# Preservation of the Classical Meanness Property of Some Graphs Based on Line Graph Operation

Authors: G. Muhiuddin; A. M. Alanazi; A. R. Kannan; V. Govindan
Journal: Journal of Mathematics
 (2021)

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## Abstract

In the present paper, we introduce the classical mean labeling of graphs and investigate their related properties. Moreover, it is obtained that the line graph operation preserves the classical meanness property for some standard graphs.


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## Body

## 1. Introduction and Preliminaries

All through this paper, by a graph we mean a simple, undirected, and finite graph. For documentations and wording, we follow [1–5]. For a point by point review on graph labeling, we refer [6]. The line graph LG of a graph G is defined to have as its vertices the edges of G, with two being adjacent if the corresponding edges share a vertex in G. The graph G°Sm is obtained from G by attaching m pendant vertices to each vertex of G. Let uα:1≤α≤n and vβα:1≤β≤m+1 be the nodes of path Pn and αth copy of the star graph Sm, respectively, then the graph Pn;Sm is obtained from n copies of Sm and the path Pn by joining uα with the central vertex v1α of the αth copy of Sm by means of an edge, for 1≤α≤n. A graph obtained by subdividing edge of G by a vertex is called subdivision graph SG and a graph obtained from the path by replacing every edge of a path by a C3 is called triangular snake graph Tn.


## 2. Literature Survey

The investigation of graceful labeling is characterized by Rosa in [7] and prime labeling is defined by Tout et al. in [8]. Somasundram and Ponraj introduced the mean labeling of graphs in [9]. Durai Baskar and Arockiaraj defined the F-harmonic mean labeling [10] and discussed its meanness for some standard graphs. The idea of F-geometric was presented by Durai Baskar et al. in [11] and F-root mean labeling was presented by Arockiaraj et al. in [12] and talked about its meanness of ladder graph in [13]. Vaidya and Barasara in [14] have discussed so many results on product cordial labeling. Vaidya and Lekha in [15] presented the idea of a bi-odd sequential labeling. The labeling of L (2, 1) in [16] is researched by Prajapati and Patel. Rajesh Kannan et al. discussed the FCM labeling of graphs and its line graphs in [17]. Propelled by and crafted by such a large number of creators in the territory of graph labeling, we present another labeling called classical mean labeling. A classical mean of two positive integers need not be an integer in general. For the classical mean is to be an integer, we may use either flooring or ceiling function. In this paper, we consider only the flooring function of our discussion and try to analyze that the line graph operation preserves the classical meanness property for some standard graphs. The labeling is one of the well studied area in Graph Theory. So, we are interested in defining new labeling called classical mean labeling. A classical mean labeling is for getting more accuracy of all the edge labeling by using the average of four different types of means of the vertex labeling of the given graph. Recently, Muhiuddin et al. studied various related concepts on graphs (see [18–22]).Graph labeling assumes an essential job in different areas of the real world system. The concepts of classical mean labeling are utilized to demonstrate numerous kinds of processes and relations in biological, social, material physical, and data systems. It is a powerful tool that makes complicated patterns to be learned easily and conveniently in various fields. A static network can be represented as a specific kind of graph by connecting nodes in some topology, and labeling can be applied for automatic routing of data in a network. The graph can be cycle, path, circuit, walk, and connected which represent a fixed network. For each network, labeling is done with a constant which helps routing to automatically detect next node in the network. The classical mean labeling is used in fast communication in sensor networks for finding the more accuracy level of sensor units.


## 3. Methodology

A functionχ is known as a classical mean labeling of a graph GV,E with p nodes and q edges if χ:VG⟶1,2,3,…,q+1 is injective and the incited edge assignment function χ∗:EG⟶1,2,3,…,q characterized as(1)χ∗uv=14χu+χv2+χuχv+2χuχvχu+χv+χu2+χv22,for all uv∈EG, is bijective. From Figure 1, a graph that concedes a classical mean labeling is said to be classical mean graph.Figure 1
A classical mean labeling ofC4°S1.Asq is the number of edges of the given graph, it cannot take a randomly large number so that such a labeling exists. However more than one classical mean labeling exists for the given graph. So, we show one among in the proof.Here, it is found that the line graph operation preserves the classical meanness property for some standard graphs.


