# A New Notion of Classical Mean Graphs Based on Duplicating Operations

Authors: A. Rajesh Kannan; P. Manivannan; K. Loganathan; Sonam Gyeltshen
Journal: Journal of Mathematics
 (2022)

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## Abstract

Classical mean labeling of a graphG with p vertices and q edges is an injective function from the vertex set of G to the set 1,2,3,…,q+1 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. 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.


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## Body

## 1. Introduction and Preliminaries

The term “graph” refers to a finite, undirected, and basic graph used throughout this paper. Let us consider a graph withp vertices and q edges, such as GV,E. [1–6] are the notations and terms we utilize. For a comprehensive look into graph labeling, we recommend [7]. The corona G1∘G2 is a graph created by linking the ith vertex of G1 to every vertex in the ith copy of G2 using one G1 copy of order p1 and p1 copies of G2. 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 v and is denoted by Nv [8].


## 2. Literature Survey

Somosundaram and Ponraj [9] proposed the idea of mean labeling. The F-harmonic mean graph was defined 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. 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 labelingΦ on a graph G with Δ=q+1 is called classical mean labeling if the injective function Φ:V⟶N−Δ+1,Δ+2,…,∞ and an induced bijective function Φ∗:V⟶N−Δ,Δ+1,…,∞ is defined by(1)Φ∗uv=14Φu+Φv2+ΦuΦv+2ΦuΦvΦu+Φv+Φu2+Φv22,for all uv∈EG. 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.Figure1 depicts classical mean labeling of 2ST4 [20].Figure 1
A classical mean labeling of2ST4.


