# New Perspectives on Classical Meanness of Some Ladder Graphs

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

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## Abstract

In this study, we investigate a new kind of mean labeling of graph. The ladder graph plays an important role in the area of communication networks, coding theory, and transportation engineering. Also, we found interesting new results corresponding to classical mean labeling for some ladder-related graphs and corona of ladder graphs with suitable examples.


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## Body

## 1. Introduction and Preliminaries

All through this paper, by a graph, we mean an undirected, simple, and finite graph. For documentations and phrasing, we follow [1–6]. For a point-by-point review on graph labeling, refer [7].LetPn be a path on n nodes denoted by u1,γ, where 1≤γ≤n, and with n−1 lines denoted by e1,δ, where 1≤δ≤n−1, where eγ is the line joining the vertices u1,γ and u1,γ+1. On each edge eδ, erect a ladder with n−γ−1 steps including the edge eγ, for γ=1,2,3,…,n−1. The resulting graph is called the one-sided step graph, and it is denoted by STn. Let P2n be a path on 2n vertices u1,γ, where 1≤γ≤2n and with 2n−1 edges e1,e2,…,e2n−1, where eγ is the line joining the vertices u1,γ and u1,γ+1. On each edge eγ, we erect a ladder with ‘γ+1’ steps including the edge eγ, for γ=1,2,3,…,n, and on each eγ, we erect a ladder with 2n+1−γ steps including eγ, for γ=n+1,n+2,…,2n−1. The graph thus obtained is called the double-sided step graph, and it is denoted by 2ST2n. Let G1 and G2 be any two graphs with p1 and p2 vertices, respectively. Then, G1×G2 is the cartesian product of two graphs. A ladder graph Ln is the graph P2×Pn. The graph G∘Sm is obtained from G by attaching m pendant vertices to each vertex of G. The triangular ladder TLn, for n≥2, is a graph obtained from two paths by u1,u2,…un and v1,v2,…vn by adding the edges uγvγ,1≤γ≤n and uγvγ+1,1≤γ≤n−1. The slanting ladder SLn is a graph obtained from two paths u1,u2,…un and v1,v2,…vn by joining each vγ, with uγ+1,1≤γ≤n−1. The graph Dn∗ having the vertices aγ,δ:1≤γ≤n,δ=1,2,3,4, and its edge set is aγ,1aγ+1,1,aγ,3aγ+1,3:1≤γ≤n−1∪aγ,1aγ,2,aγ,2aγ,3,aγ,3aγ,4,aγ,4aγ,1:1≤γ≤n.


## 2. Literature Survey

The origin of graph labeling called graceful labeling was characterized by Rosa in [8] and the mean labeling of graphs was introduced by Somasundram et al. in [9]. In [10], Arockiaraj et al. presented the idea of F-root square mean labeling of the graphs and examined its meanness [11]. Durai Baskar and Arockiaraj talked about the C-geometric meanness of some ladders in [12]. Dafik et al. researched the antimagicness of the graphs including the graph Dn∗ in [13]. Durai Baskar considered the logarithmic meanness in [14] and Rajesh Kannan et al. characterized idea of exponential mean graphs in [15]. In addition, more concepts of ladder graphs and related concepts have been studied in [16–24]. Recently, Muhiuddin et al. have applied various related concepts on graphs in different aspects (see, e.g., [25–31]).


## 3. Methodology

A labelingχ on a graph GV,E with p vertices and q edges is called a Smarandache m,k mean labeling, for an integer m≥1 and k≥2, if χ:VG⟶1,2,3,…,q+1 is injective and the induced function χ∗:EG⟶1,2,3,…,q defined by(1)χ∗uv=14χum+χvm2m+χumχvmk+2χumχvmχum+χvm+χuk+χvk2k,for all uv∈EG, is bijective.Particularly, ifm=1 and k=2, such a Smarandache m,k mean labeling is the classical mean labeling on the graph. A function χ 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(2)χ∗uv=14χu+χv2+χuχv+2χuχvχu+χv+χu2+χv22,for all uv∈EG, is bijective. A graph that concedes a classical mean labeling is said to be classical mean graph.A classical mean labeling ofC3 is shown in Figure 1.Figure 1
A classical mean labeling ofC3.Here, we found interesting new results corresponding to classical mean labeling for some ladder-related graphs and corona of ladder graphs.


