Ratio Mathematica                                                                            Volume 45, 2023 

 

              
 

 

Gaussian twin Neighborhood Prime Labeling 

on Fan Digraphs 

 
 

Palani K * 

Shunmugapriya A † 

 
 

Abstract  

Gaussian integers are complex numbers of the form 𝛾 = 𝑥 + 𝑖𝑦 where 𝑥 and 𝑦 are 
integers and𝑖2 = −1. The set of Gaussian integers is usually denoted by ℤ[𝑖]. A 
Gaussian integer 𝛾 = 𝑎 + 𝑖𝑏 ∈ ℤ[𝑖] is prime if and only if either𝛾 = ±(1 ± 𝑖), 𝑁(𝛾) =
𝑎2 + 𝑏2 is a prime integer congruent to 1 (mod 4), or 𝛾 = 𝑝 + 0𝑖 or = 0 + 𝑝𝑖 where 𝑝 
is a prime integer and |𝑝| ≡ 3(mod 4). Let 𝐷 = (𝑉,𝐴) be a digraph with |𝑉| = 𝑛. An 
injective function 𝑓: 𝑉(𝐷) → [𝛾𝑛] is said to be a Gaussian twin neighborhood prime 
labeling of 𝐷, if it is both Gaussian in and out neighborhood prime labeling. A digraph 
which admits a Gaussian twin neighborhood prime labeling is called a Gaussian twin 

neighborhood prime digraph. In this paper, we introduce some definitions of fan 

digraphs. Further, we establish the Gaussian twin neighborhood prime labeling in fan 

digraphs using Gaussian integers. 

 

Keywords: Gaussian integers, neighborhood prime, labeling, digraphs.  

 

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 24, 2022. Accepted on September 15, 2022. Published on January 30, 2023. DOI: 

10.23755/rm.v45i0.973. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY licence agreement. 

 

9

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


Palani K and Shunmugapriya A 

 

 

1. Introduction  
Graph labeling is an assignment of integers to the vertices or edges or both 

subject to certain conditions. The concept of graph labeling was introduced by Rosa in 

[1].  An useful survey on graph labeling by J.A. Gallian can be found in [2]. Spiral 

ordering of the Gaussian integer was first introduced by Hunter Lehmann and Andrew 

Park in [4]. T. J. Rajesh Kumar and T. K. Mathew Varkey [7] introduced the concept of 

Gaussian neighborhood prime labeling of a graph. K. Palani [6] et.al, introduced the 

concept of Gaussian twin neighborhood prime labeling in digraphs.  

Let 𝐷 = (𝑉,𝐴) be a digraph of order𝑛. Then 𝑉 is the set of vertices of 𝐷 
with|𝑉| = 𝑛, and A is the set of arcs of 𝐷 consisting of ordered pairs of distinct vertices. 
The in-degree 𝑑−(𝑣) of a vertex 𝑣 in a digraph 𝐷 is the number of arcs having 𝑣 as its 
terminal vertex. The out-degree 𝑑+(𝑣) of 𝑣 is the number of arcs having 𝑣 as its initial 
vertex [2]. Throughout this article we use only digraphs. 

In this article, we introduce the definition of fan and double fan digraphs by 

orienting fan and double fan graphs. Also, we investigate the existence of Gaussian twin 

neighborhood prime labeling in fan digraphs.  

 

2. Preliminaries 
The following basic definitions and properties are from [4]  

Definition 2.1. Gaussian integers are complex numbers of the form 𝛾 = 𝑥 + 𝑖𝑦 where 𝑥 
and 𝑦 are integers and𝑖2 = −1. The set of Gaussian integers is usually denoted byℤ[𝑖]. 

A Gaussian integer is even if 1 + 𝑖 divides𝛾. Otherwise it is an odd Gaussian 
integer. 

 

Definition 2.2.   A Gaussian integer 𝛾 = 𝑎 + 𝑖𝑏 ∈ ℤ[𝑖] is prime if and only if either  
(i) 𝛾 = ±(1 ± 𝑖) 
(ii) 𝑁(𝛾) = 𝑎2 + 𝑏2 is a prime integer congruent to 1 (mod 4), or 
(iii) 𝛾 = 𝑝 + 0𝑖 Or = 0 + 𝑝𝑖 where 𝑝 is a prime integer and|𝑝| ≡ 3(mod 4). 
 

Definition 2.3. The spiral ordering of the Gaussian integers is recursively defined 

ordering of the Gaussian integers. We denote the 𝑛th Gaussian integer in the spiral 
ordering by 𝛾𝑛. The ordering is defined beginning with 𝛾1 = 1 and continuing as: 

 𝛾𝑛+1 =

{
  
 

  
 
𝛾𝑛 + 𝑖                                       if Re(𝛾𝑛) ≡ 1 (mod 2),Re(𝛾𝑛) > Im(𝛾𝑛) + 1 

𝛾𝑛 − 1                if Im(𝛾𝑛) ≡ 0 (mod 2),Re(𝛾𝑛) ≤ Im(𝛾𝑛) + 1,Re(𝛾𝑛) > 1

𝛾𝑛 + 1                                       if Im(𝛾𝑛) ≡ 1 (mod 2),Re(𝛾𝑛) < Im(𝛾𝑛) + 1