## 4. Classical Meanness of Some Standard Graphs and Its Line Graph

Theorem 1.
Every pathPn for n≥1 and its line graph LPn for n≥2 are classical mean graphs.Proof.
Develop a mappingχ from the vertex set of path to 1,2,3,…,n by χvα=α, for 1≤α≤n, where vα:1≤α≤n be the nodes of the path. Therefore, for 1≤α≤n−1, χ∗vαvα+1=α. Since LPn is again a path, LPn is also a classical mean graph. Hence, every path Pn for n≥1 and its line graph LPn for n≥2 are classical mean graphs.Theorem 2.
Every cycleCn and its line graph LCn are classical mean graphs, for n≥3.Proof.
Develop a mappingχ from the vertex set of cycle to 1,2,3,…,n+1 by(2)χuα=−1+2α,1≤α≤n2+1,2n−2α+4,n2+2≤α≤n.
Therefore,(3)χ∗uαu1+α=2α−1,1≤α≤n2,n,α=n2+1,2n−2α+2,n2+2≤α≤−1+nandχ∗unu1=2.
Also, the graphLCn is again a cycle, which is given by Figure 2. Hence, every cycle Cn and its line graph LCn are classical mean graphs, for n≥3.Figure 2
A classical mean labeling ofC9.


## 5. Classical Meanness of Graphs Obtained from Vertex Identification and Its Line Graph

Theorem 3.
. The graphPn°Sm and its line graph LPn∘Sm are classical mean graphs, for n≥1 and m≤2.Proof.
Letuβα:1≤α≤nand1≤β≤m be the pendant vertices at each vα.Case i. m=1 in the graph Pn°Sm. Develop a mapping χ from the vertex set of Pn°S1 to 1,2,3,…,2n by(4)χvα=1,α=1,2α,2≤α≤nand,χu1α=2,α=1,2α−1,2≤α≤n.Therefore,(5)χ∗vα+1vα=2α,for1≤α≤n−1and,χ∗u1αvα=−1+2α,1≤α≤n.Case ii.m=2 in the graph Pn°Sm.Develop a mappingχ from the vertex set of Pn°S2 to 1,2,3,…,3n by(6)χvα=3α−1,for1≤α≤n,χu1α=3α−2,for1≤α≤nand,χu2α=3α,for1≤α≤n.Therefore,(7)χ∗vαvα+1=3α,for1≤α≤n−1,χ∗vαu1α=3α−2,for1≤α≤nand,χ∗vαu2α=3α−1,for1≤α≤n.Hence, a classical mean labeling ofP7°S1 and P7°S2 is given by Figure 3.LetVLPn°S1=v1,v2,v3,…,vn,e1,e2,e3,…,en−1 and ELPn°S1=vαeα,eαvα+1:1≤α≤n−1∪eαeα+1:1≤α≤n−2.Case iii.m=1 in the graph LPn°Sm.Develop a mappingχ from the line graph of vertex set of Pn°S1 to 1,2,3,…,3n−3 by(8)χvα=2α−1,1≤α≤3,3α−4,4≤α≤nand,χeα=2,α=1,3α,2≤α≤n−1.Therefore,(9)χ∗vαeα=3α−2,1≤α≤2,3α−3,3≤α≤−1+n,χ∗eαvα+1=3α−1,for1≤α≤n−1and,χ∗eαeα+1=3,α=1,3α+1,2≤α≤−2+n.LetVLPn°S2=xα:1≤α≤n−1∪yα,zα:1≤α≤n and ELPn°S2=xαzα,xαyα+1,xαzα+1,xαyα:1≤α≤n−1∪xαxα+1:1≤α≤n−2∪yαzα:1≤α≤n.Case iv. m=2 in the graph LPn°Sm. Develop a mapping χ from the line graph of vertex set of Pn°S2 to 1,2,3,…,6n−5 by(10)χxα=5,α=1,6α,2≤α≤n−1,χyα=1,α=1,6α−8,2≤α≤nand,χzα=6α−4,1≤α≤2,6α−5,3≤α≤n.Therefore,(11)χ∗xαxα+1=2+6α,for1≤α≤n−2,χ∗xαyα=2,α=1,6α−5,2≤α≤−1+n,χ∗xαzα=6α−3,for1≤α≤n−1,χ∗xαzα+1=6α,for1≤α≤−1+n,χ∗xαyα+1=6α−2,for1≤α≤−1+nand,χ∗yαzα=1,α=1,6α−7,2≤α≤n.Hence, from Figure4, the graph Pn°Sm and its line graph LPn°Sm are classical mean graphs, for m≤2 and n≥1.Figure 3
