Ratio Mathematica                                                                            Volume 45, 2023 

 

 
 

 

 

Near mean labeling in dicyclic snakes 

 

Palani K* 

Shunmugapriya A† 

 
  

Abstract 

K. Palani et al. defined the concept of near mean labeling in digraphs. Let 𝐷 = (𝑉, 𝐴) 
be a digraph where 𝑉the vertex is set and 𝐴 is the arc set.  Let 𝑓: 𝑉 → {0, 1, 2, … , 𝑞} be a 

1-1 map. Define  𝑓∗: 𝐴 → {1, 2, … , 𝑞} by𝑓∗(𝑒 = 𝑢𝑣⃗⃗⃗⃗ ) = ⌈
𝑓(𝑢)+𝑓(𝑣)

2
⌉. Let𝑓∗(𝑣) =

|∑ 𝑓∗(𝑣𝑤⃗⃗⃗⃗  ⃗)𝑤∈𝑉 − ∑ 𝑓
∗(𝑤𝑣⃗⃗⃗⃗  ⃗)𝑤∈𝑉 |. If𝑓

∗(𝑣) ≤ 2  ∀ 𝑣 ∈ 𝐴(𝐷), then 𝑓 is said to be a near 
mean labeling of D and 𝐷 is said to be a near mean digraph. In this paper, different 
dicyclic snakes are defined and the existence of near mean labeling in them is checked. 

 

Keywords: Near mean labeling, digraphs, di-cyclic, di-quadrilateral, di-pentagonal, 

snake. 

2010 AMS subject classification: 05C78‡ 
 

 
 

 
*PG & Research Department of Mathematics, A.P.C. Mahalaxmi College for Women, Thoothukudi-628 

002, Tamil Nadu, India); palani@apcmcollege.ac.in. 
† Department of Mathematics, Sri Sarada College for Women (Autonomous), Tirunelveli-627 011. 

(Research scholar-19122012092005, A.P.C. Mahalaxmi College for Women, Thoothukudi-628 002, 

Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, 

India); priyaarichandran@gmail.com. 
‡ Received on July 16, 2022. Accepted on September 15, 2022. Published on January 30, 2023. doi: 

10.23755/rm.v45i0.1022. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY licence agreement. 

 

238

mailto:palani@apcmcollege.ac.in
mailto:priyaarichandran@gmail.com


Palani K and Shunmugapriya A 

 

 

1. Introduction  

Graph theory has applications in many areas of the computing, social and natural 

science. The theory is also intimately related to many branches of mathematics, 

including matrix theory, numerical analysis, probability, topology and combinatory. 

Over the last 50 year graph theory has evolved into an important mathematical tool in 

the solution of a wide variety of problems in many areas of society. A graph labeling is 

an assignment of integers to the vertices or edges or both, subject to certain conditions 

have been motivated by practical problems, labelled graphs serve useful mathematical 

models for a broad range of applications such as: coding theory, including the design of 

good types codes, synch-set codes, missile guidance codes and convolutional codes with 

optimal auto correlation properties.  The concept of graph labeling was introduced by 

Rosa in 1967 [6]. A useful survey on graph labeling by J.A. Gallian (2014) can be found 

in [2]. Somasundaram and Ponraj [4] have introduced the notion of mean labeling of 

graphs. A directed graph or digraph 𝐷 consists of a finite set 𝑉 of vertices and a 
collection of ordered pairs of distinct vertices. Any such pair (𝑢, 𝑣) is called an arc or 
directed line and will usually be denoted by𝑢𝑣⃗⃗⃗⃗ . A digraph 𝐷 with 𝑝 vertices and 𝑞 arcs 
is denoted by𝐷 (𝑝, 𝑞). The indegree 𝑑−(𝑣) of a vertex 𝑣 in a digraph 𝐷 is the number of 
arcs having 𝑣 as its terminal vertex. The outdegree 𝑑+(𝑣) of 𝑣 is the number of arcs 
having 𝑣 as its initial vertex [1].  

K. Palani et al. introduced the concepts of mean and near mean digraphs in [3]. In 

this paper, different di-cyclic snakes are introduced and the existence of near mean 

labeling is investigated. 

  

2. Preliminaries 
The following definition and theorem are basics which are needed for the subsequent 

section. 
Definition 2.1: [3] Let 𝑓: 𝑉 → {0, 1, 2, … , 𝑞} be a 1-1 map. Define  𝑓∗: 𝐴 → {1, 2, … , 𝑞} 

by𝑓∗(𝑒 = 𝑢𝑣⃗⃗⃗⃗ ) = ⌈
𝑓(𝑢)+𝑓(𝑣)

2
⌉. Let𝑓∗(𝑣) = |∑ 𝑓∗(𝑣𝑤⃗⃗⃗⃗  ⃗)𝑤∈𝑉 − ∑ 𝑓

∗(𝑤𝑣⃗⃗⃗⃗  ⃗)𝑤∈𝑉 |. If𝑓
∗(𝑣) ≤

2  ∀ 𝑣 ∈ 𝐴(𝐷), then 𝑓 is said to be a near mean labeling of D and 𝐷 is said to be a near 
mean digraph. 

