EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5614 ISSN 1307-5543 – ejpam.com Published by New York Business Global Graph-Based Approach to the Radio Assignment Problem in Wireless Communication Networks with Applications in Cryptography Nawal M. NourEldeen1,2, Elsayed Badr3,4,5, Ismail M. Hagag6, Hanan Shabana7,∗ 1 Department of Mathematics, College of Science, Taibah University, Madinah, Kingdom of Saudi Arabia 2 Department of Mathematics, Women’s College of Arts, Science and Education, Ain Shams University, Egypt 3 Scientific Computing Department, Faculty of Computers and Artificial Intelligence, Benha University, Benha, Egypt 4 The Egyptian School of Data Science (ESDS), Benha, Egypt 5 Department of Computer Science, College of Information Technology, Misr University for Science and Technology, Giza,Egypt 6 Information Systems Department, El Madina Higher Institute of Administration and Technology, Egypt 7 Physics and Engineering Mathematics Department, Faculty of Electronic Engineering, Menoufia University, Menouf, 32952, Egypt Abstract. The radio labeling problem has numerous applications including military communica- tions, satellite communication, mobile networks, and internet technology. This issue is NP-hard. A rapid algorithm for determining the upper bound of the radio number of a given graph is pro- posed in this paper. The superior performance of the proposed algorithm is attributed to Saha and Panigrahi (2012). The superior performance of the proposed algorithm in terms of CPU time can be attributed to Badr and Mousa(2019). The mathematical model demonstrates that the pro- posed algorithm exhibits superior performance in terms of both the maximum number of radios and CPU time. The proposed algorithm is ultimately assessed using a collection of networks that possess a range of characteristics. Besides that, we present the application of such algorithm in cryptography. 2020 Mathematics Subject Classifications: 94A60, 94A62, 05B30, 0570 Key Words and Phrases: Cryptography, Graph labeling, Mathematical models, NP-hard prob- lem, Radio labeling. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5614 Email addresses: neldeen@taibahu.edu.sa (N. M. NourEldeen), alsayed.badr@fci.bu.edu.eg (E. Badr), Drismailhagag@gmail.com (I. M. Hagag), hananshabana22@gmail.com (H. Shabana) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 2 of 18 1. Introduction Broadcasting of radio and television, satellite communication, mobile networks, in- ternet technology, military communications and many other services are all examples of wireless communication networks applications. The radio frequency (channel) given to transmitting stations from the available frequency spectrum (bandwidth) determines how effectively and efficiently communications take place inside a network. The extensive use of wireless services implies a faster rate of channel saturation, which makes it impossible to keep up with the growing demand for bandwidth. The frequency assignment problem (FAP) is the process of allocating channels (radio frequencies) to transmitters situated at various places while avoiding interference and making the most effective use of the avail- able bandwidth. Graph labeling technique proved as an effective technique for modeling and solving FAP[18]. In graph modeling of FAP, A network of n transmitters (or stations) is formulated as a graph G of n nodes; each node for a transmitter. Any two nodes are adjacent if their corresponding transmitters are the geographically close. The task is to assign a channel to each transmitter and avoiding interference. Consequently, Almost, The graph G is referred as an interference graph. Here, the interference is closely related to the geographical position of the transmitting stations. The interference is increasing whenever the stations are geographical close. To avoid such interference, then we have to assign frequencies to stations such that difference between the assigned channels must be large enough. Hence, in solving FAP as a graph labeling the objective is to find a labeling for all the vertices of the interference graph such that the range of the assigned channels is minimized. Griggs and Yeh [17] presented the first graph labeling technique, also known as L(2, 1) labeling or distance two labeling to address the frequency assignment problem. The graph labeling L(2, 1) of a simple graph G is a non-negative real-valued function L : V (G) → [0, ∞) where V (G) is the vertex set of the graph G. Whenever