A classical mean labeling ofP7°S1 and P7°S2.Figure 4
A classical mean labeling ofLP7°S1 and LP7°S2.Theorem 4.
The graphPn;Sm and its line graph LPn;Sm are classical mean graphs, for m≤2 and n≥1.Proof.
Case i.m=1 in the graph Pn;Sm. Develop a mapping χ from the vertex set of Pn;S1 to 1,2,3,…,3n by(12)χuα=3α,αis odd and1≤α≤n,3α−2,αis even and1≤α≤n,χv1α=3α−1,1≤α≤nand,χv2α=−2+3α,αis odd and1≤α≤n,3α,αis even and1≤α≤n.Therefore,(13)χ∗uαuα+1=3α,for1≤α≤n−1,χ∗uαv1α=−1+3α,αis odd and1≤α≤n,−2+3α,αis even and1≤α≤nandχ∗v1αv2α=−2+3α,αis odd and1≤α≤n,−1+3α,αis even and1≤α≤n.Case ii. m=2 in the graph Pn;Sm. Develop a mapping χ from the vertex set of Pn;S2 to 1,2,3,…,4n by(14)χuα=4α,αis odd and1≤α≤n,−2+4α,αis even and1≤α≤n,χv1α=4α−1,1≤α≤n,χv2α=1,α=1,4α+1,2≤α≤nand,χv3α=4α−2,αis odd and1≤α≤n,4α,αis even and1≤α≤n.Therefore,(15)χ∗uαuα+1=4α,for1≤α≤n−1,χ∗uαv1α=−1+4α,αis odd and1≤α≤n,−2+4α,αis even and1≤α≤n,χ∗v1αv2α=−3+4α,for1≤α≤n−1,χ∗v1αv3α=−2+4α,αis odd and1≤α≤n,4α−1,αis even and1≤α≤n.It is clearly seen that a classical mean labeling ofP7;S1 and P6;S2 is given by Figure 5.Case iii. m=1 and n≥2 in the graph LPn;Sm. Develop a mapping χ from the line graph of vertex set of Pn;S1 to 1,2,3,…,4n−3 by(16)χuα=4α,for1≤α≤n−1,χv1α=2,α=1,4α−3,2≤α≤nand,χv2α=1,α=1,4α−5,2≤α≤n.Therefore,(17)χ∗uαuα+1=4α+1,for1≤α≤n−1,χ∗uαv1α+1=4α,for1≤α≤n−1,χ∗uαv1α=4α−2,for1≤α≤n−1and,χ∗v1αv2α=1,α=1,4α−5,2≤α≤n.Case iv. m=1 and n=1 in the graph LPn;Sm. For n=1, the graph LPn;S1 is a path and by Theorem 1, the result follows.Case v. m=2 and n≥2 in the graph LPn;Sm. Develop a mapping χ from the line graph of vertex set of Pn;S2 to 1,2,3,…,5n−3 by(18)χuα=−4+8α,1≤α≤2,5α,αis odd and3≤α≤n,1+5α,αis even and1≤α≤n,χv1α=3α,1≤α≤2,11,α=3,5α−3,4≤α≤nandαis odd,5α−4,4≤α≤nandαis even,χv2α=4α−3,1≤α≤2,5α−7,3≤α≤nandαis odd,−6+5α,3≤α≤nandαis even and,χv3α=2,α=1,−5+5α,αis odd and2≤α≤n,−3+5α,αis even and2≤α≤n.Therefore,(19)χ∗uαuα+1=6α+1,1≤α≤2,5α+2,3≤α≤n−2,χ∗uαv1α=3,α=1,4α,2≤α≤3,5α−2,4≤α≤n−1,χ∗uαv1α+1=4,α=1,5α,αis odd and2≤α≤n,5α+1,αis even and2≤α≤n,χ∗v1αv2α=4α−3,1≤α≤2,5α−6,3≤α≤nand,χ∗v1αv3α=2,α=1,5α−5,αis odd and2≤α≤n,5α−4,αis even and2≤α≤n.Case vi. m=2 and n=1 in the graph LPn;Sm.
Forn=1, the graph LP1;S2 is C3 and by Theorem 2, the result follows.
Hence, the graphPn;Sm for m≤2 and n≥1 and its line graph LPn;Sm for n≥1 are classical mean graphs given by Figure 6.Figure 5
A classical mean labeling ofP7;S1 and P6;S2.Figure 6
A classical mean labeling ofLP7;S1 and LP7;S2.