## 4. Main Results

### 4.1. Classical Meanness of Graphs Obtained from Edge Duplicating Operation

Theorem 1.
If a graphG is formed by duplicating one edge of another graph G′, then G is a classical mean graph, where G′ represents the path Pn for n≥3.Proof.
Lete′=vγ′vγ+1′ be the duplicating edge of e=vγvγ+1, 1≤γ≤n−1.Case 1.γ=−1+n or γ=1, and Δ=n+3.Let us constructΦ:V⟶N−Δ,Δ+1,…,∞ (see Table 1).Hence,(2)Φ∗v2′vδ=2δ−1,forδ=1,3.and (see Table2).Case 2.n≥4, γ=2, and Δ=n+4.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 3).Hence,(3)Φ∗v2′vδ=2δ−2,forδ=1and3,Φ∗v3′v4=6.and (see Table4).Case 3.n≥5, 3≤γ≤n−2, and Δ=n+4.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 5).Hence,(4)Φ∗vγ′vγ−1=γ−1,Φ∗vγ′vγ−1=1+γ,Φ∗vγ+1′vγ+2=3+γ,and (see Table6).Figure2 depicts classical mean labeling of G in the preceding circumstances.Table 1
Vertex labeling of Φ(v).
Φv1≤δ≤34≤δ≤nΦvδ2δ−1δ+2Φvδ′2δ−Table 2
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤34≤δ≤n−1Φ∗vδvδ+12δ−1δ+2Table 3
Vertex labeling of Φ(v).
Φvδ=12≤δ≤3δ=45≤δ≤nΦvδ−12δ−1+2δ−1+2δ3+δΦvδ′−2δ−−Table 4
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤45≤δ≤n−1Φ∗vδvδ+12δ−1δ+3Table 5
Vertex labeling of Φ(v).
Φv1≤δ≤γ−1δ=γδ=γ+1γ+2≤δ≤nΦvδδ2+γ3+δ3+δΦvγ′−1+δ−−Φvγ+1′−−3+γ−Table 6
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−2δ=γ−1δ=γγ+1≤δ≤n−1Φ∗vδvδ+1′δγγ+2δ+3Figure 2
A classical mean labeling ofG obtained by duplicating an edge of a path.Theorem 2.
If a graphG is formed by duplicating one edge of another graph G′, then G is a classical mean graph, where G′ represents the path Pn∘K1.Table 7
Vertex labeling of Φ(v).
Φv1≤δ≤23≤δ≤nΦuδδ+32δ+2Φvδ′4δ−22δ+1Table 8
Edge labeling ofΦ∗e.
Φ∗eδ=12≤δ≤n−1δ=nΦ∗uδu1+δ42δ+2−Φ∗uδvδ+122δ+12n+1Table 9
Vertex labeling of Φ(v).
Φvδ=12≤δ≤34≤δ≤nΦuδ34δ−42δ+3Φvδ17δ−122δ+2Table 10
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤23≤δ≤n−1δ=nΦ∗uδu1+δ2δ+12δ+3−Φ∗uδvδδ2δ+22n+2Table 11
Vertex labeling of Φ(v).
Φv1≤δ≤iδ=γ+1γ+2≤δ≤nΦuδ2δ2δ+42δ+3Φvδ2δ−12δ+52δ+2Table 12
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−1δ=γγ+1≤δ≤n−1Φ∗uδu1+δ2δ2γ+12δ+3Φ∗uδvδ2δ−12δ−12δ+2Table 13
Vertex labeling of Φ(v).
Φvδ=12≤δ≤34≤δ≤nΦuδ12δ+42δ+4Φvδ23δ2δ+3Table 14
Edge labeling ofΦ∗e.
Φ∗eδ=1δ=23≤δ≤n−1δ=nΦ∗uδu1+δ32δ+42δ+4−Φ∗uδvδ5δ−45δ−42δ+32δ+3Table 15
Vertex labeling of Φ(v).
Φv1≤δ≤23≤δ≤45≤δ≤nΦuδ5δ−34δ−42δ+5Φvδ3δ−23δ+12δ+4Table 16
Edge labeling ofΦ∗e.
Φ∗eδ=12≤δ≤34≤δ≤n−1δ=nΦ∗uδu1+δ42δ+32δ+5−Φ∗uδvδ13δ−12δ−42n−4Table 17
Vertex labeling of Φ(v).
Φv1≤δ≤γ−1δ=γδ=γ+1δ=γ+2γ+3≤δ≤nΦuδ2δ2δ+32δ+42δ+82δ+5Φvδ2δ−12δ2δ+62δ+92δ+4Table 18
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−2δ=γ−1δ=γδ=γ+1δ+2≤δ≤n−1Φ∗uδu1+δ2δ2δ2δ+32δ+52δ+5Φ∗uδvδ2δ−12δ−12δ+12δ+42δ+4Figure 3