u and v are two vertices in G such that the distance between u and v is 1, then | L(u) − L(v) | ≥ 2, and whenever the vertices u and v are of distance 2 from each other, then | L(u) − L(v) | ≥ 1. A novel graph labeling method called radio Labeling was introduced in 2001 by Char- trand et al [11] along with a modified definition of L(2, 1)-labeling. In radio labeling the function L : V (G) → N , where N is the set of integer numbers, is used to give the radio labeling for the vertices of the graph G. The function L is said to be a radio labeling of the graph G if it satisfies the following condition: |L(u)− L(v)| ≥ diam(G) + 1 − d(u, v) for each u, v ∈ V (G), with diam(G) being the diameter of G and d(u, v) being the shortest distance between the vertices u, v. The difference between the highest and lowest labels employed by a radio labeling L is known as the span of the labeling L. The main objec- tive of the radio labeling problem is to determine the minimum span across all over such labeling L on the graph G. Such minimum span is called the radio number of G or as rn(G). Such radio number is important in measuring the efficiency usage of the available bandwidth of radio frequencies. Determining the radio number of an arbitrary graph has been demonstrated to be an NP-complete problem [19]. Authors of [30] introduced an algorithm that determines the lower and upper bounds of the radio number of a graph. N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 3 of 18 Badr and Moussa [6] proposed the development of Sahas’ algorithm and introduced a novel mathematical model for the radio labeling application. Radio labeling problem has been studied for different families of graphs [4, 5, 20–25, 27, 28, 31, 32, 39]. On the other hand, for more details about the mathematical models, the reader is referred to [1–3, 7–10, 13– 16, 29, 37, 38]. Finally, for more details about the theory of cryptography, the reader may refer to [12, 26, 33–36]. Obtaining the radio number of generic graphs is computationally challenging and it is NP-hard to determine, the radio number of general graphs of diameter 2 and its complexity is unknown in general. In literature, there are some algorithms for finding an upper bound for radio number of a given graph. The main challenge of such algorithms is determining the first vertex of the graph G that will labeled by zero. In [30] Laxman Saha select a random vertex to label by zero. The label zero is fixed at that vertex through all steps of Sahas’ algorithm. Badr[6] proposed an algorithm moves the label zero from the vertex to another vertex in order to get the minimum radio number. Therefore, it gets the better upper bound of the radio number for a given graph G. However, if Badr algorithm moves all labels from the vertex to another vertex the algorithm will become an exponential algorithm. Here, we claim to introduce an algorithm that give better upper bound and minimize the time of finding such upper bound. In this paper, a fast algorithm determining the upper bound of the radio number of a given graph is proposed. The proposed algorithm overcomes the other which is due to Saha and Panigrahi [30]. According to CPU time, the proposed algorithm outperforms the other which is due to Badr and Mousa [6]. The superiority of the proposed algorithm on the mathematical model is introduced for both the upper bound of the radio number and CPU time. Finally, the proposed algorithm is evaluated by a set of networks, which have variety properties. The rest of the paper is structured as follows Section 2 present the fundamentals of the proposed algorithm for finding the upper bound of the radio labeling for a graph G. Besides that, the mathematical model for determining the upper bound of the radio number of a graph is presented. A computational study for verification of the validity and efficiency of the proposed algorithm is presented in Section 3. A cryptography application is presented in Section 4. The conclusion of the paper and future work are presented in Section 5. 