EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6600 ISSN 1307-5543 – ejpam.com Published by New York Business Global Paley, Cubic Paley, Quadruple Paley, and Generalized Paley Graphs with an Edge Graceful Labeling Ahmed N. Elsawy1,2,∗, Reyoud Nasser Almohammadi1 1 Department of Mathematics, College of Science, Taibah University, Madina, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt Abstract. The Paley graph Pq is a simple, connected, strongly regular graph with parameters (q, q−1 2 , q−5 4 , q−1 4 ), where V (Pq) is the finite field Fq of order q = pn, p is an odd prime, n ∈ N, and q ≡ 1 (mod 4). In Paley graphs, two vertices are adjacent if their difference is a quadratic residue (mod q). The vertex set of the generalized Paley graph m − Pq, where m ≥ 3 is an odd integer, is V (m − Pq) = Fq and the set of edges is E(m − Pq) = { (x, y) ⇔ x− y ∈ (F ∗ q ) m } . In 1985, edge-graceful labeling was first introduced by Lo [1]. A graph G with order n and size m is called edge-graceful if there exists a bijective mapping f : E(G) −→ {1, 2, 3, . . . ,m} such that the weights map fw : V (G) −→ {0, 1, 2, . . . , n− 1}, given by fw(u) = ∑ v∈N(u) f(uv) (mod n), is one-to-one and onto. In this paper, we prove that Paley graphs and generalized Paley graphs of prime order are edge graceful, edge-even graceful, and edge-odd graceful graphs. 2020 Mathematics Subject Classifications: 05C25, 05C78 Key Words and Phrases: Paley graphs, cubic Paley graphs, quadruple Paley graphs, generalized Paley graphs, edge-graceful labeling, edge-even graceful labeling, edge-odd graceful labeling 1. Introduction A finite field Fq of order q, where q is a prime power congruent to 1 modulo 4, is used in the construction of the Paley graph as its vertex set. Two vertices are adjacent if and only if their difference is a non-zero square in the field. In this paper, we are concerned with the simple form of Paley graphs where Fq = Zp is the field of integers (mod p). We are also interested in the generalized Paley graph m−Pq, where V (m−Pq) = Fq is the finite field with q = pn elements, p is an odd prime, and n ∈ N. The edge set is E(m − Pq) = { (x, y) ⇔ x− y ∈ (F ∗ q ) m } , where m ≥ 3 is an odd integer. Paley graphs have several applications, for instance, in network design [2]. Also, Paley graphs form an important family in graph theory due to their rich structural proper- ties. One of their key features is that Paley graphs are strongly regular with parameters ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6600 Email addresses: asawy@taibahu.edu.sa, ahnoby@yahoo.com (A. N. Elsawy), reyoud55@gmail.com (R. N. Almohammadi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 2 of 26 (q, q−1 2 , q−5 4 , q−1 4 ). This means that the graph has order q, is q−1 2 −regular, each pair of ad- jacent vertices has q−5 4 common neighbors, and each pair of non-adjacent vertices has q−1 4 common neighbors. Furthermore, Paley graphs are self-complementary, symmetric, and Hamiltonian, for more details (see [3], [4]). Paley graphs possess not only a Hamiltonian property but also a Hamiltonian decomposition; that is, the edge set can be represented as the union of edge-disjoint Hamiltonian cycles, see [5]. Paley graphs provide intrigu- ing characteristics that make them valuable in graph theory and allow the application of graph-theoretic techniques to the number theory involving quadratic residues (see [6], [7], [8], [9]). Another active area of research in graph theory concerns graph coloring and the clique number. Several papers have investigated the clique number of Paley graphs and their generalizations (see [10], [11], [12], [13], [14]). Graph labeling is another important branch of graph theory, in which labels are as- signed to the vertices, edges, or both, under certain conditions, see [15]. One prominent type of labeling that has attracted significant attention is edge-graceful labeling, which has important applications in various areas, including computer networks, coding theory, radar, circuit design, and communication systems. In 1967, Alexander Rosa [16] identified three types of labeling: α-labeling, β-labeling, and ρ-labeling. Later, β-labeling was renamed graceful labeling by Solomon Golomb [17]. In an edge-graceful labeling of a graph G distinct integers are assigned to the edges, and each vertex is labeled, using an injective map, by the sum of the labels on its incident edges modulo p. For more information about different types of labeling (see [18], [19], [20], [21],[22]). In this paper, we prove that Paley graphs and generalized Paley graphs of prime order are edge-graceful. 