## 6. Classical Meanness of Graphs Obtained from Other Graph Operations and Its Line Graph

Theorem 5.
. Forn≥2, MPn and its line graph LMPn are classical mean graphs.Proof.
SinceMPn is a graph LPn°S1, for n≥2 and by Theorem 3, the result follows. Let VLMPn=xα:1≤α≤n+1∪xα′:1≤α≤n−3∪yα:1≤α≤n−2 and ELMPn=yαyα+1,xα′xα+3,xα′yα+1,yαxα′:1≤α≤n−3∪yαxα+2:1≤α≤n−2∪yαxα+1:2≤α≤n−2∪xαxα+1,xαy1:1≤α≤2∪yn−2xn+1,xnxn+1.Case i. n≥3 in the graph LMPn. Develop a mapping χ from the line graph of vertex set of MPn to 1,2,3,…,7n−13 by(20)χxα=α,1≤α≤2,−14+7α,3≤α≤n,−13+7n,α=n+1,χxα′=2+7α,for1≤α≤n−3and,χyα=−2+7α,for1≤α≤n−2.Therefore,(21)χ∗yαyα+1=7α+1,for1≤α≤n−3,χ∗xα′xα+3=7α+4,for1≤α≤n−3,χ∗xα′yα+1=7α+3,for1≤α≤n−3,χ∗yαxα′=7α−1,for1≤α≤n−3,χ∗yαxα+2=7α−2,for1≤α≤n−2,χ∗yαxα+1=3,α=1,7α−5,2≤α≤n−2,χ∗xαxα+1=3α−2,for1≤α≤2,χ∗xαy1=α+1,for1≤α≤2,χ∗yn−2xn+1=7n−15andχ∗xnxn+1=−14+7n.Case ii.n=2 in the graph LMPn. For n=2, the graph LMPn is P2 and by Theorem 1, the result follows. Hence, from Figure 7, for n≥2, the graph MPn and its line graph LMPn are classical mean graphs.Figure 7
A classical mean labeling ofLMP7.Theorem 6.
. The graphSPn°K1 and its line graph LSPn°K1 are classical mean graphs, for n≥1.Proof.
LetVSPn°K1=uα,vα,xα,yβ:1≤α≤n,1≤β≤n−1 and ESPn°K1=uαxα,vαxα:1≤α≤n∪uαyα,yαuα+1:1≤α≤n−1. Develop a mapping χ from the vertex set of SPn°K1 to 1,2,3,…,4n−1 by(22)χuα=3α,1≤α≤2,4α−1,3≤α≤n,χyα=4,α=1,4α+1,2≤α≤n−1,χxα=5α−3,1≤α≤2,4α−2,3≤α≤nand,χvα=4α−3,1≤α≤2,4α−4,3≤α≤n.
Therefore,(23)χ∗uαyα=−1+4α,1≤α≤n−1,χ∗yαuα+1=4,α=1,1+4α,2≤α≤n−1,χ∗uαxα=2,α=1,4α−2,2≤α≤nand,χ∗vαxα=4α−3,1≤α≤2,4α−4,3≤α≤n.
LetVLSPn°K1=uα,uβ′,vα,wα:1≤α≤n,1≤β≤n−2 and ELSPn°K1=uαvα,vαwα:1≤α≤n∪uα′vα+1:1≤α≤n−2∪uαuα−1′:2≤α≤n−1∪uα′uα+2:1≤α≤n−2∪u1u2.Case i. n≥3 in the graph LSPn°K1. Develop a mapping χ from the line graph of vertex set of SPn°K1 to 1,2,3,…,5n−4 by(24)χuα=3,α=1,5α−6,2≤α≤n−1,5n−4,α=n,χuα′=5α+3,for1≤α≤n−2,χvα=2,α=1,5α−4,2≤α≤n−1,5α−5,α=nand,χwα=1,α=1,5α,2≤α≤n−2,5n−6,α=n−1,5n−8,α=n.Therefore,(25)χ∗uαvα=2,α=15α−6,2≤α≤n−1,5n−5,α=n,χ∗vαwα=1,α=15α−3,2≤α≤n−15n−7,α=n,χ∗uα′vα+1=6,α=15α+1,2≤α≤n−2,χ∗uαuα−1′=5α,for2≤α≤n−1,χ∗uα′uα+2=5α+3,1≤α≤n−36α−1,α=n−2and,χ∗u1u2=3.Case ii.1≤n≤2 in the graph LSPn°K1. For 1≤n≤2, the graph LSPn°K1 is a path and by Theorem 1, the result follows. Hence, from Figure 9, the graph SPn°K1 and its line graph LSPn°K1 are classical mean graphs, for n≥1.Figure 8