A classical mean labeling ofG obtained by duplicating an edge of a graph Pn∘K1.Figure 4
A classical mean labeling ofG.Proof.
Supposen≥3.Case 1.1≤γ≤n, e=uγvγ, and n≥3.Subcase (1).γ=n or γ=1, and Δ=2n+3.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(5)Φu1′=3,Φv1′=1.and (see Table7).Hence,(6)Φ∗u1′v1′=1,Φ∗u1′u1′=3.and (see Table8).Subcase (2).γ=2 and Δ=2n+4.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(7)Φu2′=7,Φv2′=6.and (see Table9).Hence,(8)Φ∗u1u2′=4,Φ∗u2′u3=7,Φ∗u2′v2′=6.and (see Table10).Subcase (3).n≥4, 3≤γ≤n−1, and Δ=2n+4.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(9)Φuγ′=2γ+3,Φvγ′=2γ+2.and (see Table11).Hence,(10)Φ∗uγ−1uγ′=2γ,Φ∗uγ′uγ+1=2γ+3,Φ∗uγ′vγ′=2γ+2,and (see Table12).Hence,Φ is a classical meanness of G.Case 2.1≤γ≤n−1, e=uγuγ+1, and n≥3.Subcase (1).γ=n−1 or γ=1, and Δ=2n+5.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(11)Φu1′=4,Φu2′=5.and (see Table13).Hence,(12)Φ∗u1′u2′=4,Φ∗u2′v2=5,Φ∗u1′v1=2,Φ∗u2′u3=7.and (see Table14).Subcase (2).n≥4, γ=2, and Δ=2n+6.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(13)Φu2′=3,Φu3′=11.and (see Table15).Hence,(14)Φ∗u1u2′=2,Φ∗u2′u3′=6,Φ∗u3′u4=11,Φ∗u2′v2=3,Φ∗u3′v3=10.and (see Table16).Subcase (3).n≥5 , 3≤γ≤n−2, and Δ=2n+5.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(15)Φuγ′=2γ−1,Φuγ+1′=2γ+7.and (see Table17)Hence (see Table18),(16)Φ∗uγ′uγ+1′=2γ+2,Φ∗uγ−1uγ′=2γ−2,Φ∗uγ+1′vγ+1=2γ+6,Φ∗uγ′uγ+2=2γ+7,Φ∗uγ′vγ=2γ−1.Figure3 depicts the classical mean labeling of G in the preceding situations.Case 3.e=u1v1, e=u1u2, and n=2.Figure4 depicts classical mean labeling of G where e=u1v1 and e=u1u2 for n=2.Theorem 3.
If a graphG is formed by duplicating one edge of another graph G′, then G is a classical mean graph, where G′ represents the path Cn for n≥3.Proof. Case 1.n≥6 and Δ=n+5.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 19).(17)Φvi′=n+2γ−1for1≤γ≤2,(18)Φvγ=γ−3,4≤γ≤14n2+14n+412n+10+n+4+n2+8n+172+2,γ−2,14n2+14n+412n+10+n+4+n2+8n+172+3≤γ≤n.Hence, (see Table20).Let us takeX=n2+14n+41/2n+10, Y=4+n, and Z=n2+8n+17/2.(19)Φ∗v1′vn=−1+n,Φ∗v1vn=−2+n,Φ∗v3v4=14X+Y+Z,Φ∗vγvγ+1=3+γ,4≤γ≤2+14X+Y+Z,−2+γ,3+14X+Y+Z≤γ≤n−1.Figure5 depicts classical mean labeling of C15 in the preceding circumstances.Case 2.n=3,4,5.Figure6 depicts classical mean labeling of a graph for Case 2.Table 19
Vertex labeling of Φ(v).
Φvδ=12≤δ≤3Φvδn−1n+2δ−2Table 20
Edge labeling ofΦ∗e.
Φ∗e1≤γ≤2Φ∗vγvγ+1n+2γ−2Φ∗vγ′vγ+1′n+2γ−1Figure 5
A classical mean labeling of the graph obtained by duplicating an edge of a graphC15.Figure 6
A classical mean labeling of the graph obtained by duplicating an edge of a graphC3,C4, and C5.