 

Definition 2.2: [5] A cyclic snake 𝑚𝐶𝑛 is obtained by replacing every edge of 𝑃𝑚 by𝐶𝑛. 

Theorem 2.3: [3] The directed cycle 𝐶𝑛⃗⃗⃗⃗  is a near mean digraph. 

 

3. Main Results 

In this section, the different dicyclic snakes are defined and the near mean labeling 

existence is verified. 

239



Near mean labeling in dicyclic snakes 

 

 

Definition 3.1: In cyclic snake𝑚𝐶𝑛, orient the edges of each cycle clockwise. The 

resulting graph is called Di-Cyclic Snake and it is denoted as𝑚𝐶𝑛⃗⃗⃗⃗ . For𝑛 = 3, 4, 5, Di-

Cyclic Snakes are called Di-Triangular Snake 𝑇𝑆𝑛⃗⃗⃗⃗⃗⃗  ⃗, Di-Quadrilateral Snake 𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗  ⃗ and Di-

Pentagonal Snake 𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗  ⃗ respectively. 
 

Definition 3.2: In 𝑚𝐶𝑛 when 𝑛 is even, orient the edges of the cycle alternately and call 

resulting graph as Alternating Di-Cyclic Snake. Denote it as𝐴𝑚𝐶𝑛⃗⃗⃗⃗ . 
 

Theorem 3.3: Di-Quadrilateral Snake 𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗  ⃗  admits near mean labeling. 

Proof:  Let 𝑉(𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗  ⃗) = {𝑢𝑖|1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑤𝑖|1 ≤ 𝑖 ≤ 𝑛 − 1} be 

the vertex set and let  𝐴(𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗  ⃗) = {{𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗  ⃗} ∪ {𝑣𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗ } ∪ {𝑢𝑖+1𝑤𝑖⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗  ⃗} ∪ {𝑤𝑖𝑢𝑖⃗⃗⃗⃗⃗⃗⃗⃗ }|1 ≤ 𝑖 ≤ 𝑛 −

1} be the arc set. 

Define 𝑓: 𝑉(𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗  ⃗) → {0, 1, 2, … , (4𝑛 − 4)} by  
𝑓(𝑢𝑖) = 4(𝑖 − 1)   for    1 ≤ 𝑖 ≤ 𝑛 
𝑓(𝑣𝑖) = 4𝑖 − 2       for   1 ≤ 𝑖 ≤ 𝑛 − 1 

 
Figure 3.1: The labeling of a Di-Quadrilateral snake 

For 𝑖 = 1 to 𝑛 − 1 

𝑓∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗  ⃗) = ⌈
𝑓(𝑢𝑖)+𝑓(𝑣𝑖)

2
⌉ = ⌈

[4(𝑖−1)]+[4𝑖−2]

2
⌉ = ⌈

8𝑖−6

2
⌉ = 4𝑖 − 3.                                (3.3.1) 

𝑓∗(𝑣𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗ ) = ⌈
𝑓(𝑣𝑖)+𝑓(𝑢𝑖+1)

2
⌉ = ⌈

[4𝑖−2]+[4(𝑖+1−1)]

2
⌉ = ⌈

8𝑖−2

2
⌉ = 4𝑖 − 1.                     (3.3.2) 

𝑓∗(𝑢𝑖+1𝑤𝑖⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗  ⃗) = ⌈
𝑓(𝑢𝑖+1)+𝑓(𝑤𝑖)

2
⌉ = ⌈

[4(𝑖+1−1)]+[4𝑖−1]

2
⌉  = ⌈

8𝑖−1

2
⌉ =

8𝑖

2
  = 4𝑖.                 (3.3.3) 

𝑓∗(𝑤𝑖𝑢𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) = ⌈
𝑓(𝑤𝑖)+𝑓(𝑢𝑖)

2
⌉ = ⌈

[4𝑖−1]+[4(𝑖−1)]

2
⌉ = ⌈

8𝑖−5

2
⌉ =

8𝑖−4

2
 = 4𝑖 − 2.                   (3.3.4) 

Now 𝑓∗(𝑢1) = |𝑓
∗(𝑢1𝑣1⃗⃗⃗⃗⃗⃗⃗⃗  ⃗) − 𝑓

∗(𝑤1𝑢1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ )| 
                     = |[4(1) − 3] − [4(1) − 2]|  [by (3.3.1) & (3.3.4)] 
                     = |1 − 2| = |−1| < 2    
Therefore, 𝑓∗(𝑢1) < 2                                                                                              (3.3.5) 
For 𝑖 = 2 to 𝑛 − 1 
𝑓∗(𝑢𝑖) = |[𝑓

∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗  ⃗) + 𝑓
∗(𝑢𝑖𝑤𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗)] − [𝑓

∗(𝑤𝑖𝑢𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) + 𝑓
∗(𝑣𝑖−1𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )]| 

            = |[4𝑖 − 3 + 4(𝑖 − 1)] − [4𝑖 − 2 + 4(𝑖 − 1) − 1]|   [by (1), (3), (4) & (2)] 
            = |8𝑖 − 7 − 8𝑖 + 7|  = |0| < 2     
Therefore, 𝑓∗(𝑢𝑖) < 2    for 𝑖 = 2 to 𝑛 − 1                                                             (3.3.6)      
𝑓∗(𝑢𝑛) = |𝑓

∗(𝑢𝑛𝑤𝑛−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) − 𝑓
∗(𝑣𝑛−1𝑢𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| 

            = |4(𝑛 − 1) − [4(𝑛 − 1) − 1|   [by (3.3.3) & (3.3.2)] 

240



Palani K and Shunmugapriya A 

 

 

            = |1| < 2    
Therefore, 𝑓∗(𝑢𝑛) < 2                                                                                              (3.3.7) 
For 𝑖 = 1 to 𝑛 − 1 
𝑓∗(𝑣𝑖) = |𝑓

∗(𝑣𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗ ) − 𝑓
∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗  ⃗)| 

            = |(4𝑖 − 1) − (4𝑖 − 3)|         [by (3.3.2) & (3.3.1)] 
            = |4𝑖 − 1 − 4𝑖 + 3|  = |2| = 2     
Therefore, 𝑓∗(𝑣𝑖) = 2 for  𝑖 = 1 to 𝑛 − 1                                                                (3.3.8) 
For 𝑖 = 1 to 𝑛 − 1 
𝑓∗(𝑤𝑖) = |𝑓

∗(𝑤𝑖𝑢𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) − 𝑓
∗(𝑢𝑖+1𝑤𝑖⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗  ⃗)| 

            = |(4𝑖 − 2) − (4𝑖)|         [by (3.3.4) & (3.3.3)] 
            = |−2| ≤ 2     
Therefore,𝑓∗(𝑤𝑖) ≤ 2 for  𝑖 = 1 to 𝑛 − 1                                                                (3.3.9) 

From equations (5) to (9),  𝑓∗(𝑢) ≤ 2 ∀ 𝑢 ∈ 𝑉(𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗  ⃗) 

Hence Di-Quadrilateral Snake 𝑄𝑆𝑛⃗⃗ ⃗⃗ ⃗⃗  ⃗  is a near mean digraph. 
 

Theorem 3.4. Di-Pentagonal Snake 𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗  ⃗ is a near mean digraph. 
Proof:  

Let 𝑉(𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗  ⃗) = {𝑢𝑖|1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑤𝑖|1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑥𝑖|1 ≤ 𝑖 ≤

𝑛 − 1} be the vertex set and let  𝐴(𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗  ⃗) = {{𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗  ⃗} ∪ {𝑣𝑖𝑤𝑖⃗⃗⃗⃗⃗⃗⃗⃗ } ∪ {𝑤𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗  ⃗} ∪ {𝑢𝑖+1𝑥𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ } ∪

{𝑥𝑖𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗  ⃗}|1 ≤ 𝑖 ≤ 𝑛 − 1} be the arc set. 

Define 𝑓: 𝑉(𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗  ⃗) → {0, 1, 2, … , (5𝑛 − 5)} by  
𝑓(𝑢𝑖) = 5(𝑖 − 1)  for    1 ≤ 𝑖 ≤ 𝑛 
𝑓(𝑣𝑖) = 5𝑖 − 3     for    1 ≤ 𝑖 ≤ 𝑛 − 1 
𝑓(𝑤𝑖) = 5𝑖 − 1    for    1 ≤ 𝑖 ≤ 𝑛 − 1 
𝑓(𝑥𝑖) = 5𝑖 − 2     for   1 ≤ 𝑖 ≤ 𝑛 − 1 
 

 
Figure 3.2. The labeling of a Di-Pentagonal snake 

For 𝑖 = 1 to 𝑛 − 1 

𝑓∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗  ⃗) = ⌈
𝑓(𝑢𝑖)+𝑓(𝑣𝑖)

2
⌉ = ⌈

[5(𝑖−1)]+[5𝑖−3]

2
⌉ = ⌈

10𝑖−8

2
⌉ = 5𝑖 − 4                            (3.4.1) 

𝑓∗(𝑣𝑖𝑤𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) = ⌈
𝑓(𝑣𝑖)+𝑓(𝑤𝑖)

2
⌉ = ⌈

[5𝑖−3]+[5𝑖−1]