A classical mean labeling ofSP6°K1. Therefore, a classical mean labeling of SP6°K1 is given by Figure 8.Figure 9
A classical mean labeling ofLSP7°K1.Theorem 7.
The triangular snakeTn and its line graph LTn are classical mean graphs, for n≥2.Proof.
LetVTn=uα:1≤α≤n−1∪vα′:1≤α≤n and ETn=uαvα,uαvα+1,vαvα+1:1≤α≤n−1. Develop a mapping χ from the vertex set of Tn to 1,2,3,…,3n−2 by(26)χuα=3α,1≤α≤n−1and,χvα=−2+3α,1≤α≤n.Therefore,(27)χ∗uαvα=3α−2,1≤α≤n−1,χ∗uαvα+1=3α,1≤α≤n−1and,χ∗vαvα+1=−1+3α,1≤α≤n−1.
Hence, a classical mean labeling ofT7 is given by Figure 10.
LetVLTn=xα:1≤α≤n∪xα′:1≤α≤n−2∪yα′:1≤α≤n−1 and ELTn=yαyα+1,xαxα−1′,xα′xα+2:1≤α≤n−2∪yαxα+1,yαxα−1′,xαyα:1≤α≤n−1∪x1x2,x2x1′.Case i.n≥3 in the graph LTn. Develop a mapping χ from the line graph of Tn to 1,2,3,…,7n−10 by(28)χxα=α,1≤α≤2,11,α=3andα=n,12,α≠3andα=n,7α−9,4≤α≤n−1,7α−10,α=n,χxα′=7α+3,for1≤α≤n−2and,χyα=5,α=1.7α−6,2≤α≤n−1.Therefore,(29)χ∗yαyα+1=6,α=1,7α−3,2≤α≤n−2,χ∗yαxα+1=3,α=1,7α−5,2≤α≤n−1,χ∗xαxα−1′=5,i=2,7α−7,3≤α≤n−2,χ∗yαxα−1′=−6+7α,for2≤α≤n−1,χ∗xαyα=2α,1≤α≤2,−8+7α,3≤α≤n−1,χ∗xα′xα+2=10,α=1,7α+3,2≤α≤n−2,χ∗x1x2=1andχ∗x2x1′=5.Case ii.n=2 in the graph LTn. For n=2, the graph LTn is a cycle C3 and by Theorem 2, the result follows. Hence, from Figure 11, the triangular snake Tn and its line graph LTn are classical mean graphs, for n≥2.Figure 10
A classical mean labeling labeling ofT7.Figure 11
A classical mean labeling labeling ofLT6.


## 7. Conclusion

In this paper, it is found that the line graph operation preserves the classical meanness property for some standard graphs. Further investigation can be done to analyze the preservation of the classical meanness property by the line graph operation for other graphs.



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