### 4.2. Classical Meanness of Graphs Obtained from Vertex Duplicating Operation

Theorem 4.
If a graphG is created by duplicating a vertex of another graph G′, then G is a classical mean graph, where G′ is the path Pn, for n≥3.Table 21
Vertex labeling of Φ(v).
Φv1≤γ≤nΦvγγ+1Table 22
Edge labeling ofΦ∗e.
Φ∗e1≤γ≤n−1Φ∗vγvγ+1γ+1Table 23
Vertex labeling of Φ(v).
Φv1≤γ≤δ−1δ≤γ≤nΦvγγγ+2Table 24
Edge labeling ofΦ∗e.
Φ∗e1≤γ≤γ−2γ=δ−1δ≤γ≤n−1Φ∗vγvγ+1γγ+1γ+2Figure 7
A classical mean labeling ofG obtained by duplicating a vertex of a path.Proof.
LetVPn be v1,v2,…,vn. Let the duplicating a vertex vδ by a vertex kδ, for some δ of the graph G′.Case 1.δ=n or δ=1.Let us takeΔ=n+2.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 21).(20)Φk1=1.Hence (see Table22),(21)Φ∗v2k1=1.Case 2.2≤δ≤n−1 and Δ=n+2.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(22)Φkδ=1+δ.and (see Table23).Hence,(23)Φ∗vδ−1kδ=δ−1,Φ∗vδ+1kδ=δ+1.and (see Table24).Figure7 depicts classical mean labeling of G in the preceding circumstances.Theorem 5.
If a graphG is created by duplicating a vertex of another graph G′, then G is a classical mean graph, where G′ is the path Pn∘K1, for n≥3.Table 25
Vertex labeling of Φ(v).
Φvδ=12≤δ≤nΦuδ22δ+2Φvδ12δ+1Table 26
Edge labeling ofΦ∗e.
Φ∗eδ=12≤δ≤n−1Φ∗uδu1+δ32δ+2Φ∗uδvδ12δ+1Φ∗e1≤γ≤2Φ∗uγ′vγ2γTable 27
Vertex labeling of Φ(v).
Φv1≤δ≤γ−1δ=γγ+1≤δ≤nΦuδ2δ2δ2δ+3Φvδ2δ−12δ+12δ+2Table 28
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−1δ=γγ+1≤δ≤n−1γ=nΦ∗uδu1+δ2δ2δ+22δ+3−Φ∗uδvδ2δ−12δ2δ+22n+2Table 29
Vertex labeling of Φ(v).
Φv1≤δ≤γδ+1≤nΦuδ2δ2δ+1Φvδ2δ−12δTable 30
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−1δ=γδ=γ+1γ+1≤δ≤n−1δ=nΦ∗uδu1+δ2δ2δ2δ+12δ+1−Φ∗uδvδ2δ−12δ−12δ2δ2δFigure 8
A classical mean labeling ofG obtained by duplicating a vertex of the graph Pn∘K1.Proof.
Case 1.v=uγ, for 1≤γ≤n.Let its duplication bev′=uγ′.Subcase 1.γ=n or γ=1 and Δ=3+2n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 25).(24)Φu1′=4.Hence (see Table26),Subcase 2.n≥3, 2≤γ≤n−1, and take Δ=4+2n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 27).(25)Φuγ′=2δ+2Hence,(26)Φ∗uγ−1uγ′=2γ−1,Φ∗uγ′uγ+1=2γ+3,Φ∗uγ′vγ=2γ+1.and (see Table28).Case 2.1≤γ≤n, v=vγ, and Δ=2+2n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(27)Φvδ′=2γ+1.and (see Table29).Hence,(28)Φ∗uγvγ′=2γ.and (see Table30).Figure8 depicts classical mean labeling of G in the preceding circumstances.Theorem 6.
If a graphG is created by duplicating a vertex of another graph G′, then G is a classical mean graph, where G′ is the path Cn, for n≥3.Proof.
Letv1,v2,…,vn be the vertices of the cycle Cn and let v=v1 and its duplicated vertex is v1′.Case 1.n≥5 and Δ=4+n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(29)Φv1′=n−1,for2≤δ≤3,Φv1=n−1,Φvγ=n+2γ−2,Φvγ=γ−3,4≤γ≤14n2+12n+282n+8+n+3+n2+6n+102+2,γ−2,14n2+12n+282n+8+n+3+n2+6n+102+3≤γ≤n.Hence,(30)Φ∗vγvγ+1=n+2γ−2,for1≤δ≤2,Φ∗v1vn=n−2,Φ∗v1′v2=n+1,Φ∗v1′vn=n−1.Let us takeX=28+12n+n2/8+2n, Y=3+n, and Z=10+6n+n2/2.(31)Φ∗v3v4=14X+Y+Z,Φ∗vγvγ+1=−3+γ,4≤γ≤14X+Y+Z+2,−2+γ,14X+Y+Z+3≤γ≤n−1.Figure9 depicts classical mean labeling of G in the preceding circumstances.Case 2.n=3,4.Figure10 depicts classical mean labeling of a graph for Case 2.Figure 9
A classical mean labeling ofG obtained by duplicating a vertex of cycle C10.Figure 10
A classical mean labeling of the graph obtained by duplicating a vertex of a graphC3 and C4.Theorem 7.
The graphS′Pn is a classical mean graph, for n≥2.Proof.
Letv1,v2,…,vn be the vertices of the path Pn and v1,v2,…,vn,v1′,v2′,…,vn′ be the vertices of S′Pn.Case 1.n is odd. Let us take Δ=−1+3n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 31).(see Table32).Hence (see Table33).(see Table34).(see Table35).Case 2.n is even and n≥4. Let us take Δ=−1+3n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 36).(see Table37).Hence (see Table38).(see Table39).(see Table40).Figure11 depicts classical mean labeling of S′P9 in the preceding circumstances.The graphS′P2 is P4 when n=2 and it is also a classical mean labeling graph.Table 31
Vertex labeling of Φ(v).
Φvγ=1γ=2γ=3Φvγ5γ−45γ−45Φvγ′44γ−54γ−5Table 32
Vertex labeling of Φ(v).
Φv2≤γ≤n/2Φv2γ6γ+1Φv2γ+16γΦv2γ′6γ−2Φv2γ+1′6γ−1Table 33
Edge labeling ofΦ∗e.
Φ∗eγ=1γ=2γ=3Φ∗vγvγ+13γ−13γ−13γ−1Φ∗vγvγ+1′5γ−45γ−47Φ∗vγ+1vγ′46γ−96γ−9Table 34
Edge labeling ofΦ∗e.
Φ∗e2≤γ≤n/2Φ∗v2γv2γ+16γΦ∗v2γ′vγ+16γ−2Φ∗v2γv2γ+1′6γ+2Table 35
Edge labeling ofΦ∗e.
Φ∗e2≤γ≤n/2−1Φ∗v2γ+1v2γ+26γ+3Φ∗v2γ+1′vγ+26γ+2Φ∗v2γ+2v2γ+2′6γ+1Table 36
Vertex labeling of Φ(v).
Φv1≤γ≤2Φvγ2γ−1Φvγ′6−γTable 37
Vertex labeling of Φ(v).
Φv2≤γ≤n/2Φv2γ−16γ−2Φv2γ6γ−3Φv2γ−1′6γ−5Φv2γ′6γ−4Table 38
Edge labeling ofΦ∗e.
Φ∗e1≤γ≤2Φ∗vγvγ+14γ−3Φ∗vγvγ+1′2γΦ∗vγ′vγ+13γTable 39
Edge labeling ofΦ∗e.
Φ∗e2≤γ≤n/2Φ∗v2γ−1v2γ6γ−3Φ∗v2γ−1′v2γ6γ−5Φ∗v2γ−1v2γ′6γ−4Table 40
Edge labeling ofΦ∗e.
Φ∗e2≤γ≤n/2−1Φ∗v2γv2γ+16γΦ∗v2γ′v2γ+16γ−1Φ∗v2γv2γ+1′6γ−2Figure 11
A classical mean labeling ofS′P9.