2
⌉ = ⌈

10𝑖−4

2
⌉ = 5𝑖 − 2                              (3.4.2) 

𝑓∗(𝑤𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗  ⃗) = ⌈
𝑓(𝑤𝑖)+𝑓(𝑢𝑖+1)

2
⌉ = ⌈

[5𝑖−1]+[5(𝑖+1−1)]

2
⌉ = ⌈

10𝑖−1

2
⌉ =

10𝑖

2
= 5𝑖              (3.4.3) 

𝑓∗(𝑢𝑖+1𝑥𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) = ⌈
𝑓(𝑢𝑖+1)+𝑓(𝑥𝑖)

2
⌉ = ⌈

[5(𝑖+1−1)]+[5𝑖−2]

2
⌉ = ⌈

10𝑖−2

2
⌉ = 5𝑖 − 1                   (3.4.4) 

241



Near mean labeling in dicyclic snakes 

 

 

𝑓∗(𝑥𝑖𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗  ⃗) = ⌈
𝑓(𝑥𝑖)+𝑓(𝑢𝑖)

2
⌉ = ⌈

[5𝑖−2]+[5(𝑖−1)]

2
⌉ = ⌈

10𝑖−7

2
⌉ =

10𝑖−6

2
= 5𝑖 − 3             (3.4.5) 

Next to find 𝑓∗(𝑢𝑖) 
Now 𝑓∗(𝑢1) = |𝑓

∗(𝑢1𝑣1⃗⃗⃗⃗⃗⃗⃗⃗  ⃗) − 𝑓
∗(𝑥1𝑢1⃗⃗⃗⃗⃗⃗⃗⃗  ⃗)| 

                     = |[5(1) − 4] − [5(1) − 3]|  [by (3.4.1) & (3.4.5)] 
                     = |1 − 2|  = |−1| < 2    
Therefore, 𝑓∗(𝑢1) < 2                                                                                          (3.4.6) 
For 𝑖 = 2 to 𝑛 − 1 
𝑓∗(𝑢𝑖) = |[𝑓

∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗  ⃗) + 𝑓
∗(𝑢𝑖𝑥𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )] − [𝑓

∗(𝑥𝑖𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗  ⃗) + 𝑓
∗(𝑤𝑖−1𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗)]| 

            = |[5𝑖 − 4 + 5(𝑖 − 1) − 1] − [5𝑖 − 3 + 5(𝑖 − 1)]| [by (3.4.1), (3.4.4), (3.4.5) 
& (3.4.3)] 

            = |10𝑖 − 10 − 10𝑖 + 8| = |−2| ≤ 2     
Therefore, 𝑓∗(𝑢𝑖) ≤ 2  for 𝑖 = 2 to 𝑛 − 1                                                            (3.4.7)      
𝑓∗(𝑢𝑛) = |𝑓

∗(𝑢𝑛𝑥𝑛−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) − 𝑓
∗(𝑤𝑛−1𝑢𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗)| 

            = |5(𝑛 − 1) − 1 − 5(𝑛 − 1)|   [by (3.4.4) & (3.4.3)] 
            = |−1| < 2     
Therefore, 𝑓∗(𝑢𝑛) < 2                                                                                          (3.4.8) 
For 𝑖 = 1 to 𝑛 − 1 
𝑓∗(𝑣𝑖) = |𝑓

∗(𝑣𝑖𝑤𝑖⃗⃗⃗⃗⃗⃗⃗⃗ ) − 𝑓
∗(𝑢𝑖𝑣𝑖⃗⃗⃗⃗⃗⃗  ⃗)| 

            = |(5𝑖 − 2) − (5𝑖 − 4)|      [by (3.4.2) & (3.4.1)] 
           = |2| = 2     
Therefore, 𝑓∗(𝑣𝑖) = 2 for  𝑖 = 1 to 𝑛 − 1                                                            (3.4.9) 
For 𝑖 = 1 to 𝑛 − 1 
𝑓∗(𝑤𝑖) = |𝑓

∗(𝑤𝑖𝑢𝑖+1⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗  ⃗) − 𝑓
∗(𝑣𝑖𝑤𝑖⃗⃗⃗⃗⃗⃗⃗⃗ )| 

            = |(5𝑖) − (5𝑖 − 2)|         [by (3.4.3) & (3.4.2)] 
            = |2| = 2    
Therefore, 𝑓∗(𝑤𝑖) = 2 for  𝑖 = 1 to 𝑛 − 1                                                            (3.4.10) 
For 𝑖 = 1 to 𝑛 − 1 
𝑓∗(𝑥𝑖) = |𝑓

∗(𝑥𝑖𝑢𝑖⃗⃗ ⃗⃗ ⃗⃗  ⃗) − 𝑓
∗(𝑢𝑖+1𝑥𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| 

            = |(5𝑖 − 3) − (5𝑖 − 1)|         [by (3.4.3) & (3.4.2)] 
            = |−2| ≤ 2    
Therefore, 𝑓∗(𝑥𝑖) ≤ 2  for   𝑖 = 1 to 𝑛 − 1                                                           (3.4.11) 

From equations (6) to (11),  𝑓∗(𝑢) ≤ 2 ∀ 𝑢 ∈ 𝑉(𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗  ⃗) 

Hence, Di-Pentagonal Snake 𝑃𝑆𝑛⃗⃗⃗⃗⃗⃗  ⃗  is a near mean digraph. 
 