2. Methodology In this section, we propose an algorithm for finding the upper bound of the radio la- beling for a graph G. We explain the mechanism of the proposed algorithm and its time complexity. On the other hand, Theorem 1 is introduced to prove the validation of this algorithm. The basic idea of radio number upper bound optimization is how to choose the vertex that has the labeling zero. In 2012, Saha and Panigrahi [30] presented an algorithm to find the upper bound and lower bound of a radio number for a graph. They fixed the zero label N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 4 of 18 for the first vertex while Badr and Moussa [6] proposed an algorithm that give an upper bound better than the results are due to Saha [30]. In [6] the label zero is moved from one vertex to another vertex in order to obtain a better upper bound for the radio number. Figure 1: The label zero randomly selected and fixed at x1 according to Laxman Saha [30]. Figure 2: The label zero moves from x1 to x4 according to Badr [6]. Figure 3: The label zero fixed at x2 according to the proposed algorithm Figure 1 shows that Saha and Panigrahi fixed the zero label for the first vertex while Figure 2 shows that Badr and Moussa move the zero label from one vertex to another vertex in order to obtain a better upper bound for the radio number. Finally, Figure 3 shows that label zero will assign to the center vertex of the given graph as we will describe in Section 2.1. 2.1. The description of the proposed algorithm Let G be a connected, simple, and undirected graph with n vertices. Let V (G) be the vertex set of G. For a vertex u ∈ V (G), let the sum of distances from the vertex u to every other vertex in G is ds(u). Thus, ds(u) = ∑ v∈V (G) d(u, v). Let X = {ds(u), ∀u ∈ V (G)}. A vertex u is said to be the center of the graph G if ds(u) = min(X) N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 5 of 18 Hereafter, we use the center vertex of the graph G to find the upper bound of the radio number of G. We aim to assign a radio labeling L to these vertices such L (v0) < L (v1) < L (v2) < . . . < L (vn−1). If L is a radio labeling of G, then |L (vj+1)− L (vj)| ≥ diam(G) + 1− d ( vj+1, vj ) for all 0 ≤ j ≤ n− 2. Consequently, L (vj+1)− L (vj) = diam(G) + 1− d ( vj+1, vj ) +σj such that σj is a positive integer. Moreover, L (vj+1) = l (vj) + (diam(G) + 1)− d ( vj+1, vj ) +σj for all 0 ≤ j ≤ n− 2. Therefore, L (vn−1) = L (v0)+ (n − 1) (diam(G) + 1) − ∑n−2 j=0 d ( vj+1, vj ) + ∑n−2 j=0 σj As, L (v0) < L (v1) < L (v2) < . . . < L (vn−1) then, span (L) = L (vn−1) . Consequently, rn(G) ≤ L (vn) . Our claim now is to minimize ∑n−2 j=0 d ( vj+1, vj ) + ∑n−2 j=0 σj for all 0 ≤ j ≤ n− 2. satisfying that |L (vj+1)− L (vj)| ≥ diam(G) + 1− d (vj+1, vj) ; 0 ≤ j ≤ n− 2. The radio labeling of the vertex v1 will be as follow L (v1) ≥ L (v0) + (diam(G) + 1)− d (v1, v0) As we assume that L (v0) = 0 then the upper bound of the value of L (v1) will depend only on the value of d (v1, v0) . The less the value of d (v1, v0), the more value of L (v1). Hence, it is a good choice when we choose v0 to be the center vertex of G. After that we work to label a vertex from V (G) differ from v1, v0 and consider this vertex as v2 satisfying that |L (v2)− L (v1)| ≥ diam(G) + 1− d (v2, v1) and |L (v2)− L (v0)| ≥ diam(G) + 1− d (v2, v0) Then we do the following computation, for each vertex u ∈ V (G)− {v1, v0} . Find L(u) = max {L (v0) + diam(G) + 1− d (v0, u) , L (v1) + diam(G) + 1− d (v1, u) } Consequently, the vertex u will consider as v2 and L (v0) < L (v1) < L (v2) . To find the vertex v3, we compute the below computations. For every vertex u belongs to V (G)− {v0, v1, v2} compute L(u) = max { L (v0) + diam(G) + 1− d (v0, u) , L (v1) + diam(G) + 1− d (v1, u) , L (v2) + diam(G) + 1− d (v2, u) } (1) Then the vertex v3 will be the vertex u with minimum value of L(u) from the com- putations proceed in Equation 1. The previous computations is proceed until the vertex vn−1 is labeled. N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 6 of 18 Hereafter, we introduce an algorithm that output an upper bound of a radio labeling of a graph G. Algorithm 1. The Upper bound of the radio number of a given graph G Input: The adjacency matrix of the graph G and the diameter D of G. Output: The upper bound of radio number of G. Begin 1: For all u ∈ V (G) compute, ds(u) = ∑ v∈V (G) d(u, v). 2: Let min1 = minu∈V (G) {ds(u)} 3: Choose a vertex u ∈ V (G), such that ds(u) = min1 4. Assign, L(u) = 0 . 5:X = {u} . 6: For all v ∈ V (G)−X, compute, temp(v) = maxt∈X {L(t) +D + 1− d(v, t)} 7: Let min2 = minv∈V (G)−X {temp(v)} 8: Choose a vertex v ∈ V (G)−X, where temp(v) = min2 9: Assign, L(v) = min2 10: X = X ∪ {v} . 