## 4.1. Classical Meanness of Graphs Obtained from Edge Duplicating Operation

Theorem 1.
If a graphG is formed by duplicating one edge of another graph G′, then G is a classical mean graph, where G′ represents the path Pn for n≥3.Proof.
Lete′=vγ′vγ+1′ be the duplicating edge of e=vγvγ+1, 1≤γ≤n−1.Case 1.γ=−1+n or γ=1, and Δ=n+3.Let us constructΦ:V⟶N−Δ,Δ+1,…,∞ (see Table 1).Hence,(2)Φ∗v2′vδ=2δ−1,forδ=1,3.and (see Table2).Case 2.n≥4, γ=2, and Δ=n+4.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 3).Hence,(3)Φ∗v2′vδ=2δ−2,forδ=1and3,Φ∗v3′v4=6.and (see Table4).Case 3.n≥5, 3≤γ≤n−2, and Δ=n+4.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 5).Hence,(4)Φ∗vγ′vγ−1=γ−1,Φ∗vγ′vγ−1=1+γ,Φ∗vγ+1′vγ+2=3+γ,and (see Table6).Figure2 depicts classical mean labeling of G in the preceding circumstances.Table 1
Vertex labeling of Φ(v).
Φv1≤δ≤34≤δ≤nΦvδ2δ−1δ+2Φvδ′2δ−Table 2
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤34≤δ≤n−1Φ∗vδvδ+12δ−1δ+2Table 3
Vertex labeling of Φ(v).
Φvδ=12≤δ≤3δ=45≤δ≤nΦvδ−12δ−1+2δ−1+2δ3+δΦvδ′−2δ−−Table 4
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤45≤δ≤n−1Φ∗vδvδ+12δ−1δ+3Table 5
Vertex labeling of Φ(v).
Φv1≤δ≤γ−1δ=γδ=γ+1γ+2≤δ≤nΦvδδ2+γ3+δ3+δΦvγ′−1+δ−−Φvγ+1′−−3+γ−Table 6
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−2δ=γ−1δ=γγ+1≤δ≤n−1Φ∗vδvδ+1′δγγ+2δ+3Figure 2
A classical mean labeling ofG obtained by duplicating an edge of a path.Theorem 2.
If a graphG is formed by duplicating one edge of another graph G′, then G is a classical mean graph, where G′ represents the path Pn∘K1.Table 7
Vertex labeling of Φ(v).
Φv1≤δ≤23≤δ≤nΦuδδ+32δ+2Φvδ′4δ−22δ+1Table 8
Edge labeling ofΦ∗e.
Φ∗eδ=12≤δ≤n−1δ=nΦ∗uδu1+δ42δ+2−Φ∗uδvδ+122δ+12n+1Table 9
Vertex labeling of Φ(v).
Φvδ=12≤δ≤34≤δ≤nΦuδ34δ−42δ+3Φvδ17δ−122δ+2Table 10
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤23≤δ≤n−1δ=nΦ∗uδu1+δ2δ+12δ+3−Φ∗uδvδδ2δ+22n+2Table 11
Vertex labeling of Φ(v).
Φv1≤δ≤iδ=γ+1γ+2≤δ≤nΦuδ2δ2δ+42δ+3Φvδ2δ−12δ+52δ+2Table 12
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−1δ=γγ+1≤δ≤n−1Φ∗uδu1+δ2δ2γ+12δ+3Φ∗uδvδ2δ−12δ−12δ+2Table 13
Vertex labeling of Φ(v).
Φvδ=12≤δ≤34≤δ≤nΦuδ12δ+42δ+4Φvδ23δ2δ+3Table 14
Edge labeling ofΦ∗e.
Φ∗eδ=1δ=23≤δ≤n−1δ=nΦ∗uδu1+δ32δ+42δ+4−Φ∗uδvδ5δ−45δ−42δ+32δ+3Table 15
Vertex labeling of Φ(v).
Φv1≤δ≤23≤δ≤45≤δ≤nΦuδ5δ−34δ−42δ+5Φvδ3δ−23δ+12δ+4Table 16
Edge labeling ofΦ∗e.
Φ∗eδ=12≤δ≤34≤δ≤n−1δ=nΦ∗uδu1+δ42δ+32δ+5−Φ∗uδvδ13δ−12δ−42n−4Table 17
Vertex labeling of Φ(v).
Φv1≤δ≤γ−1δ=γδ=γ+1δ=γ+2γ+3≤δ≤nΦuδ2δ2δ+32δ+42δ+82δ+5Φvδ2δ−12δ2δ+62δ+92δ+4Table 18
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−2δ=γ−1δ=γδ=γ+1δ+2≤δ≤n−1Φ∗uδu1+δ2δ2δ2δ+32δ+52δ+5Φ∗uδvδ2δ−12δ−12δ+12δ+42δ+4Figure 3