Theorem 3.5. Di-Cyclic snake 𝑚𝐶𝑛⃗⃗⃗⃗  is a near mean digraph for 𝑚 ≥ 1 and 𝑛 ≥ 3. 

Proof: Let 𝑢𝑖𝑗 denote the 𝑗th vertex in the 𝑖th copy of 𝐶𝑛⃗⃗⃗⃗  

Here, 𝑉(𝑚𝐶𝑛⃗⃗⃗⃗ ) = {𝑢𝑖𝑗|1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛} and  

𝐴(𝑚𝐶𝑛⃗⃗⃗⃗ ) = {𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛 − 1} ∪ {𝑢𝑖𝑛𝑢𝑖1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤ 𝑚}  

Following the procedure of near mean labeling of 𝐶𝑛⃗⃗⃗⃗  in [3], define 

𝑓: 𝑉(𝑚𝐶𝑛⃗⃗⃗⃗ ) → {0,1,2, … , 𝑚𝑛} as below 

242



Palani K and Shunmugapriya A 

 

 

 𝑓(𝑢𝑖𝑗) = {
(𝑖 − 1)𝑛 + 2(𝑗 − 1)                       for    1 ≤ 𝑗 ≤ ⌊

𝑛

2
⌋ + 1, 1 ≤ 𝑖 ≤ 𝑚

(𝑖 − 1)𝑛 + 2(𝑛 − (𝑗 − 1)) + 1    for    ⌊
𝑛

2
⌋ + 2 ≤ 𝑗 ≤ 𝑛, 1 ≤ 𝑖 ≤ 𝑚

 

where |𝑉(𝑚𝐶𝑛⃗⃗⃗⃗ )| = 𝑚𝑛 − 𝑚 + 1 and |𝐴(𝑚𝐶𝑛⃗⃗⃗⃗ )| = 𝑚𝑛 
 

 
Figure 3.3: The labeling of the Di-cyclic Snake 𝑚𝐶𝑛⃗⃗⃗⃗  when 𝑛 is even 

 

 
Figure 3.4: The labeling of the Di-cyclic Snake 𝑚𝐶𝑛⃗⃗⃗⃗  when 𝑛 is odd 

 

Now to find 𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) 

For  𝑖 = 1 to 𝑚, 𝑗 = 1 to ⌊
𝑛

2
⌋ 

𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) =  ⌈
𝑓(𝑢𝑖𝑗) + 𝑓(𝑢𝑖(𝑗+1))

2
⌉ 

  = ⌈
[(𝑖−1)𝑛+2(𝑗−1)]+[(𝑖−1)𝑛+2((𝑗+1)−1)]

2
⌉ 

  = ⌈
2(𝑖−1)𝑛+4𝑗−2

2
⌉ = (𝑖 − 1)𝑛 + 2𝑗 − 1 = 𝑛𝑖 + 2𝑗 − 𝑛 − 1. 

Therefore, 𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) = 𝑛𝑖 + 2𝑗 − 𝑛 − 1  for  𝑖 = 1 to 𝑚, 𝑗 = 1 to ⌊
𝑛

2
⌋       (3.5.1) 

243



Near mean labeling in dicyclic snakes 

 

 

𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

𝑢
𝑖(⌊

𝑛

2
⌋+2)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) = ⌈
𝑓 (𝑢

𝑖(⌊
𝑛

2
⌋+1)

) + 𝑓 (𝑢
𝑖(⌊

𝑛

2
⌋+2)

)

2
⌉ 

         = ⌈
[(𝑖−1)𝑛+2(⌊

𝑛

2
⌋+1−1)]+[(𝑖−1)𝑛+2(𝑛−(⌊

𝑛

2
⌋+2−1))+1]

2
⌉ 

         = ⌈
2(𝑖−1)𝑛+2𝑛−1

2
⌉ = ⌈

2(𝑖−1+1)𝑛−1

2
⌉ = ⌈

2𝑖𝑛−1

2
⌉ =

2𝑖𝑛

2
= 𝑛𝑖  

Therefore, 𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

𝑢
𝑖(⌊

𝑛

2
⌋+2)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) = 𝑛𝑖   for 𝑖 = 1 to 𝑚 and 𝑛 ≥ 3                           (3.5.2) 