## 4. Main Results

Theorem 1.
The one-sided step graphSTn is a classical mean graph, for n≥2.Proof.
LetPn be a path on n nodes denoted by u1,γ, where 1≤γ≤n and with n−1 lines denoted by e1,δ, where 1≤δ≤n−1, where eγ is the line joining the vertices u1,γ and u1,γ+1. On each edge eδ, erect a ladder with n−γ−1 steps including the edge eγ, for γ=1,2,3,…,n−1. The resulting graph is called the one-sided step graph, and it is denoted by STn. Let u1,a,u2,a,u3,b,u4,c,…,un,1,un,2,for1≤a≤n,1≤b≤n−1,and1≤c≤n−2, be the nodes of STn.
Construct a mappingχ from VG to 1,2,3,…,−1+n+n2:(3)χuγ,δ=1−γ+n2+δ−1,for2≤γ≤n,1≤δ≤n+2−γ,χu1,δ=n2+δ−1,for2≤δ≤n.
Therefore,(4)χ∗uγ,δuγ,δ+1=1−γ+n2+δ−1,for2≤γ≤nand1≤δ≤1−γ+n,χ∗u1,δu1,δ+1=−1+n2+δ,1≤δ≤n−1,χ∗uγ,δuγ+1,δ=1−γ+n−γ+n−1+δ,for1≤γ≤−1+nand1≤δ≤n+1−γ.
Hence, one-sided step graphSTn is a classical mean graph, for n≥2.
A classical mean labeling ofST7 is shown in Figure 2.Figure 2
A classical mean labeling ofST7.Theorem 2.
The double-sided step graph2ST2n is a classical mean graph, for n≥1.Proof.
LetP2n be a path on 2n vertices u1,γ, where 1≤γ≤2n and with 2n−1 edges e1,e2,…,e2n−1, where eγ is the line joining the vertices u1,γ and u1,γ+1. On each edge eγ, we erect a ladder with ‘γ+1’ steps including the edge eγ, for γ=1,2,3,…,n, and on each eγ, we erect a ladder with 2n+1−γ steps including eγ, for γ=n+1,n+2,…,2n−1. The graph thus obtained is called the double-sided step graph, and it is denoted by 2ST2n.
Letu1,a,u2,a,u3,b,u4,c,…,un+1,1,un+1,2, for 1≤a≤2n,1≤b≤2n−2,and1≤c≤2n−4 be the nodes of 2ST2n.
Construct a mappingχ from VG to 1,2,3,…,3n+2n2:(5)χu1,δ=1+n+2n2+2−1+δ,1≤δ≤n,3n+2n2−2δ+2n+2,1+n≤δ≤2n,for 2≤δ≤2+n−γ and 2≤γ≤n,(6)χuγ,δ=21−γ+n2+n+2−γ+2δ−2,for 2≤γ≤n and n+3−γ≤δ≤2n+3−2γ,(7)χuγ,δ=21+n−γ2+3n+1−γ−2γ+δ−n−3,χu2,1=2n2+n−2,χuγ,1=22+n−γ2−γ+n,3≤γ≤n+1,χuγ,2n+4−2γ=2n+2−γ2+1+n−γ,2≤γ≤n+1.
