Abstract
Classical mean labeling of a graph with vertices and edges is an injective function from the vertex set of to the set such that the edge labels obtained from the flooring function of the average of mean of arithmetic, geometric, harmonic, and root square of the vertex labels of each edge’s end vertices is distinct, and the set of edge labels is . One of the graph operations is to duplicate the graph. The classical meanness of graphs formed by duplicating an edge and a vertex of numerous standard graphs is discussed in this study.
1. Introduction and Preliminaries
The term “graph” refers to a finite, undirected, and basic graph used throughout this paper. Let us consider a graph with vertices and edges, such as . [1–6] are the notations and terms we utilize. For a comprehensive look into graph labeling, we recommend [7]. The corona is a graph created by linking the vertex of to every vertex in the copy of using one copy of order and copies of . A pair of nearby vertices on a graph is referred to as neighbors. The neighborhood set is defined as the set of all neighbors of a vertex and is denoted by [8].
2. Literature Survey
Somosundaram and Ponraj [9] proposed the idea of mean labeling. The -harmonic mean graph was defined by Durai Baskar and Arockiaraj [10]. Arockiaraj et al. introduced the concept of -root square mean graphs [11] and evaluated the meanness of some graphs by duplicating graph components [12]. Vaidya and Barasara thoroughly examined the harmonic mean [8] and geometric mean [13] labeling for a variety of graphs emerging from graph element duplication. In [14], Maya and Nicholas researched the duplication of divisor cordial graphs. Durai Baskar et al. researched the geometric meanness of graphs [15] and Durai Baskar and Arockiaraj talked about the -geometric mean graphs [16]. Prajapati and Gajjar developed the cordiality in the context of duplication in web and armed helm [17]. Exponential mean labeling of graphs of some standard graphs by duplication of graph elements is developed by Rajesh Kannan et al. [18]. Muhiuddin et al. defined classical mean labeling of graphs [19] and Alanazi et al. extended its meanness for specific ladder graphs [20]. We explore a conventional mean labeling of graphs based on some duplicating techniques [21–25], which has been produced by a significant number of creators in the domain of graph labeling.
3. Methodology
A labeling on a graph with is called classical mean labeling if the injective function and an induced bijective function is defined byfor all . A classical mean graph is one that facilitates classical mean labeling. In this article, we essentially address our topic’s flooring function and try to rationalize the classical meanness of some of these graphs created by duplicating procedures.
Figure 1 depicts classical mean labeling of [20].
4. Main Results
4.1. Classical Meanness of Graphs Obtained from Edge Duplicating Operation
Theorem 1. If a graph is formed by duplicating one edge of another graph , then is a classical mean graph, where represents the path for .
Proof. Let be the duplicating edge of , . Case 1. or , and . Let us construct (see Table 1). Hence, and (see Table 2). Case 2. , , and . Let us construct (see Table 3). Hence, and (see Table 4). Case 3. , , and . Let us construct (see Table 5). Hence, and (see Table 6). Figure 2 depicts classical mean labeling of in the preceding circumstances.
Theorem 2. If a graph is formed by duplicating one edge of another graph , then is a classical mean graph, where represents the path .
Proof. Suppose . Case 1. , , and . Subcase (1). or , and . Let us construct . and (see Table 7). Hence, and (see Table 8). Subcase (2). and . Let us construct . and (see Table 9). Hence, and (see Table 10). Subcase (3). , , and . Let us construct . and (see Table 11). Hence, and (see Table 12). Hence, is a classical meanness of . Case 2. , , and . Subcase (1). or , and . Let us construct . and (see Table 13). Hence, and (see Table 14). Subcase (2). , , and . Let us construct . and (see Table 15). Hence, and (see Table 16). Subcase (3). , , and . Let us construct . and (see Table 17) Hence (see Table 18), Figure 3 depicts the classical mean labeling of in the preceding situations. Case 3. , , and . Figure 4 depicts classical mean labeling of where and for .
Theorem 3. If a graph is formed by duplicating one edge of another graph , then is a classical mean graph, where represents the path for . Proof. Case 1. and . Let us construct (see Table 19). Hence, (see Table 20). Let us take , , and . Figure 5 depicts classical mean labeling of in the preceding circumstances. Case 2. . Figure 6 depicts classical mean labeling of a graph for Case 2.
4.2. Classical Meanness of Graphs Obtained from Vertex Duplicating Operation
Theorem 4. If a graph is created by duplicating a vertex of another graph , then is a classical mean graph, where is the path , for .
Proof. Let be . Let the duplicating a vertex by a vertex , for some of the graph . Case 1. or . Let us take . Let us construct (see Table 21). Hence (see Table 22), Case 2. and . Let us construct . and (see Table 23). Hence, and (see Table 24). Figure 7 depicts classical mean labeling of in the preceding circumstances.
Theorem 5. If a graph is created by duplicating a vertex of another graph , then is a classical mean graph, where is the path , for .
Proof. Case 1. , for . Let its duplication be . Subcase 1. or and . Let us construct (see Table 25). Hence (see Table 26), Subcase 2. , , and take . Let us construct (see Table 27). Hence, and (see Table 28). Case 2. , , and . Let us construct . and (see Table 29). Hence, and (see Table 30). Figure 8 depicts classical mean labeling of in the preceding circumstances.
Theorem 6. If a graph is created by duplicating a vertex of another graph , then is a classical mean graph, where is the path , for .
Proof. Let be the vertices of the cycle and let and its duplicated vertex is . Case 1. and . Let us construct . Hence, Let us take , , and . Figure 9 depicts classical mean labeling of in the preceding circumstances. Case 2. . Figure 10 depicts classical mean labeling of a graph for Case 2.
Theorem 7. The graph is a classical mean graph, for .
Proof. Let be the vertices of the path and be the vertices of . Case 1. is odd. Let us take . Let us construct (see Table 31). (see Table 32). Hence (see Table 33). (see Table 34). (see Table 35). Case 2. is even and . Let us take . Let us construct (see Table 36). (see Table 37). Hence (see Table 38). (see Table 39). (see Table 40). Figure 11 depicts classical mean labeling of in the preceding circumstances. The graph is when and it is also a classical mean labeling graph.
5. Conclusion
The classical meanness of some graphs generated from duplicating operations is addressed here, along with sufficient examples to aid comprehension. It is feasible to look into comparable outcomes for a variety of other graphs.
Data Availability
No data ware used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.