For 𝑖 = 1 to 𝑚, 𝑗 = ⌊
𝑛

2
⌋ + 2 to 𝑛 − 1  

𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) =  ⌈
𝑓(𝑢𝑖𝑗) + 𝑓(𝑢𝑖(𝑗+1))

2
⌉ 

  = ⌈
[(𝑖−1)𝑛+2(𝑛−(𝑗−1))+1]+[(𝑖−1)𝑛+2(𝑛−((𝑗+1)−1))+1]

2
⌉ 

  = ⌈
2(𝑖−1)𝑛+4𝑛−4𝑗+4

2
⌉ 

  = (𝑖 + 1)𝑛 − 2𝑗 + 2 = 𝑛𝑖 − 2𝑗 + 𝑛 + 2     

𝑓∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) = 𝑛𝑖 − 2𝑗 + 𝑛 + 2  for 𝑖 = 1 to 𝑚, 𝑗 = ⌊
𝑛

2
⌋ + 2 to 𝑛 − 1           (3.5.3) 

𝑓∗(𝑢𝑖𝑛𝑢𝑖1⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) = ⌈
𝑓(𝑢𝑖𝑛) + 𝑓(𝑢𝑖1)

2
⌉ 

   = ⌈
[(𝑖−1)𝑛+2(𝑛−(𝑛−1))+1]+[(𝑖−1)𝑛+2(1−1)]

2
⌉ 

               = ⌈
2(𝑖−1)𝑛+2(𝑛−𝑛+1)+1

2
⌉ = ⌈

2(𝑖−1)𝑛+3

2
⌉ =

2(𝑖−1)𝑛+4

2
 

               = (𝑖 − 1)𝑛 + 2 = 𝑛𝑖 − 𝑛 + 2.    
Therefore, 𝑓∗(𝑢𝑖𝑛𝑢𝑖1⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) = 𝑛𝑖 − 𝑛 + 2   for 𝑖 = 1 to 𝑚 and 𝑛 ≥ 3                            (3.5.4) 

Next to find 𝑓∗(𝑢𝑖𝑗) 

𝑓∗(𝑢11) = |𝑓
∗(𝑢11𝑢12⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) − 𝑓

∗(𝑢1𝑛𝑢11⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗)| 
   =  |[𝑛(1) + 2(1) − 𝑛 − 1] − [𝑛(1) − 𝑛 + 2]|    [by (3.5.1) & (3.5.4)] 
   =  |1 − 2|  =  |−1| < 2   
Therefore, 𝑓∗(𝑢11) < 2                                                                                        (3.5.5) 

For 𝑖 = 1 to 𝑚,   𝑗 = 2 to ⌊
𝑛

2
⌋ 

𝑓∗(𝑢𝑖𝑗) = |𝑓
∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) − 𝑓

∗(𝑢𝑖(𝑗−1)𝑢𝑖𝑗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗)| 

   =  |[𝑛𝑖 + 2𝑗 − 𝑛 − 1] − [𝑛𝑖 + 2(𝑗 − 1) − 𝑛 − 1]|  [by (3.5.1)] 
              = |−1 + 3| = |2| = 2 

Therefore, 𝑓∗(𝑢𝑖𝑗) = 2  for 𝑖 = 1 to 𝑚,   𝑗 = 2 to ⌊
𝑛

2
⌋                                        (3.5.6) 

For 𝑖 = 1 to 𝑚 − 1,  two cases for 𝑓∗(𝑢𝑖𝑗) when 𝑗 = ⌊
𝑛

2
⌋ + 1 & ⌊

𝑛

2
⌋ + 2 

Case (i): 𝑛  is odd. 

𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

) = |𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

𝑢
𝑖(⌊

𝑛

2
⌋+2)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) − 𝑓∗ (𝑢
𝑖⌊

𝑛

2
⌋
𝑢

𝑖(⌊
𝑛

2
⌋+1)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| 

  =  |𝑛𝑖 − (𝑛𝑖 + 2 ⌊
𝑛

2
⌋ − 𝑛 − 1)|    [by (3.5.2) & (3.5.1)] 

244



Palani K and Shunmugapriya A 

 

 

  = |𝑛 − 2 ⌊
𝑛

2
⌋ + 1| 

  = |𝑛 − 2 (
𝑛−1

2
) + 1| = |2| = 2                                                                                

𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

) = 2 when 𝑛  is odd                                                                        (3.5.7)                 

𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+2)

) = |(𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+2)

𝑢
𝑖(⌊

𝑛

2
⌋+3)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) + 𝑓∗(𝑢(𝑖+1)1𝑢(𝑖+1)2⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗))

− (𝑓∗ (𝑢
𝑖⌊

𝑛

2
⌋+1

𝑢
𝑖(⌊

𝑛

2
⌋+2)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) + 𝑓∗(𝑢(𝑖+1)𝑛𝑢(𝑖+1)1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ))| 