2. Paley graphs Paley graphs have many important applications in coding theory and cryptography. For example they are used in quantum secret sharing domain [23] and in construction of codes from their incidence matrices and line graphs [24]. Moreover, a new application of these graphs in cryptography was proposed in [25], introducing an efficient algorithm for the encryption and decryption of sensitive messages by converting them into binary form using Paley graphs. Definition 1. Let q = pn be a prime power, where q ≡ 1 (mod 4). A Paley graph Pq of order q is a graph with vertex set V (Pq) = Fq, where Fq is the finite field with q elements, and edge set E(Pq) = { (u, v) | u− v ∈ (F∗ q ) 2 } . Note that the condition q ≡ 1 (mod 4) is necessary for the graph to be undirected. A simple version of the Paley graph is obtained when n = 1. That is, when Fq = Zp, the field of integers (mod p). Example 1. The Paley graph P13 has vertex set V (P13) = Z13 = {0, 1, 2, . . . , 12}, where two vertices u, v ∈ Z13 are adjacent if u− v ∈ {1, 3, 4, 9, 10, 12}. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 3 of 26 Figure 1: The Paley Graph P13 of Order 13. Example 2. Consider the Paley graph P9 of order 9, then V (P9) = F9 = Z3[x]/(x 2+1) = {0, 1, 2, a+ 2, a, a+ 1, 2a+ 1, 2a+ 2, 2a}. Therefore, (F∗ 9 ) 2 = {1, 2, a, 2a}, meaning that every vertex in V (P9) is adjacent to four vertices. Hence, E(P9) = {(xi, xi + xj) | xi ∈ F9, xj ∈ (F∗ 9 ) 2}. Figure 2: The Paley Graph P9 of Order 9. Graph labeling is one of the significant fields in graph theory. It has many applications in coding theory, x-ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base management (see [26], [27]). In [21], Kamaraj and Thangakani investigated edge-even and edge-odd graceful label- A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 4 of 26 ing for Paley graphs of prime order. An edge even-graceful labeling assigns distinct even numbers to the edges and ensures that the sum of labels modulo 2k assigned to each vertex is unique. On the other hand, an edge-odd graceful labeling assigns distinct odd numbers to the edges and ensures that the sum of labels modulo 2q assigned to each vertex is unique. Kamaraj and Thangakani proved the following theorem: Theorem 1. Every Paley graph of prime order admits both an edge-even and an edge-odd graceful labeling. We introduce another lebeling of Paley graphs of prime order, which is called edge graceful labeling. Definition 2. Accordung to [1], a graph G of order n and size m is called edge-graceful if there exists a bijective mapping h : E(G) −→ {1, 2, 3, . . . ,m} such that the weight function hw : V (G) −→ {0, 1, 2, . . . , n− 1}, given by hw(u) = ∑ v∈N(u) h(uv) (mod n), is one-to-one and onto. For example, consider the Paley graph P9. As in Figure 3, we can define an edge- graceful labeling to the Paley graph P9 such that h : E(P9) −→ {1, 2, 3, . . . , 18} and hw : V (P9) −→ {0, 1, 2, . . . , 8} are bijections. Figure 3: An edge-graceful labeling for P9 In the following we provide an algorithm which produces an edge-graceful labeling for the prime order Paley graphs. Note that the case of the prime power order is open. 2.1. Edge-graceful labeling Algorithm for Paley graph of prime order Input: The Paley graph Pp with V (Pp) = Zp and E(3 − Pp) = {(u, v) | u − v ∈ (Z∗ p) 2}, where p ≡ 1 (mod 4) A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 5 of 26 (1) Rename the vertices of the graph as 0 := vp, 1 := v1, 2 := v2, . . . , p− 1 := vp−1. (2) Set r = p−1 4 , and rewrite (Z∗ p) 2 as (Z∗ p) 2 = S = {s1, s2, s3, . . . , s2r : s1 < s2 < s3 < . . . < s2r}. (3) Partition S into two sets. Let S1 = {s1, s2, s3, . . . , sr} and S2 = {sr+1, sr+2, sr+3, . . . , s2r}. Note that: p−1 is divisible by 4 and for any vertex vi ∈ Zp the vertex vi+sj is adjacent to vi for all sj ∈ S. (4) If sj ∈ S1, the vertex vi+sj is placed in clockwise direction of vi and if sj ∈ S2, the vertex vi+sj is placed in anticlockwise direction of vi. (5) Set f(vi, vi+sj ) = 0 for all i ∈ {1, 2, 3, · · · , p}, j ∈ {1, 2, 3, · · · , 2r}. (6) Set i = 1. Step 1: If i ≤ p then continue to Step 2. Else jump to Step 4. Step 2: For each sj ∈ S1, f ( vi, vi+sj ) = (j − 1) p+ i. Step 3: i = i+ 1, go back to Step 1. Step 4: For each k = 1, 2, 3, . . . , p, find the weight of the vertex vk using the following mapping: fw(vk) = ∑2r j=1 f(vk, vk+sj ) = ∑r j=1 2[(j−1)p+k]+(p−sj) = 2kr−L (mod p), where L = ∑r j=1 sj . Theorem 2. Every Paley graph