A classical mean labeling ofG obtained by duplicating an edge of a graph Pn∘K1.Figure 4
A classical mean labeling ofG.Proof.
Supposen≥3.Case 1.1≤γ≤n, e=uγvγ, and n≥3.Subcase (1).γ=n or γ=1, and Δ=2n+3.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(5)Φu1′=3,Φv1′=1.and (see Table7).Hence,(6)Φ∗u1′v1′=1,Φ∗u1′u1′=3.and (see Table8).Subcase (2).γ=2 and Δ=2n+4.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(7)Φu2′=7,Φv2′=6.and (see Table9).Hence,(8)Φ∗u1u2′=4,Φ∗u2′u3=7,Φ∗u2′v2′=6.and (see Table10).Subcase (3).n≥4, 3≤γ≤n−1, and Δ=2n+4.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(9)Φuγ′=2γ+3,Φvγ′=2γ+2.and (see Table11).Hence,(10)Φ∗uγ−1uγ′=2γ,Φ∗uγ′uγ+1=2γ+3,Φ∗uγ′vγ′=2γ+2,and (see Table12).Hence,Φ is a classical meanness of G.Case 2.1≤γ≤n−1, e=uγuγ+1, and n≥3.Subcase (1).γ=n−1 or γ=1, and Δ=2n+5.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(11)Φu1′=4,Φu2′=5.and (see Table13).Hence,(12)Φ∗u1′u2′=4,Φ∗u2′v2=5,Φ∗u1′v1=2,Φ∗u2′u3=7.and (see Table14).Subcase (2).n≥4, γ=2, and Δ=2n+6.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(13)Φu2′=3,Φu3′=11.and (see Table15).Hence,(14)Φ∗u1u2′=2,Φ∗u2′u3′=6,Φ∗u3′u4=11,Φ∗u2′v2=3,Φ∗u3′v3=10.and (see Table16).Subcase (3).n≥5 , 3≤γ≤n−2, and Δ=2n+5.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(15)Φuγ′=2γ−1,Φuγ+1′=2γ+7.and (see Table17)Hence (see Table18),(16)Φ∗uγ′uγ+1′=2γ+2,Φ∗uγ−1uγ′=2γ−2,Φ∗uγ+1′vγ+1=2γ+6,Φ∗uγ′uγ+2=2γ+7,Φ∗uγ′vγ=2γ−1.Figure3 depicts the classical mean labeling of G in the preceding situations.Case 3.e=u1v1, e=u1u2, and n=2.Figure4 depicts classical mean labeling of G where e=u1v1 and e=u1u2 for n=2.Theorem 3.
If a graphG is formed by duplicating one edge of another graph G′, then G is a classical mean graph, where G′ represents the path Cn for n≥3.Proof. Case 1.n≥6 and Δ=n+5.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 19).(17)Φvi′=n+2γ−1for1≤γ≤2,(18)Φvγ=γ−3,4≤γ≤14n2+14n+412n+10+n+4+n2+8n+172+2,γ−2,14n2+14n+412n+10+n+4+n2+8n+172+3≤γ≤n.Hence, (see Table20).Let us takeX=n2+14n+41/2n+10, Y=4+n, and Z=n2+8n+17/2.(19)Φ∗v1′vn=−1+n,Φ∗v1vn=−2+n,Φ∗v3v4=14X+Y+Z,Φ∗vγvγ+1=3+γ,4≤γ≤2+14X+Y+Z,−2+γ,3+14X+Y+Z≤γ≤n−1.Figure5 depicts classical mean labeling of C15 in the preceding circumstances.Case 2.n=3,4,5.Figure6 depicts classical mean labeling of a graph for Case 2.Table 19
Vertex labeling of Φ(v).
Φvδ=12≤δ≤3Φvδn−1n+2δ−2Table 20
Edge labeling ofΦ∗e.
Φ∗e1≤γ≤2Φ∗vγvγ+1n+2γ−2Φ∗vγ′vγ+1′n+2γ−1Figure 5
A classical mean labeling of the graph obtained by duplicating an edge of a graphC15.Figure 6
A classical mean labeling of the graph obtained by duplicating an edge of a graphC3,C4, and C5.