Therefore,(8)χ∗u1,δu1,δ+1=1+n+2n2+2−1+δ,1≤δ≤n,3n+2n2−2δ+2n,n+1≤δ≤2n−1,for 2≤δ≤2+n−γ and 2≤γ≤n−1,(9)χ∗uγ,δuγ,δ+1=2−γ+1+n2+−γ+n+2+2−2+δ,for −γ+3+n≤δ≤2+2n−2γ and 2≤γ≤n−1,(10)χ∗uγ,δuγ,δ+1=2−γ+n+12+3−γ+n+1−2−2+γ+δ−n,χ∗uγ,2n+3−2γuγ+1,2n+2−2γ=2−γ+n+12+−γ+n+1,for2≤γ≤n,χ∗un,2un,3=4,χ∗un+1,1un+1,2=1,χ∗u1,1u2,1=2n2+n−1,χ∗u1,2nu2,2n=2n2+n,χ∗uγ,2uγ+1,1=2−γ+n2−γ+n,2≤γ≤n,χ∗u1,δu2,δ=1−n+2n2+2δ−4,2≤δ≤n,−2+n+2n2−2−1−n+δ,1+n≤δ≤−1+2n,for 3≤δ≤n+2−γ and 2≤γ≤n−1,(11)χ∗uγ,δuγ+1,δ−1=2−γ+1+n2−−γ+2+n+2δ−4,for 3+n−γ≤δ≤2+2n−2γ and 2≤γ≤n−1,(12)χ∗uγ,δuγ+1,δ−1=2−γ+1+n2+−γ+3+n−2γ−2δ+2n+2,χ∗uγ,1uγ,2=2−γ+1+n+12+3−γ+1+n,for2≤γ≤n,χ∗uγ,2n+3−2γuγ,2n+4−2γ=2−γ+1+n2+3−γ+1+n+1,for2≤γ≤n.
Hence, the double-sided step graph2ST2n is a classical mean graph, for n≥1.
A classical mean labeling of2ST10 is shown in Figure 3.Figure 3
A classical mean labeling of2ST10.Theorem 3.
Forn≥2 and m≤4, the graph Pm×Pn is a classical mean graph.Proof.
TakeVPm×Pn=vγδ:1≤δ≤n,1≤γ≤m and EPm×Pn=vγδvγ+1δ:1≤δ≤n,1≤γ≤m−1∪vγδvγδ+1:1≤δ≤n−1,1≤γ≤m.Case (i).m=2.Construct a mappingχ from VP2×Pn to 1,2,3,…,3n−1:(13)χvγδ=γ+3δ−1,for1≤γ≤2and1≤δ≤n.Therefore,(14)χ∗v1δv2δ=3δ−2,for1≤δ≤n,χ∗vγδvγδ+1=γ+3δ−2,for1≤γ≤2and1≤δ≤n−1.Case (ii).m=3.Construct a mappingχ from VP3×Pn to 1,2,3,…,5n−2:(15)χvγ1=γ,1≤γ≤2,4,γ=3,χvγδ=γ+5δ−1,for1≤γ≤3and2≤δ≤n.Therefore,(16)χ∗vγ1vγ+11=γ,for1≤γ≤2,χ∗vγ1vγ2=γ+2,for1≤γ≤3,χ∗vγδvγ+1δ=γ+5δ−5,for1≤γ≤2and2≤δ≤n,χ∗vγδvγδ+1=γ+5δ−3,for1≤γ≤3and2≤δ≤n−1.Case (iii).m=4.
Consider the graphP4×Pn, for n≥3.
Construct a mappingχ from VP4×Pn to 1,2,3,…,7n−3:(17)χvγ1=γ,1≤γ≤2,γ+1,3≤γ≤4,χvγ2=γ+7,for1≤γ≤4,χvγδ=γ+7δ−1,for1≤γ≤4and3≤δ≤n.
Therefore,(18)χ∗v1δv1δ+1=8δ−5,1≤δ≤2,7δ−3,3≤δ≤n−1,χ∗vγδvγδ+1=γ+7δ−4,for2≤γ≤4and1≤δ≤n−1,χ∗vγ1vγ+11=γ,1≤γ≤2,4,γ=3,χ∗vγ2vγ+12=γ+7,for1≤γ≤3,χ∗vγδvγ+1δ=γ+7δ−7,for1≤γ≤3and3≤δ≤n.
Hence, forn≥2 and m≤4, the graph Pm×Pn is a classical mean graph.
A classical mean labeling ofP2×P4, P3×P8, and P4×P6 are shown in Figure 4.Figure 4
A classical mean labeling ofP2×P4, P3×P8, and P4×P6.
(a)(b)(c)Corollary 1.
Every Ladder graphLn=P2×Pn is a classical mean graph.A classical mean labeling ofP2×P5 is shown in Figure 5.Figure 5
A classical mean labeling ofP2×P5.Theorem 4.
Form≤2 and n≥2, the graph Ln∘Sm is a classical mean graph.Proof.