of prime order admits an edge-graceful labeling. Proof. To prove that the algorithm defines an edge-graceful labeling, we need to prove that both functions f and fw are bijections. (i) Consider the function fw : V (Pp) −→ Zp defined by fw(vk) = ∑2r j=1 f(vk, vk+sj ) = 2kr − L (mod p). Now, we prove that fw is one-to-one, which implies tat it is a bijection. Let vx and vy be two vertices in V (Pp), if fw(vx) = fw(vy) then 2xr−L = 2yr − L (mod p) which leads to x = y. (ii) The function f : E(Pp) −→ {1, 2, 3, . . . , rp} is defined as f(vx, vx+sj ) = (j− 1)p+ x. To prove that f is a bijection, we need only to prove the injectivity of f . Let (vx, vx+si) and (vy, vy+sj ) be two edges, with x, y ∈ {1, 2, 3, · · · , p}, si, sj ∈ S = (Z∗ p) 2, and f(vx, vx+sj ) = f(vy, vy+si), which implies that (i− 1)p+x = (j− 1)p+ y. In case of i = j or x = y the proof is trivial. The last case if x ̸= y and j ̸= i, here we will find that x− y = (j− i)p but |x− y| < p and in the same time |p(i− j)| ≥ p which is a contradiction. So, from these three cases, we can be sure that the function f is a one-to-one function. □ Example 3. We apply the previous algorithm to show that: the Paley graph P13 is an edge-graceful graph, where V (P13) = {v1, v2, · · · , v13} and |E(P13)| = 13·12 4 = 39. Figure 4 illustrates the edge-graceful labeling for P13. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 6 of 26 Figure 4: An edge-graceful labeling for P13 Input: The graph P13. Here we have p = 13, r = 3, and (Z∗ 13) 2 = {12, 22, 32, · · · , 122} = {1, 4, 9, 3, 12, 10}. (1) Rename the vertices of the graph as 0 := v13, 1 := v1, 2 := v2, . . . , 12 := v12. (2) Rewrite (Z∗ 13) 2 as (Z∗ 13) 2 = S = {s1, s2, s3, s4, s5, s6 : s1 < s2 < s3 < s4 < s5 < s6} = {1, 3, 4, 9, 10, 12}. (3) Partition S into two sets. Let S1 = {s1, s2, s3} = {1, 3, 4} and S2 = {s4, s5, s6} = {9, 10, 12}. Note that: for any vertex vi ∈ Z13 the vertices vi+1, vi+3, vi+4, vi+9, vi+10, vi+12 are adjacent to vi. (4) The vertices vi+1, vi+3, vi+4 are placed in clockwise direction of vi and the vertices vi+9, vi+10, vi+12 are placed in anticlockwise direction of vi. (5) Set f(vi, vi+sj ) = 0 for all i ∈ {1, 2, 3, . . . , 13}, j ∈ {1, 2, 3, . . . , 6}. (6) Set i = 1. Step 1: i = 1 ≤ 13, then continue to Step 2. Step 2: For each s1, s2, s3 ∈ S1, f ( v1, v1+sj ) = (j − 1) p+ 1. That is, f(v1, v2) = (1 − 1)13 + 1 = 1, f(v1, v4) = (2 − 1)13 + 1 = 14, f(v1, v5) = (3− 1)13 + 1 = 27. Step 3: i = 1 + 1 = 2, go back to Step 1. Step 1: i = 2 ≤ 13, then continue to Step 2. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 7 of 26 Step 2: For each s1, s2, s3 ∈ S1, f ( v2, v2+sj ) = (j − 1) p+ 2. That is, f(v2, v3) = (1 − 1)13 + 2 = 2, f(v2, v5) = (2 − 1)13 + 2 = 15, f(v2, v6) = (3− 1)13 + 2 = 28. Step 3: i = 2+1 = 3, go back to Step 1. We repeat Step1, Step2, and Step3 and get: f(v3, v4) = 3, f(v3, v6) = 16, f(v3, v7) = 29, f(v4, v5) = 4, f(v4, v7) = 17, f(v4, v8) = 30, f(v5, v6) = 5, f(v5, v8) = 18, f(v5, v9) = 31, f(v6, v7) = 6, f(v6, v9) = 19, f(v6, v10) = 32, f(v7, v8) = 7, f(v7, v10) = 20, f(v7, v11) = 33, f(v8, v9) = 8, f(v8, v11) = 21, f(v8, v12) = 34, f(v9, v10) = 9, f(v9, v12) = 22, f(v9, v13) = 35, f(v10, v11) = 10, f(v10, v13) = 23, f(v10, v1) = 36, f(v11, v12) = 11, f(v11, v1) = 24, f(v11, v2) = 37, f(v12, v13) = 12, f(v12, v2) = 25, f(v12, v3) = 38, f(v13, v1) = 13, f(v13, v3) = 26, f(v13, v4) = 39. Step 4: For each k = 1, 2, 3, . . . , 13, the weight of the vertex vk can be found by: fw(vk) = ∑6 j=1 f(vk, vk+sj ) = 6k− ∑3 j=1 sj = 6k−8 (mod p). fw (v1) = 6·1−8 = −2 ≡ 11 (mod 13). Equivalently, fw (v1) = f(v1, v2)+f(v1, v4)+f(v1, v5)+f(v10, v1)+f(v11, v1)+f(v13, v1) = 1 + 14 + 27 + 36 + 24 + 13 ≡ 11 (mod 13). By the same function modulo 13 , we get: fw(v2) = 4, fw(v3) = 10, fw(v4) = 3, fw(v5) = 9, fw(v6) = 2, fw(v7) = 8, fw(v8) = 1, fw(v9) = 7, fw(v10) = 0, fw(v11) = 6, fw(v12) = 12, fw(v13) = 5. 3. Cubic Paley graphs A natural generalization of Paley graph is the cubic Paley graph, which has the same vertex set, and two vertices are adjacent if their difference is a cubic residue in Fq. Definition 3. [28] Let q = pn, where p is an odd prime, and n ∈ N, such that q ≡ 1 (mod 3). The graph 3−Pq, with V (3−Pq) = Fq and E(3−Pq) = { (u, v) | u− v ∈ (F∗ q ) 3 } is called the cubic Paley graph of order q. Note that: W. Ananchuen and L. Caccetta, in [28], stated that the condition q ≡ 1 (mod 3) is necessary to ensure that −1 is a cubic residue and, consequently, that the graph 3 − Pq is well-defined. However, we disagree with this claim, since (−1)3 = −1 in any field with any order, implying that −1 is always a cubic residue. Therefore, the graph 3− Pq is well-defined without the condition q ≡ 1 (mod 3). Example 4. The cubic Paley graph 3 − P19 of order 19 has V (3 − P19) = Z19 and E(3− P19) = {(u, v) | u− v ∈ {1, 7, 8, 11, 12, 18}}. See Figure 5. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 8 of 26 Figure 5: The cubic Paley graph 3− P19 Now we provide an algorithm that produces an edge-graceful labeling for cubic Paley graphs of prime order 3.1. Edge-graceful labeling Algorithm for cubic Paley graphs of prime order Input: The cubic Paley graph 3 − Pp with V (3 − Pp) = Zp and E(3 − Pp) = {(u, v) | u− v ∈ (Z∗ p) 3}, where p is any odd prime. (1) Rename the vertices of the graph as 0 := vp, 1 := v1, 2 := v2, . . . , p− 1 := vp−1. (2) Set r = p−1 2d , where d = gcd(3, p− 1), and rewrite (Z∗ p) 3 as (Z∗ p) 3 = S = {s1, s2, s3, . . . , s2r : s1 < s2 < s3 < . . . < s2r}. (3) Partition S into two sets. Let S1 = {s1, s2, s3, . . . , sr} and S2 = {sr+1, sr+2, sr+3, . . . , s2r}. Note that: p−1 is divisible by 2d and for any vertex vi ∈ Zp the vertex vi+sj is adjacent to vi for all sj ∈ S. (4) If sj ∈ S1, the vertex vi+sj is placed in clockwise direction of vi and if sj ∈ S2, the vertex vi+sj is placed in anticlockwise direction of vi. (5) Set f(vi, vi+sj ) = 0 for all i ∈ {1, 2, 3, · · · , p}, j ∈ {1, 2, 3, · · · , 2r}. (6) Set i = 1. Step 1: If i ≤ p then continue to Step 2. Else jump to Step 4. Step 2: For each sj ∈ S1, f ( vi, vi+sj ) = (j − 1) p+ i. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 9 of 26 Step 3: i = i+ 1, go back to Step 1. Step 4: For each k = 1, 2, 3, . . . , p, find the weight of the vertex vk using the following mapping: fw(vk) = ∑2r j=1 f(vk, vk+sj ) = ∑r j=1 2[(j−1)p+k]+(p−sj) = 2kr−L (mod p), where L = ∑r j=1 sj . Theorem 3. Every cubic Paley graph of prime order admits an edge-graceful labeling. Proof. See Theorem 5 with m = 3. Example 5. We apply the algorithm to show that the cubic Paley graph 3− P19 is edge- graceful with V (3−P19) = {v1, v2, · · · , v19} and |E(3− P19)| = 19·18 6 = 57. Figure 6 shows the edge-graceful labeling for the cubic Paley graph 3− P19. Figure 6: An Edge-Graceful Labeling of 3− P19 A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 10 of 26 Input: The graph 3− P19. We have p = 19,m = 3, d = 3, and (Z∗ 19) 3 = {13, 23, · · · , 183} = {1, 8, 7, 11, 18, 12}. (1) Rename the vertices of the graph as 0 := v19, 1 := v1, 2 := v2, . . . , 18 := v18. (2) Rewrite (Z∗ 19) 3 as (Z∗ 19) 3 = S = {s1, s2, s3, s4, s5, s6 : s1 < s2 < s3 < s4 < s5 < s6} = {1, 7, 8, 11, 12, 18} (3) Partition S into two sets. Let S1 = {s1, s2, s3} = {1, 7, 8} and S2 = {s4, s5, s6} = {11, 12, 18}. Note that: for any vertex vi ∈ Z19 the vertices vi+1, vi+7, vi+8, vi+11, vi+12, vi+18 are adjacent to vi. (4) The vertices vi+1, vi+7, vi+8 are placed in clockwise direction of vi and the vertices vi+11, vi+12, vi+18 are placed in anticlockwise direction of vi. (5) Set f(vi, vi+sj ) = 0 for all i ∈ {1, 2, 3, · · · , 19}, j ∈ {1, 2, 3, · · · , 6}. (6) Set i = 1. Step 1: i = 1 ≤ 19, then continue to Step 2. Step 2: For each s1, s2, s3 ∈ S1, f ( v1, v1+sj ) = (j − 1) p+ 1. That is, f(v1, v2) = (1 − 1)19 + 1 = 1, f(v1, v8) = (2 − 1)19 + 1 = 20, f(v1, v9) = (3− 1)19 + 1 = 39. Step 3: i = 1 + 1 = 2, go back to Step 1. Step 1: i = 2 ≤ 19, then continue to Step 2. Step 2: For each s1, s2, s3 ∈ S1, f ( v2, v2+sj ) = (j − 1) p+ 2. That is, f(v2, v3) = (1 − 1)19 + 2 = 2, f(v2, v9) = (2 − 1)19 + 2 = 21, f(v2, v10) = (3− 1)19 + 2 = 40. Step 3: i = 2 + 1 = 3, go back to Step 1. We repeat Step1, Step2, and Step3 and get: f(v3, v4) = 3, f(v3, v10) = 22, f(v3, v11) = 41, f(v4, v5) = 4, f(v4, v11) = 23, f(v4, v12) = 42, f(v5, v6) = 5, f(v5, v12) = 24, f(v5, v13) = 43, f(v6, v7) = 6, f(v6, v13) = 25, f(v6, v14) = 44, f(v7, v8) = 7, f(v7, v14) = 26, f(v7, v15) = 45, f(v8, v9) = 8, f(v8, v15) = 27, f(v8, v16) = 46, f(v9, v10) = 9, f(v9, v16) = 28, f(v9, v17) = 47, f(v10, v11) = 10, f(v10, v17) = 29, f(v10, v18) = 48, f(v11, v12) = 11, f(v11, v18) = 30, f(v11, v19) = 49, f(v12, v13) = 12, f(v12, v19) = 31, f(v12, v1) = 50, f(v13, v14) = 13, f(v13, v1) = 32, f(v13, v2) = 51, f(v14, v15) = 14, f(v14, v2) = 33, f(v14, v3) = 52, f(v15, v16) = 15, f(v15, v3) = 34, f(v15, v4) = 53, f(v16, v17) = 16, f(v16, v4) = 35, f(v16, v5) = 54, f(v17, v18) = 17, f(v17, v5) = 36, f(v17, v6) = 55, f(v18, v19) = 18, f(v18, v6) = 37, f(v18, v7) = 56, f(v19, v1) = 19, f(v19, v7) = 38, f(v19, v8) = 57. Step 4: For each k = 1, 2, 3, · · · , 19, find the weight of the vertex vk using the following mapping: fw(vk) = ∑6 j=1 f(vk, vk+sj ) = 6k − ∑3 j=1 sj = 6k − 16 (mod p). So fw (v1) = 6 · 1− 16 = −10 ≡ 9 (mod 19). Equivalently, fw (v1) = f(v1, v2)+f(v1, v8)+f(v1, v9)+f(v12, v1)+f(v13, v1)+f(v19, v1) = 1 + 20 + 39 + 50 + 32 + 19 ≡ 9 (mod 19). By the same function modulo 19, we get: fw(v2) = 15, fw(v3) = 2, fw(v4) = 8, fw(v5) = 14, fw(v6) = 1, fw(v7) = 7, fw(v8) = 13, fw(v9) = 0, fw(v10) = 6, fw(v11) = 12, fw(v12) = 18, fw(v13) = 5, fw(v14) = 11, fw(v15) = 17, fw(v16) = 4, fw(v17) = 10, fw(v18) = 16, fw(v19) = 3. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 11 of 26 4. Quadruple Paley graphs Similarly like cubic Paley graphs, quadruple Paley graphs are defined with the same vertex set and two vertices are adjacent if their difference is a quadruple residue in Fq. Definition 4. [28] Let q = pn, where p is an odd prime number, and n ∈ N, such that q ≡ 1 (mod 8). The graph 4−Pq, with V (4−Pq) = Fq and E(4−Pq) = { (u, v) | u− v ∈ (F∗ q ) 4 } , is called the quadruple Paley graph of order q. Note that: The condition q ≡ 1 (mod 8) is necessary to ensure that −1 is a quadruple and, consequently, the graph 4− Pq is well-defined. Example 6. The quadruple Paley graph 4 − P17 of order 17 has V (4 − P17) = Z17 and E(4− P17) = {(u, v) | u− v ∈ {1, 4, 13, 16}}, as in Figure 7. Figure 7: The Quadruple Paley Graph 4− P17 Example 7. The quadruple Paley graph 4 − P9 with vertex set V (4 − P9) = F9 and (F∗ 9 ) 4 = {1, 2} so the edge set E(4 − P9) = {(0, 1), (0, 2), (1, 2), (a, a + 1), (a, a + 2), (a + 1, a+ 2), (2a, 2a+ 1), (2a, 2a+ 2), (2a+ 1, 2a+ 2)}. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 12 of 26 Figure 8: The Quadruple Paley Graph 4− P9 Now we provide an algorithm which produces an edge-graceful labeling for quadruple Paley graphs of prime order. 4.1. Edge-graceful labeling Algorithm for quadruple Paley graphs of prime order Input: The quadruple Paley graph 4−Pp with V (4−Pp) = Zp and E(4−Pp) = {(u, v) | u− v ∈ (Z∗ p) 4}, where p ≡ 1 (mod 8) (1) Rename the vertices of the graph as 0 := vp, 1 := v1, 2 := v2, . . . , p− 1 := vp−1. (2) Set r = p−1 8 , and rewrite (Z∗ p) 4 as (Z∗ p) 4 = S = {s1, s2, s3, . . . , s2r : s1 < s2 < s3 < . . . < s2r}. (3) Partition S into two sets. Let S1 = {s1, s2, s3, . . . , sr} and S2 = {sr+1, sr+2, sr+3, . . . , s2r}. Note that: p − 1 is divisible by 8 and for any vertex vi ∈ Zp the vertex vi+sj is adjacent to vi for all sj ∈ S. (4) If sj ∈ S1, the vertex vi+sj is placed in clockwise direction of vi and if sj ∈ S2, the vertex vi+sj is placed in anticlockwise direction of vi. (5) Set f(vi, vi+sj ) = 0 for all i ∈ {1, 2, 3, · · · , p}, j ∈ {1, 2, 3, · · · , 2r}. (6) Set i = 1. Step 1: If i ≤ p then continue to Step 2. Else jump to Step 4. Step 2: For each sj ∈ S1, f ( vi, vi+sj ) = (j − 1) p+ i. Step 3: i = i+ 1, go back to Step 1. Step 4: For each k = 1, 2, 3, . . . , p, find the weight of the vertex vk using the following mapping: fw(vk) = ∑2r j=1 f(vk, vk+sj ) = ∑r j=1 2[(j−1)p+k]+(p−sj) = 2kr−L (mod p), where L = ∑r j=1 sj . Theorem 4. Every quadruple Paley graph of prime order admits an edge-graceful labeling. Proof. See Theorem 5 with m = 4. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 13 of 26 Figure 9: An Edge-Graceful Labeling of 4− P17. Example 8. We apply the algorithm to show that: the quadruple Paley graph 4− P17 is edge-graceful with V (4 − P17) = {v1, v2, · · · , v17} and |E(4− P17)| = 17·16 8 = 34. Here, p = 17, r = 2, and S1 = {1, 4}. So for each sj ∈ S1, f ( vk, vk+sj ) = (j − 1) 17 + k, and fw(vk) = 4k − ∑2 j=1 sj = 4k − 5 (mod 17). Figure 9 shows the edge-graceful labeling for the cubic Paley graph 4− P17. 5. Generalized Paley graphs Generalized Paley graphs were first introduced by Cohen [29], and reintroduced by Lim and Praeger [30] in 2009, and by Elsawy [3] in 2009. The generalized Paley graph extends the concept of the Paley graph and its higher-order variants, providing a broader class of graphs based on higher power residues in finite fields. In the following we extend the results of edge-graceful labeling on Paley, cubic Paley, quadruple Paley graphs to generalized Paley graphs Definition 5. Let m,n be positive integers and p be an odd prime such that: ”if m is even then q ≡ 1 (mod 2m) and if m is odd then p is any odd prime”. The generalized Paley graph m−Pq has vertex set V (m−Pq) = Fq, where Fq is the finite field of order q = pn, and two vertices are adjacent if their difference belongs to (F ∗ q ) m. Remark 1. The generalized Paley graphs m−Pq are non-directed graphs