32 # If we have a complete permutation , verify u0 to V1 33 if len(perm) == 54: 34 v_values = compute_v_values(*perm) 35 # Verify if u0 to V1 are distinct 36 if len(set(v_values)) == 20: 37 print(f"Permutation: {perm}, u0 to V1: {v_values}") 38 return 39 40 # Try to add each unused number to the permutation 41 for num in numbers: 42 if not used[num - 1]: # Check if number is not used 43 # Mark the number as used 44 used[num - 1] = True 45 perm.append(num) 46 47 # Recursively try the next number 48 backtrack(perm, used) 49 50 # Undo the choice (backtrack) 51 perm.pop() 52 used[num - 1] = False 53 54 # Start the backtracking process 55 def find_valid_permutations(): 56 used = [False] * 54 # Keeps track of used numbers 57 backtrack([], used) 58 59 # Start finding the valid permutations A. D. C. Angel et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6425 33 of 33 60 find_valid_permutations()