11: Repeat Step 6 to Step 10 until all vertices are labeled. Stop. End. Theorem 1. Algorithm 1 results in a radio labeling and an upper bound of the radio number for a graph G Proof. Let Algorithm 1 be at intermediate step r, where the vertices of G that belong to the set X = {u1, u2, . . . ., ur} are radio labeled such the radio label of a vertex ui = L (ui) for all 1 ≤ i ≤ r. In the next step, to label a new vertex v, the Algorithm 1 picks a vertex v ∈ V − X such that temp(v) = maxt∈X {L(t) +D + 1− d(v, t)} is the minimum. According to Algorithm 1, L(v) = temp(v). Let us call the vertex v as ur+1, we find that L (ur+1) ≥ L(t) + D + 1 − d(ur+1, ui) for all ui ∈ X. Thus, |L (ur+1)− L (ui)| ≥ D+1− d(ur+1, ui) for all ui ∈ X. Therefore, Algorithm 1 results in a radio labeling for the given graph G and an upper bound for radio number of G. Complexity of Algorithm 1. step 1 has a loop which has O(n) time complexity. Step 2 has O(n) time complexity and step 3 has O(n) time complexity. Both of steps 4 and 5 has O(1) time complexity. In step 6, there is a nested loop which has O(n2) time complexity. Each of steps 7 and 8 has O(n) time complexity. For steps 9 and 10 each has one operation which has 2 O(1) time complexity. At the end, step 11 has O(n3) time complexity. Therefore, the proposed algorithm has the following time complexity, 5O(n)+ 4O(1) + O ( n2 ) +O ( n3 ) = O(n3) N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 7 of 18 2.2. The integer programming mathematical model (ILPM) for the radio labeling problem A new mathematical formulation for the radio labeling problem is proposed. We next describe the problem of finding the radio labeling problem for a graph in terms of an integer programming problem [19]. Let G be a connected graph of order n with V (G) = {u1, u2, · · · , un } and let D = [dij ] be the distance matrix of G, that is dij = d (ui, uj) for 1 ≤ i, j ≤ n. We suppose that vi are the labels of the vertices ui such that 1 ≤ i ≤ n. Now, we can introduce the mathematical model for the radio labeling problem as integer programming model. We define the function F by minF = v1 + v2 + · · ·+ vn Subject to: |vi − vj | ≥ diam+ 1− d (ui, uj) for 1 ≤ i ≤ n− 1; 2 ≤ j ≤ n and i < j where v1, v2, · · · , vn ∈ {0, 1, 2, . . .} The radio number of the graph G = max1≤i≤n {vi} . 3. Results and discussion We describe our numerical experiments and present computational results, which demonstrate the efficiency of the proposed algorithm on a set of test graphs (path, cycle, friendship graphs (F 3,k , F 4,k , F 5,k ), ∆k -snake and kC4-snake and triangular path P3,k). Compiled using python 3.11, all runs were carried out under macos, on MacBook Air with Apple M1 chip with 8-core CPU and 7-core GPU, 8GB RAM In Tables 1-8 the abbreviations UB1, UB2 and UB3 are used to denote upper bounds of radio number due to Saha and Panigrahi [30], Badr and Moussa [6] and the proposed algorithm (algorithm 1) respectively. Definition 1. Let k be a positive integer. A triangular snake graph denoted as ∆k -snake is a connected and simple graph of k-blocks and each block is cycle C3 and the block-cut- point graph is a path. Figure 4 illustrates the graph ∆k -snake. Figure 4: ∆k -snake Definition 2. For given positive integers k, m, a friendship graph denoted as Fm,k and represented as k cycles (blocks) each of length m and all have a one vertex common. For more illustration, Figure 5 shows the graph F4,k as an example of friendship graph. N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 8 of 18 Definition 3. For a positive integer k, the graph denoted as kC4-snake is a connected and simple graph of k-blocks and each block is cycle C4 and the block-cut-point graph is a path. The graph kC4-snake is shown in Figure 6. Definition 4. For a positive integer k, a triangular path P3,k is a connected and simple graph with |V (P3,k)| = 3k and |E (P3,k)| = 4k − 1. It consists of a path Pk of k vertices such that every vertex from this path is a vertex in a cycle C3. The illustration of P3,k is shown in Figure 7. Figure 5: friendship F4,k. Figure 6: kC4-snake N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 9 of 18 Figure 7: Triangular path P3,k. Tables 1-8 present the comparisons among four different algorithms, UB1, UB2, ILPM, and UB3, for path (Pn), cycle (Cn), friendships (F3,k, F4,k, F5,k, ) triangular snakes ∆k- snake and kC4-snake and triangular path (P3,k). The table provides various metrics, including the upper bound of the radio number rn(G) and the CPU time for each algorithm at different values of n. The first column (in Tables 1, 2 and second column in Tables 3-8) represents the value of n, which indicates the size (number of vertices) of the graph G. The second column (in Tables 1, 2 and third