because (F ∗ q ) m = −(F ∗ q ) m, in other words −1 ∈ (F ∗ q ) m. Remark 2. The generalized Paley graphs m − Pq are regular of degree q−1 d , where d is the greatest common divisor of m and q − 1 (see [3]). A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 14 of 26 Remark 3. The size of m− Pq is q(q−1) 2d , and the positive integer q−1 d is even. Remark 4. For m = 2, 3, or 4 we get Paley, cubic Paley, or quadruple Paley graphs respectively. Now, we provide an algorithm which produces an edge-graceful labeling for the prime order generalized Paley graphs 5.1. Edge-graceful labeling Algorithm for generalized Paley graphs of prime order Input: The generalized Paley graph m − Pp has V (m − Pp) = Zp and E(m − Pp) = {(u, v) | u − v ∈ (Z∗ p) m}, such that: if m is even then p ≡ 1 (mod 2m) and if m is odd then p is any odd prime. (1) Rename the vertices of the graph as 0 := vp, 1 := v1, 2 := v2, . . . , p− 1 := vp−1. (2) Set r = p−1 2d , where d = gcd(m, p− 1), and rewrite (Z∗ p) m as (Z∗ p) m = S = {s1, s2, s3, . . . , s2r : s1 < s2 < s3 < . . . < s2r}. (3) Partition S into two sets. Let S1 = {s1, s2, s3, . . . , sr} and S2 = {sr+1, sr+2, sr+3, . . . , s2r}. Note that: p−1 is divisible by 2d and for any vertex vi ∈ Zp the vertex vi+sj is adjacent to vi for all sj ∈ S. (4) If sj ∈ S1, the vertex vi+sj is placed in clockwise direction of vi and if sj ∈ S2, the vertex vi+sj is placed in anticlockwise direction of vi. (5) Set f(vi, vi+sj ) = 0 for all i ∈ {1, 2, 3, · · · , p}, j ∈ {1, 2, 3, · · · , 2r}. (6) Set i = 1. Step 1: If i ≤ p then continue to Step 2. Else jump to Step 4. Step 2: For each sj ∈ S1, f ( vi, vi+sj ) = (j − 1) p+ i. Step 3: i = i+ 1, go back to Step 1. Step 4: For each k = 1, 2, 3, . . . , p, find the weight of the vertex vk using the following mapping: fw(vk) = ∑2r j=1 f(vk, vk+sj ) = ∑r j=1 2[(j−1)p+k]+(p−sj) = 2kr−L (mod p), where L = ∑r j=1 sj . Theorem 5. Paley graphs and their generalizations of prime order are edge-graceful graphs. Proof. To prove that the algorithm defines an edge-graceful labeling, we need to prove that both functions f and fw are bijections. (i) Consider the function fw : V (m−Pp) −→ Zp such that fw(vk) = ∑2r j=1 f(vk, vk+sj ) = 2kr − L (mod p). Now, we prove that fw is one-to-one, which implies tat it is a bijection. Let vx and vy be two vertices in V (m − Pp), if fw(vx) = fw(vy) then 2xr − L = 2yr − L (mod p) which leads to x = y. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 15 of 26 (ii) The function f : E(m − Pp) −→ {1, 2, 3, . . . , rp} is defined as f(vx, vx+sj ) = (j − 1)p+ x. To prove that f is a bijection, we need only to prove that it is one-to-one. Let (vx, vx+si) and (vy, vy+sj ) be two edges, with x, y ∈ {1, 2, 3, · · · , p}, si, sj ∈ S = (Z∗ p) m, and f(vx, vx+sj ) = f(vy, vy+si), which implies that (i−1)p+x = (j−1)p+y. In case of i = j or x = y the proof is trivial. The last case if x ̸= y and j ̸= i, here we will find that x− y = (j− i)p but |x− y| < p and in the same time |p(i− j)| ≥ p which is a contradiction. So, from these three cases, we can be sure that the function f is a one-to-one function. □ Example 9. Consider the cubic Paley graph 3− P13. Here, p = 13,m = 3, d = 3, r = 2, and S1 = {1, 5}. So for each sj ∈ S1, f ( vk, vk+sj ) = (j − 1) 13 + k, and fw(vk) = 4k− ∑2 j=1 sj = 4k− 6 (mod 13). The edge-graceful labeling of the graph 3−P13 is shown in Figure 10. Figure 10: An Edge-Graceful Labeling of 3− P13. Example 10. Consider the generalized Paley graph 5−P7. Here, p = 7,m = 5, 7, or 11, d = 1, r = 3, and S1 = {1, 2, 3}. So for each sj ∈ S1, f ( vk, vk+sj ) = (j − 1) 7 + k, and fw(vk) = 6k − ∑3 j=1 sj = 6k − 6 (mod 7). The edge-graceful labeling of the graph 5 − P7 is shown in Figure 11. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 16 of 26 Figure 11: An Edge-Graceful Labeling of 5− P7, 7− P7, 11− P7. Example 11. Consider the Paley graph P17. Here, p = 17,m = d = 2, r = 4, and S1 = {1, 2, 4, 8}. So for each sj ∈ S1, f ( vk, vk+sj ) = (j − 1) 17 + k, and fw(vk) = 8k− ∑4 j=1 sj = 8k− 15 (mod 17). The edge-graceful labeling of the graph P17 is shown in Figure 12. Figure 12: An Edge-Graceful Labeling of P17. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 17 of 26 Example 12. Consider the generalized Paley graph 7 − P29. Here, p = 29,m = 7, d = 7, r = 2, and S1 = {1, 12}. So for each sj ∈ S1, f ( vk, vk+sj ) = (j − 1) 29 + k, and fw(vk) = 4k− ∑2 j=1 sj = 4k−13 (mod 29). The edge-graceful labeling of the graph 7−P29 is shown in Figure 13. Figure 13: An Edge-Graceful Labeling of 7− P29. Example 13. Consider the generalized Paley graph 9 − P31. Here, p = 31,m = 9, d = 3, r = 5, and S1 = {1, 2, 4, 8, 15}. So for each sj ∈ S1, f ( vk, vk+sj ) = (j − 1) 31 + k, and fw(vk) = 10k− ∑5 j=1 sj = 10k− 30 (mod 31). The edge-graceful labeling of the graph 9− P31 is shown in Figure 14. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 18 of 26 Figure 14: An Edge-Graceful Labeling of 9− P31. 6. Edge even graceful labeling of generalized Paley graphs Definition 6. A graph G of order n and size m is said to be edge-even graceful if there exists a bijection f : E(G) −→ {2, 4, 6, . . . , 2m} such that the function fw : V (G) −→ {0, 2, 4, . . . , 2k − 2} , k = max(n,m) given by fw(u) = ∑ uv∈N(u) f(uv) (mod 2k) is an injection. Using the following algorithm, we show that the generalized Paley graphs m − Pp of prime order are edge-even graceful. 6.1. Edge-even graceful labeling Algorithm for generalized Paley graph of prime order Input: The generalized Paley graph m − Pp with Zp as its vertices and two vertices are joined by an edge if their difference belongs to (Z∗ p) m such that: if m is even then p ≡ 1 A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 19 of 26 (mod 2m) and if m is odd then p is any odd prime. (1) Rename the vertices of the graph as 0 := vp, 1 := v1, 2 := v2, . . . , p− 1 := vp−1. (2) Set r = p−1 2d where d = gcd(m, p− 1), and rewrite (Z∗ p) m as (Z∗ p) m = S = {s1, s2, s3, . . . , s2r : s1 > s2 > s3 > . . . > s2r}. (3) Partition S into two sets. Let S1 = {s1, s2, s3, . . . , sr} and S2 = {sr+1, sr+2, sr+3, . . . , s2r}. (4) If sj ∈ S1, the vertex vi+sj is placed in anticlockwise direction of vi and if sj ∈ S2, the vertex vi+sj is placed in clockwise direction of vi. (5) Set f(vi, vi+sj ) = 0 for all i ∈ {1, 2, 3, . . . , p}, j ∈ {1, 2, 3, . . . , 2r}. (6) Set i = 1. Step 1: If i ≤ p then continue to Step 2. Else jump to Step 4. Step 2: For each sj ∈ S1, f ( vi, vi+sj ) = 2[(j − 1) p+ i]. Step 3: i = i+ 1, go back to Step 1. Step 4: For each k = 1, 2, 3, . . . , p, find the weight of the vertex vk using the fol- lowing mapping: fw(vk) = ∑2r j=1 f(vk, vk+sj ) (mod 2t), where t = max(p, pr). Take M = ∑r j=1 sj , then fw(vk) = ∑r j=1 4[(j − 1) p+ k] + 2(p− sj) = 4rk − 2M (mod 2rp). Theorem 6. Paley graphs and their generalizations of prime order are edge-even graceful graphs. Proof. To prove that the algorithm defines an edge-even graceful labeling, we need to prove that both functions f and fw are injection. (i) Consider the function fw : V (m−Pp) −→ {0, 2, 4, . . . , 2rp− 2} defined by fw(vk) =∑2r j=1 f(vk, vk+sj ) = 4kr − 2M (mod 2rp). Let vx and vy be two vertices in V (m− Pp), if fw(vx) = fw(vy) then 4xr−2M = 4yr−2M (mod 2rp) which leads to x = y. (ii) The function f : E(m−Pp) −→ {2, 4, 6, . . . , 2rp} is defined as f(vx, vx+sj ) = 2[(j − 1)p+ x]. To prove that f is a bijection, we need only to prove that it is one-to-one. Let (vx, vx+si) and (vy, vy+sj ) be two edges, with x, y ∈ {1, 2, 3, · · · , p}, si, sj ∈ S1, and f(vx, vx+sj ) = f(vy, vy+si), which implies that 2[(i− 1)p+ x] = 2[(j − 1)p+ y]. In case of i = j or x = y the proof is trivial. The last case if x ̸= y and j ̸= i, here we will find that x− y = (j− i)p but |x− y| < p and in the same time |p(i− j)| ≥ p which is a contradiction. So, from these three cases, we can be sure that the function f is a one-to-one function and because |E(m− Pp)| = rp, f is a bijection. □ Example 14. Consider the quadruple Paley graph 4 − P17. Here, p = 17,m = 4, d = 4, r = 2, and S1 = {16, 13}. So for each sj ∈ S1, f ( vk, vk+sj ) = 2[(j − 1) 17 + k], and fw(vk) = 8k− 2 ∑2 j=1 sj = 8k− 58 (mod 68). The edge-even graceful labeling of the graph 4− P17 is shown in Figure 15. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 20 of 26 Figure 15: An Edge-Even Graceful Labeling of 4− P17. Example 15. Consider the generalized Paley graph 5 − P7. Here, p = 7,m = 5, d = 1, r = 3, and S1 = {6, 5, 4}. So for each sj ∈ S1, f ( vk, vk+sj ) = 2[(j − 1) 7 + k], and fw(vk) = 12k − 2 ∑2 j=1 sj = 12k − 30 (mod 42). The edge-even graceful labeling of the graph 5− P7 is shown in Figure 16. Figure 16: An Edge-Even Graceful Labeling of 7− P7, 5− P7, 11− P7. Example 16. Consider the cubic Paley graph 3−P19. Here, p = 19,m = 3, d = 3, r = 3, and S1 = {18, 12, 11}. So for each sj ∈ S1, f ( vk, vk+sj ) = 2[(j − 1) 19+ k], and fw(vk) = 12k−2 ∑2 j=1 sj = 12k−82 (mod 114). The edge-even graceful labeling of the graph 3−P19 is shown in Figure 17. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 21 of 26 Figure 17: An Edge-Even Graceful Labeling of 3− P19. 