A New Notion of Classical Mean Graphs Based on Duplicating Operations

Classical mean labeling of a graph G with p vertices and q edges is an injective function from the vertex set of G to the set 1 , 2 , 3 , ... , q + 1 (cid:31) (cid:30) 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 1 , 2 , 3 , ... ,q (cid:31) (cid:30) . 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.


Introduction and Preliminaries
e term "graph" refers to a nite, undirected, and basic graph used throughout this paper. Let us consider a graph with p vertices and q edges, such as G(V, E). [1][2][3][4][5][6] are the notations and terms we utilize. For a comprehensive look into graph labeling, we recommend [7]. e corona G 1 ∘ G 2 is a graph created by linking the i th vertex of G 1 to every vertex in the i th copy of G 2 using one G 1 copy of order p 1 and p 1 copies of G 2 . A pair of nearby vertices on a graph is referred to as neighbors. e neighborhood set is de ned as the set of all neighbors of a vertex v and is denoted by N(v) [8].

Literature Survey
Somosundaram and Ponraj [9] proposed the idea of mean labeling. e F-harmonic mean graph was de ned by Durai Baskar and Arockiaraj [10]. Arockiaraj et al. introduced the concept of F-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 F-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. de ned classical mean labeling of graphs [19] and Alanazi et al. extended its meanness for speci c ladder graphs [20]. We explore a conventional mean labeling of graphs based on some duplicating techniques [21][22][23][24][25], which has been produced by a signi cant number of creators in the domain of graph labeling.

Methodology
A labeling Φ on a graph G with Δ q + 1 is called classical mean labeling if the injective function for all uv ∈ E(G). 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 2ST 4 [20].

Classical Meanness of Graphs Obtained from Edge Duplicating Operation
Theorem 1. If a graph G is formed by duplicating one edge of another graph G ′ , then G is a classical mean graph, where G ′ represents the path P n for n ≥ 3. Table 1). Hence, and (see Table 2). Case 2. n ≥ 4, c � 2, and Δ � n + 4. Let us construct Φ: Table 3). Hence, and (see Table 4). Case 3. n ≥ 5, 3 ≤ c ≤ n − 2, and Δ � n + 4. Let us construct Φ: Table 5). Hence, and (see Table 6). Figure 2 depicts classical mean labeling of G in the preceding circumstances. □ Theorem 2. If a graph G is formed by duplicating one edge of another graph G ′ , then G is a classical mean graph, where G ′ represents the path P n ∘ K 1 .
and (see Table 17) Hence (see Table 18),  □ Theorem 3. If a graph G is formed by duplicating one edge of another graph G ′ , then G is a classical mean graph, where G ′ represents the path C n for n ≥ 3.
Journal of Mathematics

Classical Meanness of Graphs Obtained from Vertex Duplicating Operation
Theorem 4. If a graph G is created by duplicating a vertex of another graph G ′ , then G is a classical mean graph, where G ′ is the path P n , for n ≥ 3.
Proof. Let V(P n ) be v 1 , v 2 , . . . , v n . Let the duplicating a vertex v δ by a vertex k δ , for some δ of the graph G ′ .
Hence (see Table 22), Case 2. 2 ≤ δ ≤ n − 1 and Δ � n + 2. Let us construct Φ: and (see Table 23). Hence, and (see Table 24). Figure 7 depicts classical mean labeling of G in the preceding circumstances. □ Theorem 5. If a graph G is created by duplicating a vertex of another graph G ′ , then G is a classical mean graph, where G ′ is the path P n ∘ K 1 , for n ≥ 3.
and (see Table 29). Hence, and (see Table 30). Figure 8 depicts classical mean labeling of G in the preceding circumstances. □ Theorem 6. If a graph G is created by duplicating a vertex of another graph G ′ , then G is a classical mean graph, where G ′ is the path C n , for n ≥ 3. Let v 1 , v 2 , . . . , v n be the vertices of the cycle C n and let v � v 1 and its duplicated vertex is v 1 ′ . Case 1. n ≥ 5 and Δ � 4 + n. Let us construct Φ: V ⟶ N − Δ + 1, Δ + 2, . . . , ∞ { }.

Conclusion
e 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.