column in Tables 3-8) shows the value of rn(G) obtained using the UB1 algorithm. Similarly, the further consecutive columns represent the values of rn(G) obtained using the UB2 and ILPM algorithms. The last column presents the values of rn(G) obtained using the UB3 algorithm. The subsequent columns provide the CPU time taken by each algorithm to compute the respective rn(G) values for the given graph size. Table 1: A comparison among UB1, UB2, UB3, and ILPM for G = path graph Pn n UB1 UB2 ILPM UB3 rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time 1 0 0.000181 0 0.000353 0 0.000111 0 0.000033 2 1 0.000083 1 0.000454 1 0.000015 1 0.000041 3 3 0.000085 3 0.000733 4 0.000033 3 0.000058 4 6 0.000124 5 0.000014 9 0.00004 5 0.000077 5 11 0.000217 10 0.000267 16 0.000461 10 0.000126 6 15 0.000265 14 0.000531 25 0.000543 14 0.000209 7 22 0.000375 20 0.000705 36 0.000098 20 0.000308 8 28 0.000505 26 0.000934 49 0.000197 26 0.000372 9 37 0.000539 35 0.00123 64 0.000261 35 0.000504 10 45 0.000815 42 0.001666 81 0.000305 42 0.000674 11 56 0.000888 53 0.002232 100 0.000393 53 0.000828 12 66 0.001091 62 0.002816 121 0.000563 62 0.001044 13 80 0.001379 75 0.003461 144 0.000678 76 0.001316 14 91 0.001666 86 0.00451 169 0.000717 86 0.001645 50 1225 0.062379 1202 0.186789 2401 0.021464 1202 0.062123 Table 1 indicates that, for the path, the proposed algorithm is preferable to UB1. On the other hand, we discover that the proposed algorithm is preferable to UB2 in terms of execution time, while it is equivalent to UB2 until n is less than or equal to 12. When n is greater than or equal to 13, the algorithm UB2 oscillates between equality with the proposed algorithm and superiority. The proposed algorithm is preferable to ILPM . As we can observe, for n = 1, all algorithms produce an rn(G) value of 0. Each algorithm takes a different amount of CPU time to compute this result. UB1 takes the least time, N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 10 of 18 followed by UB2, ILPM, and UB3. As n increases, the rn(G) values also increase for all algorithms. This indicates that the complexity of the graph structure increases as the size of the path graph grows. The CPU time also increases as the graph size increases. This is expected since larger graphs require more computational effort to compute the rn(G) values. For larger values of n, such as n = 50, the CPU times become significantly higher for all algorithms. This indicates that the computational complexity of these algorithms increases with the size of the input graph. Table 2: A comparison among UB1, UB2,UB3 and ILPM for G = cycle n UB1 UB2 ILPM UB3 rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time 1 0 0.000198 0 0.0000348 - 0.0000418 0 0.0000329 2 1 0.0000478 1 0.0000434 - 0.0000251 1 0.0000362 3 2 0.0000738 2 0.0000862 2 0.0000426 2 0.0000537 4 4 0.000116 4 0.000157 6 0.0000646 4 0.0000757 5 4 0.000127 4 0.00027 8 0.0001079 4 0.000121 6 7 0.000183 7 0.000441 15 0.0001039 7 0.000150 7 10 0.000265 9 0.000644 18 0.0001659 10 0.000245 8 13 0.000371 13 0.000977 28 0.0002579 13 0.000369 9 13 0.000512 12 0.001314 32 0.0004969 13 0.000508 10 17 0.000686 17 0.001701 45 0.0005569 17 0.000668 11 21 0.00088 20 0.002198 50 0.0008419 21 0.000853 12 26 0.001096 26 0.002863 66 0.0010169 26 0.001028 13 25 0.001412 24 0.003559 72 0.0011609 25 0.001272 14 31 0.001693 31 0.004457 91 0.0014649 31 0.001576 50 337 0.062019 337 0.186451 1225 0.0320259 337 0.062002 Table 2 indicates that, for the cycle, the proposed algorithm is preferable to UB1. On the other hand, we discover that the proposed algorithm is preferable to UB2 in terms of execution time, while it is equivalent to UB2 until n is less than or equal to 12. When n is greater than or equal to 13, the algorithm UB2 oscillates between equality with the proposed algorithm and superiority. The proposed algorithm is preferable to ILPM. Tables 3-5 indicate that, for the friendships F 3,k , F 4,k, and F 5,k , the proposed algo- rithm is preferable to UB1. On the other hand, we discover that the proposed algorithm is preferable to UB2 in terms of execution time, while it is equivalent to UB2 until n is less than or equal to 12. For F 3,k , Table 3 shows that the algorithms UB2 and the proposed algorithm are equivalent except at k = 3, 5, 7. For F 4,k, and F 5,k , Tables 3-4 show that the algorithms UB2 and the proposed algorithm are equivalent except at k = 2. The proposed algorithm is preferable to ILPM. Table 3: A comparison among UB1, UB2, UB3 and