## 4.2. Classical Meanness of Graphs Obtained from Vertex Duplicating Operation

Theorem 4.
If a graphG is created by duplicating a vertex of another graph G′, then G is a classical mean graph, where G′ is the path Pn, for n≥3.Table 21
Vertex labeling of Φ(v).
Φv1≤γ≤nΦvγγ+1Table 22
Edge labeling ofΦ∗e.
Φ∗e1≤γ≤n−1Φ∗vγvγ+1γ+1Table 23
Vertex labeling of Φ(v).
Φv1≤γ≤δ−1δ≤γ≤nΦvγγγ+2Table 24
Edge labeling ofΦ∗e.
Φ∗e1≤γ≤γ−2γ=δ−1δ≤γ≤n−1Φ∗vγvγ+1γγ+1γ+2Figure 7
A classical mean labeling ofG obtained by duplicating a vertex of a path.Proof.
LetVPn be v1,v2,…,vn. Let the duplicating a vertex vδ by a vertex kδ, for some δ of the graph G′.Case 1.δ=n or δ=1.Let us takeΔ=n+2.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 21).(20)Φk1=1.Hence (see Table22),(21)Φ∗v2k1=1.Case 2.2≤δ≤n−1 and Δ=n+2.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(22)Φkδ=1+δ.and (see Table23).Hence,(23)Φ∗vδ−1kδ=δ−1,Φ∗vδ+1kδ=δ+1.and (see Table24).Figure7 depicts classical mean labeling of G in the preceding circumstances.Theorem 5.
If a graphG is created by duplicating a vertex of another graph G′, then G is a classical mean graph, where G′ is the path Pn∘K1, for n≥3.Table 25
Vertex labeling of Φ(v).
Φvδ=12≤δ≤nΦuδ22δ+2Φvδ12δ+1Table 26
Edge labeling ofΦ∗e.
Φ∗eδ=12≤δ≤n−1Φ∗uδu1+δ32δ+2Φ∗uδvδ12δ+1Φ∗e1≤γ≤2Φ∗uγ′vγ2γTable 27
Vertex labeling of Φ(v).
Φv1≤δ≤γ−1δ=γγ+1≤δ≤nΦuδ2δ2δ2δ+3Φvδ2δ−12δ+12δ+2Table 28
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−1δ=γγ+1≤δ≤n−1γ=nΦ∗uδu1+δ2δ2δ+22δ+3−Φ∗uδvδ2δ−12δ2δ+22n+2Table 29
Vertex labeling of Φ(v).
Φv1≤δ≤γδ+1≤nΦuδ2δ2δ+1Φvδ2δ−12δTable 30
Edge labeling ofΦ∗e.
Φ∗e1≤δ≤γ−1δ=γδ=γ+1γ+1≤δ≤n−1δ=nΦ∗uδu1+δ2δ2δ2δ+12δ+1−Φ∗uδvδ2δ−12δ−12δ2δ2δFigure 8
A classical mean labeling ofG obtained by duplicating a vertex of the graph Pn∘K1.Proof.
Case 1.v=uγ, for 1≤γ≤n.Let its duplication bev′=uγ′.Subcase 1.γ=n or γ=1 and Δ=3+2n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 25).(24)Φu1′=4.Hence (see Table26),Subcase 2.n≥3, 2≤γ≤n−1, and take Δ=4+2n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 27).(25)Φuγ′=2δ+2Hence,(26)Φ∗uγ−1uγ′=2γ−1,Φ∗uγ′uγ+1=2γ+3,Φ∗uγ′vγ=2γ+1.and (see Table28).Case 2.1≤γ≤n, v=vγ, and Δ=2+2n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(27)Φvδ′=2γ+1.and (see Table29).Hence,(28)Φ∗uγvγ′=2γ.and (see Table30).Figure8 depicts classical mean labeling of G in the preceding circumstances.Theorem 6.
If a graphG is created by duplicating a vertex of another graph G′, then G is a classical mean graph, where G′ is the path Cn, for n≥3.Proof.