Case (i).m=1.Construct a mappingχ from VLn∘S1 to 1,2,3,…,5n−1:(19)χuγ=3,γ=1,5γ−3,2≤γ≤n,χvγ=4,γ=1,5γ−2,2≤γ≤n,χu1γ=5γ−4,for1≤γ≤n,χx1γ=2,γ=1,5γ−1,2≤γ≤n.Therefore,(20)χ∗uγuγ+1=5γ−1,for1≤γ≤n−1,χ∗vγvγ+1=5γ,for1≤γ≤n−1,χ∗uγvγ=3,γ=1,5γ−3,2≤γ≤n,χ∗uγw1γ=5γ−4,for1≤γ≤n,χ∗vγx1γ=2,γ=1,5γ−2,2≤γ≤n.Case (ii).m=2.
Construct a mappingχ from VLn∘S2 to 1,2,3,…,7n−1:(21)χuγ=3,γ=1,−2+7γ,γis even and2≤γ≤n,7γ−5,γis odd and2≤γ≤n,χvγ=4,γ=1,7γ−4,γis even and2≤γ≤n,7γ−1,γis odd and2≤γ≤n,χw1γ=1,γ=1,−3+7γ,γis even and2≤γ≤n,−6+7γ,γis odd and2≤γ≤n,χw2γ=2,γ=1,−1+7γ,γis even and2≤γ≤n,−4+7γ,γis odd and2≤γ≤n,χx1γ=2γ+3,1≤γ≤2,−6+7γ,γis even and2≤γ≤n,−3+7γ,γis odd and2≤γ≤n,χx2γ=8,γ=1,−5+7γ,γis even and2≤γ≤n,−2+7γγis odd and2≤γ≤n.
Therefore,(22)χ∗uγuγ+1=5,γ=1,7γ−1,2≤γ≤n−1,χ∗vγvγ+1=7γ,for1≤γ≤n−1,χ∗uγvγ=7γ−4,for1≤γ≤n,χ∗uγw1γ=1,γ=1,7γ−3,2≤γ≤nandγis even,7γ−6,1≤γ≤nandγisodd,χ∗uγw2γ=7γ−2,1≤γ≤nandγis even,7γ−5,1≤γ≤nandγisodd,χ∗vγx1γ=7γ−6,1≤γ≤nandγis even,7γ−3,4≤γ≤nandγisodd,χ∗vγx2γ=7γ−5,2≤γ≤nandγis even,7γ−2,1≤γ≤nandγisodd.
Hence, form≤2 and n≥2, the graph Ln∘Sm is a classical mean graph.
A classical mean labeling ofL5∘S1 is shown in Figure 6.
A classical mean labeling ofL5∘S2 is shown in Figure 7.Figure 6
A classical mean labeling ofL5∘S1.Figure 7
A classical mean labeling ofL5∘S2.Theorem 5.
The triangular ladder graphTLn is a classical mean graph, for n≥2.Proof.
Construct a mappingχ from VTLn to 1,2,3,…,4n−2:(23)χuγ=4γ−3,for1≤γ≤n,χvγ=4γ−1,1≤γ≤n−1,4n−2,γ=n.
Therefore,(24)χ∗uγuγ+1=4γ−2,for1≤γ≤n−1,χ∗uγvγ=4γ−3,for1≤γ≤n,χ∗vγvγ+1=4γ,for1≤γ≤n−1,χ∗uγvγ+1=−1+4γ,for1≤γ≤n−1.
Hence, the triangular ladder graphTLn is a classical mean graph, for n≥2.
A classical mean labeling ofTL8 is shown in Figure 8.Figure 8
A classical mean labeling ofTL8.Theorem 6.
Form≤2 and n≥2, the graph TLn∘Sm is a classical mean graph.Proof.
Case (i).m=1.Construct a mappingχ from VLn∘S1 to 1,2,3,…,6n−2:(25)χuγ=5γ−3,1≤γ≤2,6γ−4,3≤γ≤n,χvγ=6γ−2,for1≤γ≤n,χw1γ=7γ−6,1≤γ≤2,6γ−5,3≤γ≤n,χx1γ=3,γ=1,6γ−3,2≤γ≤n.Therefore,(26)χ∗uγuγ+1=−2+6γ,for1≤γ≤n−1,χ∗vγvγ+1=6γ,for1≤γ≤n−1,χ∗uγvγ+1=6γ−1,for1≤γ≤n−1,χ∗uγvγ=2,γ=1,6γ−4,2≤γ≤n,χ∗uγw1γ=6γ−5,for1≤γ≤n,χ∗vγx1γ=3,γ=1,6γ−3,2≤γ≤n.Case (ii).m=2.