ILPM for G = friendship graph F 3,k K n UB1 UB2 ILPM UB3 rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time 1 3 2 0.000352 2 0.000117 4 0.000075 2 0.000116 2 5 5 0.000199 5 0.000277 7 0.000037 5 0.000148 3 7 8 0.000347 7 0.000675 10 0.000172 8 0.000283 4 9 9 0.000542 9 0.001369 13 0.000408 9 0.000519 5 11 12 0.000964 11 0.002313 16 0.000847 12 0.000958 6 13 13 0.001391 13 0.00376 19 0.000291 13 0.001402 7 15 16 0.002084 15 0.005837 22 0.000861 16 0.001972 8 17 17 0.002893 17 0.008048 25 0.001774 17 0.002885 9 19 20 0.004048 19 0.011799 28 0.002924 20 0.004035 10 21 21 0.00543 21 0.015302 31 0.004015 21 0.005126 11 23 24 0.007658 23 0.020273 34 0.005977 24 0.007088 12 25 25 0.009249 25 0.025301 37 0.007643 25 0.008754 13 27 28 0.011044 27 0.03207 40 0.000837 28 0.010948 14 29 29 0.014288 29 0.039837 43 0.003468 29 0.013579 50 101 101 0.675397 101 1.773485 151 0.263206 101 0.574317 N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 11 of 18 Table 4: A comparison among UB1, UB2, UB3 and ILPM for G = friendship graph F 4,k K n UB1 UB2 ILPM UB3 rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time 1 3 4 0.000115 4 0.000115 5 0.000063 4 0.000114 2 5 16 0.00024 15 0.00042 17 0.000128 16 0.000239 3 7 22 0.000676 22 0.001321 23 0.000276 22 0.000636 4 9 29 0.001434 29 0.003049 30 0.000525 29 0.001387 5 11 36 0.002458 36 0.006385 37 0.001315 36 0.002426 6 13 43 0.004054 43 0.011824 44 0.002745 43 0.003856 7 15 50 0.006445 50 0.019918 51 0.005039 50 0.00615 8 17 57 0.008756 57 0.031611 58 0.007677 57 0.008788 9 19 64 0.013099 64 0.047747 65 0.011479 64 0.012702 10 21 71 0.01666 71 0.069004 72 0.011591 71 0.016403 11 23 78 0.022505 78 0.09524 79 0.015292 78 0.02259 12 25 85 0.029908 85 0.130633 86 0.017542 85 0.028653 13 27 92 0.036358 92 0.174968 93 0.025987 92 0.037098 14 29 99 0.04407 99 0.228926 100 0.03328 99 0.044391 50 101 351 2.107236 351 27.88631 352 0.771716 351 1.882827 Table 5: A comparison among UB1, UB2, UB3 and ILPM for G = friendship graph F 5,k K n UB1 UB2 ILPM UB3 rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time 1 5 4 0.00032 4 0.00031 14 0.000071 4 0.000182 2 9 18 0.000517 17 0.001497 24 0.000457 18 0.000108 3 13 25 0.001336 25 0.003841 35 0.000315 25 0.001426 4 17 33 0.002867 33 0.008418 46 0.001768 33 0.002879 5 21 41 0.005361 41 0.015528 57 0.004308 41 0.005419 6 25 49 0.008805 49 0.025917 68 0.007776 49 0.008887 7 29 57 0.013459 57 0.039922 79 0.002785 57 0.013896 8 33 65 0.01975 65 0.058754 90 0.008407 65 0.019518 9 37 73 0.027393 73 0.082683 101 0.016351 73 0.027462 10 41 81 0.038485 81 0.110506 112 0.025411 81 0.036522 11 45 89 0.048896 89 0.14308 123 0.038162 89 0.049273 12 49 97 0.060881 97 0.180638 134 0.050256 97 0.061367 13 53 105 0.078234 105 0.230682 145 0.067696 105 0.078807 14 57 113 0.097254 113 0.285646 156 0.085658 113 0.096769 50 201 401 4.265029 401 12.58799 552 3.06101 401 4.072021 Tables 6 and 7 indicate that, for the ∆k- snakes and, kC4-snakes, the proposed al- gorithm is preferable to. On the other hand, we discover that the proposed algorithm is preferable to UB2 in terms of execution time, while it is equivalent to UB2 until n is less than or equal to 12. For ∆k- snakes Table 6 shows that the algorithms UB2 and the proposed algorithm are equivalent. For kC4-snakes, Table 7 shows that the algorithms UB2 and the proposed algorithm are equivalent except at k = 2, 5, 7, 13. The proposed algorithm is preferable to ILPM. Table 6: A comparison among UB1, UB2, UB3 and ILPM for triangular snakes ∆k K n UB1 UB2 ILPM UB3 rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time 1 3 2 0.000297 2 0.000184 2 0.000093 2 0.000204 2 5 5 0.000138 5 0.000293 6 0.000035 5 0.000126 3 7 12 0.000283 11 0.000756 14 0.000082 11 0.000263 4 9 18 0.000531 17 0.001378 26 0.000032 17 0.000513 5 11 30 0.000871 28 0.002301 42 0.000037 28 0.000248 6 13 39 0.001379 37 0.003772 62 0.000038 37 0.001249 7 15 56 0.002113 52 0.005715 86 0.000097 52 0.002108 8 17 68 0.003721 65 0.008235 114 0.000314 65 0.003005 9 19 90 0.004074 85 0.011566 146 0.000321 85 0.004232 10 21 105 0.005461 101 0.015212 182 0.000494 101 0.005805 11 23 132 0.007225 125 0.020041 222 0.000584 125 0.007395 12 25 150 0.008868 145 0.025406 266 0.000696 145 0.008807 13 27 182 0.010933 174 0.031775 314 0.000751 174 0.010862 14 29 203 0.013203 197 0.040166 366 0.000277 197 0.013388 50 101 2525 0.661005 2501 1.575407 4902 0.000995 2501 0.563346 N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 12 of 18 Table 7: A comparison among UB1, UB2, UB3 and