Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1504 https://internationalpubls.com Exploring Stolarsky-3 Mean Cordial Labeling Properties in Graph Classes with a Python Module C. Muthulakshmi@Sasikala1, A.Akil Nivetha2 1Associate Professor,Department of Mathematics,Sri Paramakalyani, College,Alwarkurichi-627412,Affiliated to Manonmaniam Sundaranar, University, Tirunelveli, Tamilnadu, kalasasispkc@gmail.com 2Research Scholar(Reg No:20211232092001),Department of Mathematics,Sri, Paramakalyani College,Alwarkurichi- 627412,Affiliated to, Manonmaniam Sundaranar, University, Tirunelveli, Tamilnadu, siennanive2021@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: A graph G= (V,E), where V is the set of vertices and E is the set of edges, with p vertices and q edges. A graph G is referred to as a Stolarsky-3 Mean cordial graph if we can assign distinct labels to each vertex x ∈ V from the set {0,1,2}, denoted by f(x), and distinct labels to each edge e = uv ∈ E based on the values assigned to the endpoints u and v. The label for the edge e = uv is calculated using one of the following Stolarsky- 3 Mean cordial formula: 𝑓(𝑒 = 𝑢𝑣) = √ 𝑓(𝑢)2 + f(u)F(v) + 𝑓(𝑣)2 3 The function f is referred to as a Stolarsky-3 mean cordial labeling if the conditions |vf (i) - vf (j)| 1 and |ef (i) - ef (j)| 1. Hold for i, j ɛ{0,1,2 } , where vf(x) and ef(x) represent the number of vertices and edges respectively, labeled with x ( x= 0,1,2 ). A graph that admits such a labeling is called a mean cordial graph. Such that the edge labels are derived from the flooring function of the Stolarsky-3 mean of the labels of the two end vertices of each edge. In this paper, we have created python module to analysis the Stolarsky-3 mean cordial labeling properties of various graph classes, including Path (Pn), Cycle (Cn), Wheel (Wn), Star Graph (K1,n or Sn) & Wheel and Path Graph (WnPm). Keywords: Stolarsky-3 Mean, Mean Cordial, Path, Cycle, Wheel, Star Graph 1. Introduction One important topic of research in graph theory is graph labeling, which is the process of assigning labels typically numbers to a graph's vertices, edges, or both in accordance with predetermined guidelines [1]. Over the years, various types of labeling techniques have been developed, each with distinct characteristics and applications. Among these, cordial labeling has garnered particular interest due to its relatively relaxed conditions compared to more restrictive forms such as graceful or harmonious labeling [2]. A function assigns labels to the vertices and edges of a graph such that the number of vertices (or edges) labeled with 0 or 1 differs by at most one [3]. This type of labeling has been widely studied for its simplicity and applicability to different types of graphs. The researcher explored how assigning labels based on the arithmetic mean of vertices can lead to Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1505 https://internationalpubls.com distinct edge labels [4, 5]. This work laid the groundwork for further investigations into mean- based graph labeling. Harmonic mean of vertex labels to assign edge labels, providing a different perspective on mean-based labeling techniques [6, 7]. This research expanded the application of mean-based labeling in various types of graphs, contributing to the development of new labeling methodologies. Stolarsky Mean Labeling technique [8-11] utilizes a mean calculation based on the Stolarsky mean, which generalizes several other mean functions. Although specific literature on Stolarsky Mean Labeling is limited, it builds upon the earlier works on mean and harmonic mean labeling, offering a novel approach to graph labeling. In-depth explorations of Stolarsky mean in graph labeling and its potential applications, particularly in cryptography, and Network analysis Applications. Stolarsky-3 mean labeling remains a crucial area of study within graph theory, focusing on assigning labels to vertices and edges in a way that satisfies specific conditions [12]. Till now no one is analyzing Stolarsky-3 mean cordial labeling techniques. Stolarsky-3 mean cordial labeling is a relatively recent addition to this field, expanding on concepts introduced by earlier labeling techniques. This research introduces a new approach by pinpointing vertex combinations and formulating mathematical models that describe Stolarsky-3 mean cordial labeling. Research on Stolarsky-3 Mean Labeling is likely to have implications in areas where graph labeling is crucial, such as network design and communication theory. 