7. Edge odd graceful labeling of generalized Paley graphs Definition 7. A graph G of order n and size m is said to be edge-odd graceful if there exists a bijection f : E(G) −→ {1, 3, 5, . . . , 2m− 1} such that the function fw : V (G) −→ {0, 1, 2, . . . , 2m− 1} , given by fw(u) = ∑ uv∈N(u) f(uv) (mod 2m) is an injection. 7.1. Edge-odd graceful labeling Algorithm for generalized Paley graph of prime order Input: The generalized Paley graph m − Pp with Zp as its vertices and two vertices are joined by an edge if their difference belongs to (Z∗ p) m such that: if m is even then p ≡ 1 (mod 2m) and if m is odd then p is any odd prime. (1) Rename the vertices of the graph as 0 := vp, 1 := v1, 2 := v2, . . . , p− 1 := vp−1. (2) Set r = p−1 2d where d = gcd(m, p− 1), and rewrite (Z∗ p) m as (Z∗ p) m = S = {s1, s2, s3, . . . , s2r : s1 > s2 > s3 > . . . > s2r}. (3) Partition S into two sets. Let S1 = {s1, s2, s3, . . . , sr} and S2 = {sr+1, sr+2, sr+3, . . . , s2r}. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 22 of 26 (4) If sj ∈ S1, the vertex vi+sj is placed in anticlockwise direction of vi and if sj ∈ S2, the vertex vi+sj is placed in clockwise direction of vi. (5) Set f(vi, vi+sj ) = 0 for all i ∈ {1, 2, 3, . . . , p}, j ∈ {1, 2, 3, . . . , 2r}. (6) Set i = 1. Step 1: If i ≤ p then continue to Step 2. Else jump to Step 4. Step 2: For each sj ∈ S1, f ( vi, vi+sj ) = 2[(j − 1) p+ i]− 1. Step 3: i = i+ 1, go back to Step 1. Step 4: For each k = 1, 2, 3, . . . , p, find the weight of the vertex vk using the following mapping: fw(vk) = ∑2r j=1 f(vk, vk+sj ) (mod 2rp). Take M = ∑r j=1 sj , then fw(vk) = ∑r j=1 4[(j − 1) p+ k]− 2 + 2(p− sj) = 4rk − 2(M + r) (mod 2rp). Theorem 7. Paley graphs and their generalizations of prime order are edge-odd graceful graphs. Proof. Following the same steps in the previews theorem we can easily prove that the algorithm defines an edge-odd graceful labeling. □ Example 17. Consider the cubic Paley graph 3−P13. Here, p = 13,m = 3, d = 3, r = 2, and S1 = {12, 8}. So for each sj ∈ S1, f ( vk, vk+sj ) = 2[(j − 1) 13 + k]− 1, and fw(vk) = 8k−2r−2 ∑2 j=1 sj = 8k−44 (mod 52). The edge-odd graceful labeling of the graph 3−P13 is shown in Figure 18. Figure 18: An Edge-Odd Graceful Labeling of 3− P13. Example 18. Consider the quadruple Paley graph 4 − P17. Here, p = 17,m = 4, d = 4, r = 2, and S1 = {16, 13}. So for each sj ∈ S1, f ( vk, vk+sj ) = 2[(j − 1) 17 + k]− 1, and fw(vk) = 8k − 2r − 2 ∑2 j=1 sj = 8k − 62 (mod 68). The edge-odd graceful labeling of the graph 4− P17 is shown in Figure 19. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 23 of 26 Figure 19: An Edge-Odd Graceful Labeling of 4− P17. 8. Paley Graph of prime power order An edge-graceful, edge-even graceful, and edge-odd graceful labelling for Paley graph of order q = pn, where n > 1, remains an open challenge. The Paley graph Pq of order q = pn, where n > 1, has the vertex set V (Pq) = Fq. Here, Fq ̸= {0, 1, 2, . . . , q}, and so the algorithms introduced in this paper are not applicable for this case. For example, if p = 5, n = 2, we get the smallest case without edge-graceful, edge-even graceful, and edge-odd graceful labelling, where V (P25) = F25 = {0, 1, 2, 3, 4, a, 2a, 3a, 4a, a + 1, a + 2, a + 3, a + 4, 2a + 1, 2a + 2, 2a + 3, 2a + 4, 3a + 1, 3a + 2, 3a + 3, 3a + 4, 4a + 1, 4a + 2, 4a + 3, 4a + 4 } = Z5[x]/(x 2 + 2). And (F∗ 25) 2 = {1, 2, 3, 4, a+2, a+3, 2a+1, 2a+4, 3a+1, 3a+4, 4a+2, 4a+3}, so E(P25) = {(xi, xi + xj) ∀xi ∈ F25 and ∀xj ∈ (F∗ 25) 2}. (see Figure 20). The case of P9 is shown in Figure 3. A. N. Elsawy, R. N. Almohammadi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6600 24 of 26 Figure 20: The Paley graph P25. The structure of this field is more complex, and the simple modular arithmetic used in the proof for no longer holds directly. Therefore, we need to understand the structure of the field deeply to construct an edge-graceful, edge-even graceful, and edge-odd graceful labelling algorithm. For more information about the structure of the field, see [3]. Also, an other challenge is to find different types of labellings for Paley graph and its generalizations. 9. Conclusions We introduced three algorithms which produce an edge-graceful, edge-even graceful, and edge-odd graceful labelling for Paley graphs and their generalizations. We proved that Paley graphs and their generalizations including cubic and quadruple Paley graphs of prime order are edge-graceful, edge-even graceful, and edge-odd graceful graphs. 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