Construct a mappingχ from VLn∘S2 to 1,2,3,…,8n−2:(27)χuγ=2,γ=1,8γ−3,2≤γ≤n,χvγ=6,γ=1,8γ−5,2≤γ≤n,χw1γ=1,γ=1,8γ−4,2≤γ≤n,χw2γ=3,γ=1,8γ−2,2≤γ≤n,χx1γ=4,γ=18γ−7,2≤γ≤n,χx2γ=9,γ=1,8γ−6,2≤γ≤n.
Therefore,(28)χ∗uγuγ+1=6,γ=1,8γ,2≤γ≤n−1,χ∗vγvγ+1=8,γ=1,8γ−2,2≤γ≤n−1,χ∗uγvγ=8γ−5,for1≤γ≤n,χ∗uγvγ+1=5,γ=1,8γ−1,2≤γ≤n−1,χ∗uγw1γ=1,γ=1,8γ−4,2≤γ≤n,χ∗uγw2γ=2,γ=1,8γ−3,2≤γ≤n,χ∗vγx1γ=4,γ=1,8γ−5,2≤γ≤n,χ∗vγx2γ=7,γ=1,8γ−6,2≤γ≤n.
Hence, form≤2 and n≥2, the graph TLn∘Sm is a classical mean graph.
A classical mean labeling ofTL5∘S1 is shown in Figure 9.
A classical mean labeling ofTL4∘S2 is shown in Figure 10.Figure 9
A classical mean labeling ofTL5∘S1.Figure 10
A classical mean labeling ofTL4∘S2.Theorem 7.
Forn≥2, the slanting ladder graph SLn is a classical mean graph.Proof.
Construct a mappingχ from VSLn to 1,2,3,…,3n−2:(29)χu1=1,χuγ=−4+3γ,for2≤γ≤n,χvγ=3γ,for1≤γ≤−1+n,χvn=−2+3n.
Therefore,(30)χ∗uγuγ+1=1,γ=1,3γ−3,2≤γ≤n−1,χ∗vγvγ+1=3γ+1,for1≤γ≤n−2,χ∗vn−1vn=3n−3,χ∗vγuγ+1=3γ−1,for1≤γ≤n−1.
Hence, forn≥2, the slanting ladder graph SLn is a classical mean graph.
A classical mean labeling ofSL8 is shown in Figure 11.Figure 11
A classical mean labeling ofSL8.Theorem 8.
Form≤2 and n≥2, the graph SLn∘Sm is a classical mean graph.Proof.
LetESLn∘Sm=uγuγ+1,vγvγ+1:1≤γ≤n−1∪uγvγ−1:2≤γ≤n∪uγuδγ:1≤γ≤n,1≤δ≤m∪vγvδγ:1≤γ≤n,1≤δ≤m.Case (i).n≥3 and m=1.Construct a mappingχ from VSLn∘S1 to 1,2,3,…,5n−2:(31)χuγ=1+γ,1≤γ≤2,−6+5γ,3≤γ≤n,χvγ=6,γ=1,5γ,2≤γ≤n−1,−2+5n,γ=n,χw1γ=−2+3γ,1≤γ≤2,−7+5γ,3≤γ≤n,χx1γ=7,γ=1,1+5γ,2≤γ≤n−1,−3+5n,γ=n.Therefore,(32)χ∗uγuγ+1=−1+3γ,1≤γ≤2,−4+5γ,3≤γ≤n−1,χ∗vγvγ+1=2+5γ,1≤γ≤n−2,−4+5n,γ=n−1,χ∗vγuγ+1=−1+5γ,for1≤γ≤n−1,χ∗uγw1γ=1,γ=1,−7+5γ,2≤γ≤n,χ∗vγx1γ=6,γ=1,5γ,2≤γ≤n−1,−3+5n,γ=n.Case (ii).n≥3 and m=2.Construct a mappingχ from VSLn∘S2 to 1,2,3,…,7n−2(33)χuγ=2+γ,1≤γ≤2,−9+7γ,γis odd and3≤γ≤n−1,−6+7γ,γis even and3≤γ≤n−1,−10+7n,nis even andγ=n,−9+7n,nis odd andγ=n,χvγ=8,γ=1,−1+7γ,γis odd and2≤γ≤n−32+7γ,γis even and2≤γ≤n−3−13+7n,nis even andγ=n−2,−15+7n,nis odd andγ=n−2,7n−5,γ=n−1,7n−3,γ=n,χw1γ=1,γ=1−5+5γ,2≤γ≤3,−7+7γ,γis even and4≤γ≤n−1,−10+7γ,γis odd and4≤γ≤n−1,−11+7n,nis even andγ=n,−10+7n,nis odd andγ=n,χw2γ=−5+7γ,1≤γ≤2,−5+7γ,γis even and3≤γ≤n−1,−8+7γ,γis