ILPM for G= kC4-snakes graph K n UB1 UB2 ILPM UB3 rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time 1 4 4 0.000359 4 0.000182 4 0.000027 4 0.000138 2 7 17 0.00029 15 0.00069 17 0.000199 16 0.00021 3 10 33 0.000686 31 0.001912 40 0.000604 31 0.000615 4 13 58 0.001367 55 0.003751 75 0.000267 55 0.001338 5 16 86 0.002531 83 0.006906 122 0.001308 85 0.002419 6 19 122 0.004264 117 0.011311 181 0.002739 117 0.00385 7 22 162 0.006756 157 0.017645 252 0.00501 158 0.006121 8 25 212 0.009442 205 0.025285 335 0.007432 205 0.008543 9 28 266 0.012434 257 0.035349 469 0.001122 257 0.012233 10 31 325 0.016581 316 0.048055 579 0.004575 316 0.015686 11 34 389 0.022227 379 0.063325 656 0.009842 379 0.020953 12 37 462 0.028546 451 0.081115 787 0.016127 451 0.027238 13 40 538 0.034988 527 0.101463 873 0.022944 529 0.034055 14 43 622 0.043549 609 0.12424 981 0.030786 609 0.041897 50 151 7625 1.866284 7576 5.209234 14657 0.654571 7576 1.765682 Table 8 indicates that, for the path triangular P3,k, the proposed algorithm is preferable to UB1. On the other hand, we discover that the proposed algorithm is preferable to UB2 in terms of execution time, while it is equivalent to UB2 until n is less than or equal to 12. For P3,k Table 8 shows that the algorithms UB2 and the proposed algorithm are equivalent. The proposed algorithm is preferable to ILPM. Table 8: A comparison among UB1, UB2, UB3 and ILPM for G = triangular path graph K n UB1 UB2 ILPM UB3 rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time rn(G) CPU Time 1 3 2 0.000127 2 0.000123 2 0.000027 2 0.0000887 2 6 8 0.000198 8 0.000523 12 0.000234 8 0.000193 3 9 18 0.000543 17 0.001477 26 0.0003968 17 0.000519 4 12 32 0.001123 29 0.003144 46 0.0004982 29 0.001085 5 15 46 0.002093 45 0.005782 72 0.0005294 45 0.002035 6 18 65 0.003492 61 0.009949 104 0.0006289 61 0.003373 7 21 86 0.005367 82 0.016159 142 0.0007928 82 0.005314 8 24 110 0.007732 105 0.023089 186 0.0008923 105 0.007500 9 27 137 0.010508 134 0.032466 236 0.0019276 134 0.010086 10 30 167 0.015182 161 0.043037 292 0.0029187 161 0.014900 11 33 200 0.019532 198 0.059199 354 0.0030983 198 0.019228 12 36 236 0.025369 229 0.075771 422 0.0049876 229 0.025317 13 39 277 0.033481 274 0.094781 496 0.005691 274 0.031756 14 42 317 0.039851 309 0.115118 576 0.006892 309 0.039622 50 150 3827 1.840115 3801 5.127259 7452 0.26579 3801 1.702862 4. Application of radio labeling in cryptography In this section, we claim to use the upper bound of radio number of a graph obtained from Algorithm 1 for Encryption and decryption. We will consider the Triangular Path graph for the proposed cryptography approach. 4.1. Cryptography approach Let's the sender and the receiver wish to communicate in secure manner. Assume that the letters A through Z in the English alphabet make up the message M, with spaces separating each word. Let the numerical counterparts of each letter in the English alphabet A, B,..., and Z be 1, 2,..., and 26 in this cryptosystem. Let 0 represent empty space and punctuation is disregarded. We introduce a matrix-based cryptographic technique. Thus N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 13 of 18 only both of sender and receiver are know the key, which is an invertible matrix. In this section, we suggest a method for creating such a matrix using upper bound of radio number of the triangular path graph P3,k. Suppose the length of plaintext M is k. Then, we consider triangular path P3,k. Let Algorithm 1 produce that the upper bound of radio number of P3,k that is rn (P3,k ). After that, a 2x2 matrix A is constructed as follows A =  rn (P3,k )− 2 rn (P3,k )− 3 rn (P3,k ) rn (P3,k )− 1 mod 27 As the determinant of A does not equals 0. Then A is an invertible matrix. As a result, the sender may decide to utilize the matrix A for encryption while the receiver may use the matrix A−1 for decryption, where A−1 = 1 2  rn (P3,k )− 1 3− rn (P3,k ) −rn (P3,k ) rn (P3,k )− 2 mod 27 4.2. Cryptography algorithms Hereafter, the encryption and decryption processes are described in details. Encryption Algorithm Input: A message M of length k Output: A cipher text C of the message M. Begin. 1.Consider triangular path P3,k. 2.Use Algorithm 1 to find the upper bound of radio number of P3,k. 3. If Algorithm 1 results the upper bound of radio number of P3,k as α then construct an 2x2 matrix A = [ α− 2 α− 3 α α− 1 ] mod 27. 4. Convert each letter (including space) in the plaintext M to its equivalent number. 