Research Article
New Perspectives on Classical Meanness of Some Ladder Graphs
A.M.Alanazi ,
1
G. Muhiuddin ,
1
A.R.Kannan,
2
andV.Govindan
3
1
Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia
2
Department of Mathematics, Mepco Schlenk Engineering College, Sivakasi 626 005, Tamil Nadu, India
3
Department of Mathematics, Sri Vidya Mandir Arts & Science College, Katteri, Uthangarai 636902, Tamilnadu, India
Correspondence should be addressed to G. Muhiuddin; chishtygm@gmail.com
Received 3 April 2021; Accepted 14 June 2021; Published 30 June 2021
Academic Editor: A. H. Kara
Copyright © 2021 A. M. Alanazi et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, we investigate a new kind of mean labeling of graph. e 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.
1.IntroductionandPreliminaries
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 la-
beling, refer [7].
Let P
n
be a path on n nodes denoted by u
1,c
, where
1 ≤ c ≤ n, and with n − 1 lines denoted by e
1,δ
, where
1 ≤ δ ≤ n − 1, where e
c
is the line joining the vertices u
1,c
and
u
1,c+1
. On each edge e
δ
, erect a ladder with n −(c − 1) steps
including the edge e
c
, for c � 1, 2, 3, ... ,n − 1. e resulting
graph is called the one-sided step graph, and it is denoted by
ST
n
. Let P
2n
be a path on 2n vertices u
1,c
, where 1 ≤ c ≤ 2n
and with 2n − 1 edges e
1
,e
2
, ... ,e
2n− 1
, where e
c
is the line
joining the vertices u
1,c
and u
1,c+1
. On each edge e
c
, we erect
a ladder with ‘c + 1’ steps including the edge e
c
, for
c � 1, 2, 3, ... ,n, and on each e
c
, we erect a ladder with 2n +
1 − c steps including e
c
, for c � n + 1,n + 2, ... , 2n − 1. e
graph thus obtained is called the double-sided step graph,
and it is denoted by 2ST
2n
. Let G
1
and G
2
be any two graphs
with p
1
and p
2
vertices, respectively. en, G
1
× G
2
is the
cartesian product of two graphs. A ladder graph L
n
is the
graph P
2
× P
n
. e graph G ∘ S
m
is obtained from G by
attaching m pendant vertices to each vertex of G. e tri-
angular ladder TL
n
, for n ≥ 2, is a graph obtained from two
paths by u
1
,u
2
, ... u
n
and v
1
,v
2
, ... v
n
by adding the edges
u
c
v
c
, 1 ≤ c ≤ n and u
c
v
c+1
, 1 ≤ c ≤ n − 1. e slanting ladder
SL
n
is a graph obtained from two paths u
1
,u
2
, ... u
n
and
v
1
,v
2
, ... v
n
by joining each v
c
, with u
c+1
, 1 ≤ c ≤ n − 1. e
graph D
∗
n
having the vertices a
c,δ
: 1 ≤ c ≤ n, δ � 1, 2, 3, 4 ,
and its edge set is a
c,1
a
c+1,1
,a
c,3
a
c+1,3
: 1 ≤ c ≤ n − 1
∪ a
c,1
a
c,2
,a
c,2
a
c,3
,a
c,3
a
c,4
,a
c,4
a
c,1
: 1 ≤ c ≤ n .
2.LiteratureSurvey
e 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 D
∗
n
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 labeling χ onagraph G(V, E) with p vertices and q edges is
called a Smarandache (m, k) mean labeling, for an integer
m ≥ 1and k ≥ 2,if χ: V(G) ⟶ 1, 2, 3, ... ,q + 1 is injective
Hindawi
Journal of Mathematics
Volume 2021, Article ID 9926350, 14 pages
https://doi.org/10.1155/2021/9926350