odd and3≤γ≤n−1,−7+7n,nis even andγ=n,−8+7n,nis odd andγ=n,χx1γ=9,γ=1,7γ,2≤γ≤n−3andγiseven,−3+7γ,2≤γ≤n−3andγisodd,−12+7n,nis even andγ=n−2,−17+7n,nis odd andγ=n−2,−8+7n,nis even andγ=n−1,−7+7n,nis odd andγ=n−1,−4+7n,γ=n,χx2γ=11,γ=1,7γ+1,γis even and2≤γ≤n−3,−2+7γ,γis odd and2≤γ≤n−3,−9+7n,nis even andγ=n−2,−16+7n,nis odd andγ=n−2,−6+7n,γ=n−1,−2+7n,γ=n.Therefore,(34)χ∗uγuγ+1=−1+4γ,1≤γ≤2,−5+7γ,3≤γ≤n−2,−14+7n,nis even andγ=n−1,−12+7n,nis odd andγ=n−1,χ∗vγvγ+1=11,γ=1,3+7γ,2≤γ≤n−3,−10+7n,nis even andγ=n−2,−11+7n,nis odd andγ=n−2,−5+7n,γ=n−1,χ∗vγuγ+1=5,γ=1,−1+7γ,2≤γ≤n−1,χ∗uγw1γ=1,γ=1,−8+6γ,2≤γ≤3,−7+7γ,γis even and4≤γ≤n−1,−10+7γ,γis odd and4≤γ≤n−1,−11+7n,nis even andγ=n,−10+7n,nis odd andγ=n,χ∗uγw2γ=−2+4γ,1≤γ≤2,−6+7γ,γis even and3≤γ≤n−1,−9+7γ,γis odd and3≤γ≤n−1,−9+7n,γ=n,χ∗vγx1γ=8,γ=1,7γ,γis even and2≤γ≤n−3,−3+7γ,γis odd and2≤γ≤n−3,−13+7n,nis even andγ=n−2,−17+7n,nis odd andγ=n−2,−7+7n,γ=n−1,−4+7n,γ=n,χ∗vγx2γ=9,γ=1,1+7γ,γis even and2≤γ≤n−3,−2+7γ,γis odd and2≤γ≤n−3,−12+7n,nis even andγ=n−2,−16+7n,nis odd andγ=n−2,−6+7γ,γ=n−1,−3+7n,γ=n.Case (iii).n=2 and m=1,2.
A classical mean labeling ofSL2∘S1 and SL2∘S2 are shown in Figure 12.
A classical mean labeling ofSL6∘S1 is shown in Figure 13.
A classical mean labeling ofSL5∘S2 is shown in Figure 14.Figure 12
A classical mean labeling ofSL2∘S1 and SL2∘S2.
(a)(b)Figure 13
A classical mean labeling ofSL6∘S1.Figure 14
A classical mean labeling ofSL5∘S2.Theorem 9.
Forn≥2, the graph Dn∗ is a classical mean graph.Proof.
TakeVDn∗=aγ,δ:1≤γ≤n,δ=1,2,3,4 and EDn∗=aγ,1aγ+1,1,aγ,3aγ+1,3:1≤γ≤n−1∪aγ,1aγ,2,aγ,2aγ,3,aγ,3aγ,4,aγ,4aγ,1:1≤γ≤n.
Construct a mappingχ from VDn∗ to 1,2,3,…,6n−1:(35)χaγ,1=−2+6γ,for1≤γ≤n,χaγ,2=−5+6γ,for1≤γ≤n,χaγ,3=−3+6γ,for1≤γ≤n,χaγ,4=−1+6γ,for1≤γ≤n.
Therefore,(36)χ∗aγ,1aγ+1,1=6γ,for1≤γ≤n−1,χ∗aγ,3aγ+1,3=−1+6γ,for1≤γ≤n−1,χ∗aγ,1aγ,2=−4+6γ,for1≤γ≤n,χ∗aγ,2aγ,3=−5+6γ,for1≤γ≤n,χ∗aγ,3aγ,4=−3+6γ,for1≤γ≤n,χ∗aγ,4aγ,1=−2+6γ,for1≤γ≤n.
Hence, forn≥2, the graph Dn∗ is a classical mean graph.
A classical mean labeling ofD4∗ is shown in Figure 15.Figure 15
A classical mean labeling ofD4∗.Theorem 10.