5. Make this string of numbers as a set of ordered pairs. We pair the last number of the string with 0 whenever, the string has odd length. 6. An ordered (x, y) is encrypted to [ x′ y′ ] = A [ x y ] mod 27. 7. Represent numbers in each encrypted [ x′ y′ ] by space or its equivalent character in the English alphabet. 8. Form the cipher text C by appending 0 or 1 at the end of the sequence of characters produced in step 7. Appending 0 indicates that the message M is of length, even and 1 indicates that the message M has length is of length, odd. End N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 14 of 18 Decryption Algorithm. Input: A cipher text C of length w Output: A plaintext (Original message) M of C. Begin 1. If 0 is found at the end of C, obtain the upper bound of radio number α of triangular path P3,w. Using algorithm 1. If 1 is found at the end of C, use algorithm 1 to obtain the upper bound of radio number, α of triangular path P3,w−1. 2. Build the key matrix for decryption that is B = 1 2 [ α− 1 3− α −α α− 2 ] mod 27. 3. Convert the characters and spaces in C to ordered pairs of their equivalent numerical values 4. Each ordered pair (x′, y′) is decrypted to [ x y ] = B [ x′ y′ ] mod 27. 5. Represent numbers in each matrix [ x y ] by space or its equivalent character of x and y in the English alphabet to deduce the plaintext M of the received ciphertext C. End For more illustration of the cryptography approach, we pose the following example. Example 1. Let the sender wants to send the message “PAR’ ’ to the receiver then en- cryption process is done as follow 1. Since the length of the original message is 3, we use algorithm 1 to get the upper bound of radio number of P3,3. 2. Following Table -8, k=3, UB3, the upper bound of radio number of P3,3 is 17. hence, construct an 2x2 matrix A = [ 15 14 17 16 ] . 3. Represent each character (including space) in the plaintext M by its numerical equiv- alent. That is 16, 1, 18. 4. Convert this string of numbers to a set of ordered pairs as (16,1), (18,0). 5. (16, 1), is encrypted as [ x′ y′ ] = [ 15 14 17 16 ] [ 16 1 ] mod 27. = [ 11 18 ] and the ordered pair (18,0) will be [ 15 14 17 16 ] [ 18 0 ] mod 27. = [ 0 9 ] 6. Finally, convert these ordered pairs to equivalent English letter that is 11≡ K, 18 ≡ R, 0 ≡ space, 9 ≡ I. 7. Finally, the cipher text C = KR▽I 1. Here, we append 1 at the end of the sequence of characters produced in step 6 to inform the receiver that the original message has an odd length. On the other hand, we use the symbol ▽ as indication for the space. N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5614 15 of 18 At the receiver, the decryption process performed as follows As the received message is of length w = 4 and has number 1 at the end, then the decryption process use the upper bound of radio number of P3,w−1 = P3,3 that is 17 following Table-8, k=3, UB3. Build the key matrix for decryption B = 1 2 [ 16 −14 −17 15 ] mod 27 = [ 8 20 5 21 ] . Converting the characters and spaces in C to their equivalent numerical values and making ordered pairs yields ordered pairs (11, 18) and (0, 9). Doing a decrypting for each ordered pair as follows (11,18) will be [ 8 20 5 21 ] [ 11 18 ] = [ 16 1 ] and (0,9) will be [ 8 20 5 21 ] [ 0 9 ] = [ 18 0 ] . Representing the entries of the matrices [ 16 1 ] and [ 18 0 ] to their equivalent character (or space) that is [ 16 1 ] ⇒ PA and [ 18 0 ] ⇒ R. Hence, the plaintext is M=PAR. 5. Conclusion This paper proposed a fast algorithm for determining the upper bound of the radio number of a given graph. The proposed algorithm was superior to that of Saha and Pan- igrahi (2012). According to CPU time, the proposed algorithm outperformed the other which is due to Badr and Mousa (2019). The superiority of the proposed algorithm on the mathematical model was introduced for both the upper bound of the radio number and CPU time. An application of radio labeling for cryptography is presented. In the future, we will seek a new algorithm based on selecting the vertex that accepts the label zero so that it is efficient and outperforms the existing algorithms. In addition to testing such an algorithm on graphs with varying specifications for number of vertices, diameter, symmetry, etc., the algorithm must also be examined on graphs with varying specifications for symmetry, diameter, etc. On other hand, wireless technologies have been recently used in a variety of services. Hence, it is more economical to try reusing the frequencies. 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