= |[(𝑛𝑖 − 2 (⌊
𝑛

2
⌋ + 2) + 𝑛 + 2) + 𝑛(𝑖 + 1) + 2 − 𝑛 − 1] − [𝑛𝑖 + 𝑛(𝑖 + 1) − 𝑛 + 2]|                      

                     [by (3.5.3), (3.5.1), (3.5.2) & (3.5.4)] 

= |[(𝑛𝑖 − 2 (⌊
𝑛

2
⌋) − 4 + 𝑛 + 2) + (𝑛𝑖 + 𝑛 + 2 − 𝑛 − 1)] − [𝑛𝑖 + 𝑛𝑖 + 2]| 

 = |[2𝑛𝑖 − 2 (
𝑛−1

2
) + 𝑛 − 1] − [2𝑛𝑖 + 2]| 

= |−2| ≤ 2                                                                                        

𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+2)

) ≤ 2 when 𝑛  is odd.                                                                       (3.5.8) 

Case (ii): 𝑛  is even. 

𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

) = |[𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

𝑢
𝑖(⌊

𝑛

2
⌋+2)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) + 𝑓∗(𝑢(𝑖+1)1𝑢(𝑖+1)2⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗)]   

− [𝑓∗ (𝑢
𝑖⌊

𝑛

2
⌋
𝑢

𝑖(⌊
𝑛

2
⌋+1)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) + 𝑓∗(𝑢(𝑖+1)𝑛𝑢(𝑖+1)1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )]| 

= |[𝑛𝑖 + (𝑛(𝑖 + 1) + 2(1) − 𝑛 − 1)] − [(𝑛𝑖 + 2 ⌊
𝑛

2
⌋ − 𝑛 − 1) + (𝑛(𝑖 + 1) − 𝑛 + 2)]|                      

                        [by (3.5.2), (3.5.1) & (3.5.4) ] 

  = |[𝑛𝑖 + 𝑛𝑖 + 𝑛 + 2 − 𝑛 − 1] − [𝑛𝑖 + 2 (
𝑛

2
) − 𝑛 − 1 + 𝑛𝑖 + 2]| 

   = |0| < 2                                                                                                 

𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

) < 2 when 𝑛  is even                                                                       (3.5.9) 

𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+2)

) = |𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+2)

𝑢
𝑖(⌊

𝑛

2
⌋+3)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) − 𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

𝑢
𝑖(⌊

𝑛

2
⌋+2)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| 

  =  |(𝑛𝑖 − 2 (⌊
𝑛

2
⌋ + 2) + 𝑛 + 2) − 𝑛𝑖|    [by (3) & (4)] 

  = |𝑛 − 2 ⌊
𝑛

2
⌋ − 4 + 2| 

  = |𝑛 − 2 (
𝑛

2
) − 2| 

  = |−2| ≤ 2        

𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+2)

) ≤ 2 when 𝑛  is even                                                                  (3.5.10) 

Cases (i) and (ii) imply 𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

) ≤ 2  for 𝑖 = 1 to 𝑚 − 1                      (3.5.11)  

and 𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+2)

) ≤ 2 for 𝑖 = 1 to 𝑚 − 1                                                      (3.5.12)   

245



Near mean labeling in dicyclic snakes 

 

 

Now to find 𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+1)

) and 𝑓∗ (𝑢
𝑖(⌊

𝑛

2
⌋+2)

) for 𝑖 = 𝑚 

𝑓∗ (𝑢
𝑚(⌊

𝑛

2
⌋+1)

) = |𝑓∗ (𝑢
𝑚(⌊

𝑛

2
⌋+1)

𝑢
𝑚(⌊

𝑛

2
⌋+2)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) − 𝑓∗ (𝑢
𝑚⌊

𝑛

2
⌋
𝑢

𝑚(⌊
𝑛

2
⌋+1)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗)| 

               =  |𝑛𝑚 − (𝑛𝑚 + 2 ⌊
𝑛

2
⌋ − 𝑛 − 1)|    [by (2) & (1)] 

 = |𝑛 − 2 ⌊
𝑛

2
⌋ + 1| 

 = {
|𝑛 − 2 (

𝑛−1

2
) + 1|  if 𝑛 is odd

|𝑛 − 2 (
𝑛

2
) + 1|   if 𝑛 is even

 

= {
|2|   if 𝑛 is odd
|1|  if 𝑛 is even

 

Therefore, 𝑓∗ (𝑢
𝑚(⌊

𝑛

2
⌋+1)

) ≤ 2                                                                           (3.5.13) 

𝑓∗ (𝑢
𝑚(⌊

𝑛

2
⌋+2)

) = |𝑓∗ (𝑢
𝑚(⌊

𝑛

2
⌋+2)

𝑢
𝑚(⌊

𝑛

2
⌋+3)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) − 𝑓∗ (𝑢
𝑚(⌊

𝑛

2
⌋+1)