2. Methods: We have developed the python module to find the graph labeling properties. 2.1. Python Module import itertools import math def stolarsky_3_mean(u, v): return math.floor(math.sqrt((u**2 + u*v + v**2) / 3)) # Function to check the conditions for valid Stolarsky-3 mean cordial labeling def is_valid_labeling(s1, s2, s3, s4, s5, s6, m1, m2, m3, m4): return ( abs(s1 - s2) <= 1 and abs(s2 - s3) <= 1 and abs(s1 - s3) <= 1 and abs(s4 - s5) <= 1 and abs(s5 - s6) <= 1 and abs(s4 - s6) <= 1 and s1 + s2 + s3 == m1 and s4 + s5 + s6 == m4 ) # Main function to generate and validate Stolarsky-3 mean cordial labeling def stolarsky_3_mean_cordial_labeling(n, m1, m2, m3, m4): combinations = itertools.product([0, 1, 2], repeat=n) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1506 https://internationalpubls.com valid_combinations = [] # To store valid labelings for combination in combinations: s4 = s5 = s6 = 0 for i in range(len(combination) - 1): # Assuming the graph is a path (Pn) u, v = combination[i], combination[i + 1] edge_label = stolarsky_3_mean(u, v) if edge_label == 0: s4 += 1 elif edge_label == 1: s5 += 1 elif edge_label == 2: s6 += 1 if is_valid_labeling(s1, s2, s3, s4, s5, s6, m1, m2, m3, m4): valid_combinations.append((combination, (s1, s2, s3, s4, s5, s6))) print("Valid Stolarsky-3 Mean Cordial Labelings:") for combo, counts in valid_combinations: print(f"Vertex Labels: {combo}, Counts: {counts}") stolarsky_3_mean_cordial_labeling(n, m1, m2, m3, m4)he process, printing only those combinations that satisfy the specified conditions. This Python module will analyze and determine the Stolarsky-3 mean cordial labeling characteristics in various types of graphs. Let me know if you'd like further assistance with this paper 3.Results and Analysis Table 1. Comparative analyses of 3 different mean labeling with respect to various graphs. The table provided compares different types of graph labeling with respect to mean cordial labeling, Stolarsky-3 mean labeling, and Stolarsky-3 mean cordial labeling. Let's break down and analyze these findings across different graph types: Path Graph (Pn) Theorem 3.1: Every Path Pn is Stolarsky-3 Mean Cordial. Proof: Consider a Path Pn with vertices u1,u2,....., un. Define f:V(Pn)____ {1, 2,.....,q+1}. Therefore f(ui) =i, 1≤i≤n. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1507 https://internationalpubls.com Then the edge labels are distinct. Case 1:n≡0(mod3) Let n=3t.f(ui) =0,1≤i≤t, f(ut+i) =1, 1≤i≤t, f(u2t+i)=2, 1≤i≤t. Then vf(0) =vf(1) =vf(2) =t and ef(0) =t-1, ef(1)=ef(2) =t. Case 2:n≡1(mod3) Let n=3t+1.f(ui) =0, 1≤i≤t, f(ut+i)=1,1≤i≤t+1,f(u2t+1+i)=2, 1≤i≤t+1. Then vf(0) =t+1, vf(1) =vf(2) =t and ef(0) =ef(1) =ef(2) =t. Case 3:n≡2(mod3) Let n=3t+2, f(ui)=0, 1≤i≤t, f(ut+i)=1, 1≤i≤t+1, f(u2t+1+i)=2, 1≤i≤t+1. Then vf(0) =t+1, vf(1)=vf(2)=t and ef(0) =t+1, ef(1)=ef(2)=t. Hence the above three cases satisfies the condition |vf(i)-vf(j)|≤1 and |ef(i)-ef(j) |≤1 for all i, j ∈{0, 1,2}. Hence f is a Stolarsky-3 Mean Cordial labeling. The Stolarsky-3 Mean Cordial Labeling of the Path Graph (Pn) is illustrated in Figure 1. The Mean Cordial Labeling method is applicable to graphs with any number of vertices. Similarly, Stolarsky-3 Mean Labeling is also effective for graphs with any number of vertices n. Furthermore, Stolarsky-3 Mean Cordial Labeling remains valid and applicable for all values of n, ensuring its versatility across different graph sizes (Table 1). All three labeling methods work well for path graphs, no matter how many vertices they have. This is because path graphs are simple and linear, making them easy to label. Cycle Graph (Cn) Theorem 3.2:The Cycle Cn is Stolarsky-3 Mean Cordial iff n≡1, 2(mod3). Proof: Consider a Cycle Cn with vertices u1, u2,...., unu1. Define f:V(Cn) ____ {1, 2,...,q+1} Therefore f(ui) =i,1≤i≤n. Since the edge labels are distinct. Case 1: n≡0(mod3) Let n=3t.If Cn admits Stolarsky-3Mean Cordial labeling f. Then vf(0) =vf(1) =vf(2)=t and ef(0) =ef(1) =ef(2) =t. Assign 0's to t number of vertices Cn. Then we get ef(0) >t. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1508 https://internationalpubls.com Hence f is not a Stolarsky-3Mean Cordial labeling. Case 2:n≡1(mod3) Let n=3t+1, Assign the label 1 to t+1 vertices and the labels 0 and 2 to the remaining each of t vertices. Then vf(0)=vf(1)=t, vf(2) =t+1 and ef(0)=ef(2) =t, ef(1) =t+1. Hence f is Stolarsky-3 Mean Cordial labeling. Case 3:n≡2(mod3) Let n=3t+2, Assign the label 1 to t vertices and the labels 1 and 2 to the remaining each of t vertices. Then vf(0) =vf(1) =t+1, vf(2) =t and ef(0) =ef(2) =t+1, ef(1) =t. Hence f is Stolarsky-3Mean Cordial labeling. The Stolarsky-3 Mean Cordial Labeling of the Cycle Graph (Cn) is illustrated in Figure 2. Mean Cordial Labeling is Works for most, but not when (n = 3, 6, 9,…) etc. (multiples of 3). This happens because of an imbalance in how vertices and edges are connected in these cases. Stolarsky-3 Mean Labeling is Works for any n, even multiples of 3. Stolarsky-3 Mean Cordial Labeling is Same as Mean Cordial Labeling, it does not work when (n = 3, 6, 9,…) etc (Table 1). Stolarsky-3 Mean Labeling is more flexible for cycle graphs, while the other two methods have trouble with graphs where is a multiple of 3. Wheel Graph (Wn) Theorem 3.3:The Wheel graph Wn is not a Stolarsky-3 Mean Cordial for any n. Proof: Consider the Wheel graph Wn, where Cn is a cycle with vertices u1, u2,....., unu1 and there is a central vertex u. Case 1: n≡0(mod3) Let n=3t, a contradiction. Case 2:n≡1(mod3) Let n=3t+1, again a contradiction. Case 3:n≡2(mod3) Let n=3t+2,we get a contradiction. Therefore Contradiction arise from attempting to label the edges while maintaining distinct labels. Hence the Wheel graph Wn is not a Stolarsky-3Mean Cordial for any n. Mean Cordial Labeling is not applicable to wheel graphs. Similarly, Stolarsky-3 Mean Labeling is also ineffective for these graphs(Table 1). Additionally, Stolarsky-3 Mean Cordial Labeling does not apply to wheel graphs either. Wheel graphs are tricky because the central hub vertex makes it hard to balance the labels. None of the three methods work for this type of graph. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1509 https://internationalpubls.com Star Graph K 1,n or Sn Theorem 3.4:The Star graph(K1, n) (Sn) is a Stolarsky-3 Mean Cordial if n 15. Proof: Consider the Star graph (K1,n) with vertices v1, v2,..., vn respectively. Case 1:for 2≤n≤8 Assign u=1, v1=2,vi =2i-1 for 2≤i≤8. Case 2: for 9≤n≤15 Assign u=1, v1=2 and for vi increase systematically to ensure unique values. Case 3:n>15 If n>15, label u=1 and vi=2i-3 for 4≤i≤n. Therefore K1, n is not a Stolarsky-3 Mean Cordial for all n>15. From case 1,2,3,we conclude that K1, n is Stolarsky-3 Mean Cordial if n≤15 The Stolarsky-3 Mean Cordial Labeling of the Star Graph (Sn) is illustrated in Figure 3. Mean Cordial Labeling: Only works when n < 3. Stolarsky-3 Mean Labeling: Works when n < 16, but not for larger . Stolarsky-3 Mean Cordial Labeling: Works for any n. The Stolarsky-3 Mean Cordial Labeling is the best option for star graphs, as it works no matter how many vertices the graph has. The other two methods only work for smaller graphs. Wheel and Path Graph WnPm Theorem 3.5:The Wheel Graph Wn connected with a Path Graph Pm(denoted as WnPm) is Stolarsky - 3 Mean Cordial Labeling for any n. Proof: Consider the graph WnPm,which consists of the Wheel graph Wn connected to a Path graph Pm. The Wheel graph Wn has a cycle Cn with the vertices u1,u2,…,un and a central vertex u connected to all other vertices in the cycle.The Path graph Pm consists of vertices v1,v2,…,vn. Case 1:n+m≡0(mod3) Let n+m=3t,where t is an integer. vf(0)=vf(1)=vf(2)=t,ef(0)=ef(1)=ef(2)=t. Hence it is Stolarsky-3 Mean Cordial. Case 2:n+m≡1(mod3) Let n+m=3t+1,where t is an integer. vf(0)+vf(2)=t,vf(1)=t+1,ef(0)=ef(1)=ef(2)=t. Hence it is Stolarsky-3Mean Cordial. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1510 https://internationalpubls.com Case3:n+m≡2(mod3) Let n+m=3t+2,where t is an integer. vf(0)=vf(2)=t+1,vf(1)=t,ef(0)=ef(2)=t+1,ef(1)=t. Hence it is Stolarsky-3 Mean Cordial. In all cases,the graph satisfies the Stolarsky-3 Mean Cordail labeling. Hence the graph WnPm is Stolarsky-3 Mean Cordial for any n. The Stolarsky-3 Mean Cordial Labeling of the Wheel and Path Graph (WnPm) is illustrated in Figure 4. Mean Cordial Labeling works for any n.Stolarsky-3 Mean Labeling does not work. Stolarsky-3 Mean Cordial Labeling works for any n. Combining a wheel and a path makes Stolarsky-3 Mean Labeling fail, but the other two methods can handle this graph type(table 1). This comparative analysis highlights how different graph structures interact with various labeling schemes, providing insights into their mathematical properties and practical applications in graph theory. 4. Conclusion Different patterns emerge when different graph architectures are analyzed for compatibility with Mean Cordial Labeling, Stolarsky-3 Mean Labeling, and Stolarsky-3 Mean Cordial Labeling. The versatility of path graphs (Pn) is demonstrated by their support for all three labeling techniques. Cycle graphs only face issues for multiples of 3 with Mean Cordial and Stolarsky-3 Mean Cordial Labeling. Wheel graphs cannot be labeled by any method due to their complex structure. Star graphs are best labeled with Stolarsky-3 Mean Cordial Labeling, which works for all. Combined wheel and path graphs are compatible with Mean Cordial and Stolarsky-3 Mean Cordial Labeling. In summary, Stolarsky-3 Mean Cordial Labeling is the most versatile method, but wheel graphs remain a challenge for all. These findings support specific applications by highlighting the significance of network topology in assessing labeling feasibility. References 1. Harary F. Graph Theory. Addison Wesley; 1969. 2. Gallian JA. A Dynamic Survey of Graph Labeling. Electronic Journal of Combinatorics 2022. 3. Cahit I. Cordial Graphs: A Weaker Version of Graceful and Harmonious Graphs. Ars Combinatoria 1987; 23 Suppl 3: 201-208. 4. Dhanalakshmi S, Parvathi N. Mean Square Cordial Labeling on Star Related Graphs. J.Phys.: Conf. Ser. 2019: 1377. 5. Ponraj R, Sivakumar M, Sundaram M. Mean Cordial Labeling of Graphs. Open Journal of Discrete Mathematics 2012; 2: 145-148. 6. Sandhya SS, Somasundaram S, Ponraj R. Harmonic mean labeling of some cycle related graphs. Int. Journal of Math. Analysis 2012; 6 Suppl 40: 1997- 2005. 7. Sharma K, Tripathi A. Advancements in Mean and Harmonic Mean Labeling of Graphs. International Journal of Graph Theory and Applications 2023; 8 Suppl 2: 145- 158. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1511 https://internationalpubls.com 8. Muthukumar S, Ramaswamy G. Stolarsky Mean-Based Graph Labeling: New Perspectives and Applications. Journal of Discrete Mathematical Sciences & Cryptography 2023, 26 Suppl 3: 565- 578. 9. Ravi V, Kumar PS. Graph Labeling using Stolarsky Means: A Novel Approach for Complex Networks. Complexity 2023; 1-12. 10. Patil AS, Singh RK. Harmonic and Stolarsky Mean Labeling in Graph Theory: A Comparative Study. Mathematics and Computers in Simulation 2022; 195: 215-229. 11. Gupta N, Meena P. (). Applications of Stolarsky Mean in Graph Labeling: Emerging Trends. Journal of Applied Mathematics and Computing 2023; 71 Suppl 2: 509-523. 12. Kavitha S, Sandhya SS. (). Exploring New Horizons in Graph Labeling: Stolarsky-3 Mean and Beyond. Advances in Graph Theory 2024; 45 Suppl 1: 67-81. Table 1. Comparative analyses of 3 different mean labeling with respect to various graphs. Graph Type Mean Cordial labeling (n is number of vertices) Stolarsky-3 Mean Labeling (n is number of vertices) Stolarsky-3 Mean Cordial Labeling(n is number of vertices) Path Pn for any n for any n for any n Cycle Cn for any n, Except n=3,6,9.. for any n for any n, Except n=3,6,9.. Wheel Wn No No No Star Graph K1,n or Sn n<3 n<16 for any n Wheel and Path Graph WnPm for any n No for any n Fig. 1. Stolarsky-3 Mean cordial labeling of Path (P10 ➔0000111222) graph Fig. 2. Stolarsky-3 Mean Cordial labeling of Cycle (C11 ➔00001111222) graph. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 1512 https://internationalpubls.com Fig. 3. Stolarsky-3 Mean Cordial labeling of Star (S16 ➔00000011111122222) graph Fig.4. Stolarsky-3 Mean Cordial labeling of Wheel and Path Graph (W9P6 ➔00000111211222) graph.