Letv1,v2,…,vn be the vertices of the cycle Cn and let v=v1 and its duplicated vertex is v1′.Case 1.n≥5 and Δ=4+n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞.(29)Φv1′=n−1,for2≤δ≤3,Φv1=n−1,Φvγ=n+2γ−2,Φvγ=γ−3,4≤γ≤14n2+12n+282n+8+n+3+n2+6n+102+2,γ−2,14n2+12n+282n+8+n+3+n2+6n+102+3≤γ≤n.Hence,(30)Φ∗vγvγ+1=n+2γ−2,for1≤δ≤2,Φ∗v1vn=n−2,Φ∗v1′v2=n+1,Φ∗v1′vn=n−1.Let us takeX=28+12n+n2/8+2n, Y=3+n, and Z=10+6n+n2/2.(31)Φ∗v3v4=14X+Y+Z,Φ∗vγvγ+1=−3+γ,4≤γ≤14X+Y+Z+2,−2+γ,14X+Y+Z+3≤γ≤n−1.Figure9 depicts classical mean labeling of G in the preceding circumstances.Case 2.n=3,4.Figure10 depicts classical mean labeling of a graph for Case 2.Figure 9
A classical mean labeling ofG obtained by duplicating a vertex of cycle C10.Figure 10
A classical mean labeling of the graph obtained by duplicating a vertex of a graphC3 and C4.Theorem 7.
The graphS′Pn is a classical mean graph, for n≥2.Proof.
Letv1,v2,…,vn be the vertices of the path Pn and v1,v2,…,vn,v1′,v2′,…,vn′ be the vertices of S′Pn.Case 1.n is odd. Let us take Δ=−1+3n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 31).(see Table32).Hence (see Table33).(see Table34).(see Table35).Case 2.n is even and n≥4. Let us take Δ=−1+3n.Let us constructΦ:V⟶N−Δ+1,Δ+2,…,∞ (see Table 36).(see Table37).Hence (see Table38).(see Table39).(see Table40).Figure11 depicts classical mean labeling of S′P9 in the preceding circumstances.The graphS′P2 is P4 when n=2 and it is also a classical mean labeling graph.Table 31
Vertex labeling of Φ(v).
Φvγ=1γ=2γ=3Φvγ5γ−45γ−45Φvγ′44γ−54γ−5Table 32
Vertex labeling of Φ(v).
Φv2≤γ≤n/2Φv2γ6γ+1Φv2γ+16γΦv2γ′6γ−2Φv2γ+1′6γ−1Table 33
Edge labeling ofΦ∗e.
Φ∗eγ=1γ=2γ=3Φ∗vγvγ+13γ−13γ−13γ−1Φ∗vγvγ+1′5γ−45γ−47Φ∗vγ+1vγ′46γ−96γ−9Table 34
Edge labeling ofΦ∗e.
Φ∗e2≤γ≤n/2Φ∗v2γv2γ+16γΦ∗v2γ′vγ+16γ−2Φ∗v2γv2γ+1′6γ+2Table 35
Edge labeling ofΦ∗e.
Φ∗e2≤γ≤n/2−1Φ∗v2γ+1v2γ+26γ+3Φ∗v2γ+1′vγ+26γ+2Φ∗v2γ+2v2γ+2′6γ+1Table 36
Vertex labeling of Φ(v).
Φv1≤γ≤2Φvγ2γ−1Φvγ′6−γTable 37
Vertex labeling of Φ(v).
Φv2≤γ≤n/2Φv2γ−16γ−2Φv2γ6γ−3Φv2γ−1′6γ−5Φv2γ′6γ−4Table 38
Edge labeling ofΦ∗e.
Φ∗e1≤γ≤2Φ∗vγvγ+14γ−3Φ∗vγvγ+1′2γΦ∗vγ′vγ+13γTable 39
Edge labeling ofΦ∗e.
Φ∗e2≤γ≤n/2Φ∗v2γ−1v2γ6γ−3Φ∗v2γ−1′v2γ6γ−5Φ∗v2γ−1v2γ′6γ−4Table 40
Edge labeling ofΦ∗e.
Φ∗e2≤γ≤n/2−1Φ∗v2γv2γ+16γΦ∗v2γ′v2γ+16γ−1Φ∗v2γv2γ+1′6γ−2Figure 11
A classical mean labeling ofS′P9.


## 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.



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