Forn≥1, the diamond ladder graph Dln is a classical mean graph.Proof.
VDln=xγ,yγ:1≤γ≤n∪zγ:1≤γ≤2n and EDln=xγyγ:1≤γ≤n∪xγxγ+1,yγyγ+1:1≤γ≤n−1∪xγz2γ−1,xγz2γ,yγz2γ−1,yγz2γ:1≤γ≤n∪z2γz2γ+1:1≤γ≤n−1. Thus, VDln=4n and EDln=8n−3.
Construct a mappingχ from VDln to 1,2,3,…,8n−2:(37)χxγ=−5+8γ,for1≤γ≤n,χyγ=−3+8γ,for1≤γ≤n,χzγ=1,γ=1,4γ−−1γ+1+12−2,γis even and2≤γ≤2n,4γ−−1γ+1+12−2,γis odd and3≤γ≤2n.
Therefore,(38)χ∗xγxγ+1=−2+8γ,for1≤γ≤n−1,χ∗yγyγ+1=8γ,for1≤γ≤n−1,χ∗xγyγ=−5+8γ,for1≤γ≤n,χ∗z2γz2γ+1=−1+8γ,for1≤γ≤n−1,χ∗xγz2γ−1=−7+8γ,for1≤γ≤n,χ∗xγz2γ=−4+8γ,for1≤γ≤n,χ∗yγz2γ−1=−6+8γ,for1≤γ≤n,χ∗yγz2γ=−3+8γ,for1≤γ≤n.
Hence, forn≥1, the diamond ladder graph Dln is a classical mean graph.
A classical mean labeling ofDl4 is shown in Figure 16.Figure 16
A classical mean labeling ofDl4.Theorem 11.
Forn≥4, the latitude ladder graph LLn is a classical mean graph.Proof.
Here,VLLn=uγ:1≤γ≤n and ELLn=uγuγ+1:1≤γ≤n−1∪unu1∪uγun+2−γ:2≤γ≤n/2.
Construct a mappingχ from VLLn to 1,2,3,…,3n/2:(39)χuγ=3γ−2,1≤γ≤n2−1,3γ−1,γ=n2,3n2,γ=1+n2,3γ+3n−+3,2+n2≤γ≤n.
Therefore,(40)χ∗uγuγ+1=3γ−1,1≤γ≤n2,3n−3γ+1,n2+1≤γ≤n,χ∗uγun+2−γ=3γ−3,for2≤γ≤n2.
Hence, forn≥4, the latitude ladder graph LLn is a classical mean graph.
A classical mean labeling ofLL14 is shown in Figure 17.Figure 17
A classical mean labeling of latitude graphLL14.


## 5. Conclusion

The Cartesian product is one among graph operations. Based on this operation, the classical mean labeling of various graphs such as the one-sided step graphSTn, double-sided step graph 2ST2n, graph Pm×Pn, ladder graph Ln, graph Ln∘Sm, triangular ladder graph TLn, graph TLn∘Sm, slanting ladder graph SLn, graph SLn∘Sm, graph Dn∗, diamond ladder graph Dln, and latitude ladder graph LLn are established. It would be very interesting to analyze that the classical meanness of various ladder-related graphs. Investigating classical mean labeling of other classes of graphs is still open and this is for future work. One can also explore the exclusive applications of classical mean labeling in real-life problems.



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