𝑢
𝑚(⌊

𝑛

2
⌋+2)

⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗)| 

                           = |(𝑛𝑚 − 2 (⌊
𝑛

2
⌋ + 2) + 𝑛 + 2) − 𝑛𝑚|    [by (3.5.3) & (3.5.2)] 

   = |𝑛 − 2 ⌊
𝑛

2
⌋ − 2| 

   = {
|𝑛 − 2 (

𝑛−1

2
) − 2|  if 𝑛 is odd

|𝑛 − 2 (
𝑛

2
) − 2|   if 𝑛 is even

 

= {
|−1|   if 𝑛 is odd
|−2|  if 𝑛 is even

 

Therefore, 𝑓∗ (𝑢
𝑚(⌊

𝑛

2
⌋+2)

) ≤ 2                                                                             (3.5.14) 

For 𝑖 = 1 to 𝑚, 𝑗 = ⌊
𝑛

2
⌋ + 3 to 𝑛 − 1 

𝑓∗(𝑢𝑖𝑗) = |𝑓
∗(𝑢𝑖𝑗𝑢𝑖(𝑗+1)⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗) − 𝑓

∗(𝑢𝑖(𝑗−1)𝑢𝑖𝑗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗)| 

   =  |[𝑛𝑖 − 2𝑗 + 𝑛 + 2] − [𝑛𝑖 − 2(𝑗 − 1) + 𝑛 + 2]|  [by (3)] 
              = |−2| 

Therefore, 𝑓∗(𝑢𝑖𝑗) ≤ 2  for 𝑖 = 1 to 𝑚, 𝑗 = ⌊
𝑛

2
⌋ + 3 to 𝑛 − 1                            (3.5.15) 

For 𝑖 = 1 to 𝑚 and 𝑗 = 𝑛 

𝑓∗(𝑢𝑖𝑛) = |𝑓
∗(𝑢𝑖𝑛𝑢𝑖1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ) − 𝑓

∗(𝑢𝑖(𝑛−1)𝑢𝑖𝑛⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ )| 

   =  |[𝑛𝑖 − 𝑛 + 2] − [𝑛𝑖 − 2(𝑛 − 1) + 𝑛 + 2]|  [by (4)] 
               = |𝑛𝑖 − 𝑛 + 2 − 𝑛𝑖 + 2𝑛 − 2 − 𝑛 − 2| = |−2| 
Therefore, 𝑓∗(𝑢𝑖𝑛) ≤ 2                                                                                          (3.5.16) 

Equations (3.5.5), (3.5.6) and (3.5.11) to (3.5.16) imply 𝑓∗(𝑢𝑖𝑗) ≤ 2 for1 ≤ 𝑖 ≤ 𝑚, 1 ≤

𝑗 ≤ 𝑛. 

Thus, Di-Cyclic snake 𝑚𝐶𝑛⃗⃗⃗⃗  is a near mean digraph for 𝑚 ≥ 1 and𝑛 ≥ 3. 
 

Theorem 3.6.  Alternating Di-Cyclic snakes are non near mean digraphs. 

246



Palani K and Shunmugapriya A 

 

 

Proof: In an alternating Di-Cyclic snake, either 𝑑+(𝑢21) = 0 and 𝑑
−(𝑢21) = 4 (or)  

𝑑+(𝑢21) = 4  and𝑑
−(𝑢21) = 0. 

Therefore, Corresponding to every𝑓: 𝑉 → {0, 1, 2, … , 𝑞},  ∑ 𝑓∗(𝑢21𝑤⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ )𝑤∈𝑉 = 0 and 
 ∑ 𝑓∗(𝑤𝑢21⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ )𝑤∈𝑉  is a sum of at least three positive integers (or) ∑ 𝑓

∗(𝑤𝑢21⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗ )𝑤∈𝑉 = 0 and 
∑ 𝑓∗(𝑢21𝑤⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ )𝑤∈𝑉  is a sum of at least three positive integers. 
Therefore, 𝑓∗(𝑢21) > 2.  
Therefore, No function 𝑓: 𝑉 → {0, 1, 2, … , 𝑞} is a near mean labeling. 
Thus, an alternating di-cyclic snake is a non near mean digraph. 

 

4. Conclusions 

In this article, different dicyclic snakes are introduced. Also, existence of near mean 

labeling is verified to dicyclic snakes and its generalisation. Most of the labeling are 

proved only for graphs. In this way, we develop the concept of labeling into digraphs 

 

References 
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Xidian University, 14(3): 1298-1307, 2020. 

[4] Ponraj R and Somasundaram S, Mean labeling of graphs, National Academy of         

Science Letters, 26: 210-213, 2003. 

[5] Raval K K and Prajapati U M, Vertex even and odd mean labeling in the context of      

some cyclic snake graphs, Journal of Emerging Technologies and Innovative     

Research (JETIR), 4(6), 2017.  

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