𝛾𝑛 + 𝑖                                                           if Im(𝛾𝑛) ≡ 0 (mod 2),Re(𝛾𝑛) = 1

𝛾𝑛 − 𝑖                 if Re(𝛾𝑛) ≡ 0 (mod 2),Re(𝛾𝑛) ≥ Im(𝛾𝑛) + 1,Im(𝛾𝑛) > 0

𝛾𝑛 + 1                                                          if Re(𝛾𝑛) ≡ 0 (mod 2), Im(𝛾𝑛) = 0

 

and [𝛾𝑛] denotes the set of first 𝑛 Gaussian integers in the spiral ordering. 
 

Properties 2.4. 1. A Gaussian integer 𝛾 = 𝑥 + 𝑖𝑦 is called a prime Gaussian integer if 
its only divisors are ±1,±𝑖,±𝛾 or ±𝛾𝑖. 

10



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

2. Two Gaussian integers 𝑥 and 𝑦 are relatively prime if their only common divisors are 
the units inℤ[𝑖]. 
3. Let 𝛾 be a Gaussian integer and let 𝑢 be a unit. Then 𝛾 and 𝛾 + 𝑢 are relatively prime.  
4. In the spiral ordering, consecutive Gaussian integers are relatively prime. 

5. In the spiral ordering, consecutive odd Gaussian integers are relatively prime. 

6. Let 𝛼 be a prime Gaussian integer and 𝛾 be a Gaussian integer. Then 𝛾 and 𝛾 + 𝛼 are 
relatively prime if and only if𝛼 ∤ 𝛾. 
7. Let 𝛾 be an odd Gaussian integer, let 𝑡 be a positive integer and 𝑢 be a unit. Then 𝛾 
and 𝛾 + 𝑢(1 + 𝑖)𝑡 are relatively prime. 

The following definitions are taken from [6] 

 

Definition 2.5. Let 𝐷 = (𝑉,𝐴) be a digraph with |𝑉| = 𝑛. An injective function 
𝑓: 𝑉(𝐷) → [𝛾𝑛] is called Gaussian in-neighborhood prime labeling of 𝐷, if for every 
vertex 𝑣 ∈ 𝑉 (𝐷) with𝑑−(𝑣) >  1, the Gaussian integers in the set {𝑓(𝑢):𝑢 ∈ 𝑁−(𝑣)} 
are relatively prime where 𝑁−(𝑣) = {𝑢 ∈  𝑉 (𝐷) ∶ 𝑢𝑣 ⃗⃗⃗⃗⃗⃗ ∈ 𝐴(𝐷)}. 
 

Definition 2.6. Let 𝐷 = (𝑉,𝐴) be a digraph with|𝑉| = 𝑛. An injective function 
𝑓: 𝑉(𝐷) → [𝛾𝑛] is called Gaussian out-neighbourhood prime labeling of𝐷, if for 
every vertex 𝑣 ∈ 𝑉 (𝐷) with𝑑+(𝑣) >  1, the Gaussian integers in the set {𝑓(𝑢):𝑢 ∈
𝑁+(𝑣)} are relatively prime where𝑁+(𝑣) = {𝑢 ∈  𝑉 (𝐷) ∶ 𝑣𝑢 ⃗⃗⃗⃗⃗⃗ ∈ 𝐴(𝐷)}. 
 

Definition 2.7. Let 𝐷 = (𝑉,𝐴) be a digraph with|𝑉| = 𝑛. A function 𝑓: 𝑉(𝐷) → [𝛾𝑛] is 
said to be a Gaussian twin neighbourhood prime labeling of𝐷, if it is both Gaussian 
in and out neighborhood prime labeling. A digraph which admits Gaussian twin 

neighborhood prime labeling is called a Gaussian twin neighborhood prime digraph. 

 

Observation 2.8.  

1. If 𝐷 is a digraph such that 𝑁+(𝑣) or 𝑁−(𝑣) are either 𝜑 or singleton set, then 𝐷 
admits a Gaussian twin neighborhood prime labeling. 

2. A neighborhood prime digraph 𝐷 in which every vertex is such that either its in-
degree or out-degree at most 1 is Gaussian twin neighborhood prime. 

The following definitions are referred from [8]. 

 

Definition 2.9. Fan graph is defined as the graph𝑃𝑛 + 𝐾1, 𝑛 ≥ 2 where 𝐾1 is the empty 
graph on one vertex and 𝑃𝑛, a path graph on 𝑛 vertices. 

 

Definition 2.10. A double fan 𝐷𝐹𝑛 consists of two fan graphs with a common path. In 
other words𝐷𝐹𝑛 = 𝑃𝑛 + 𝐾2̅̅ ̅,𝑛 ≥ 2. 
 

3. Fan Digraphs 

In this section, some new digraphs are introduced by orienting fan graphs in 

different possible ways and named accordingly. Also we investigate the existence of the 

Gaussian twin neighborhood prime labeling of those digraphs. 

11



Palani K and Shunmugapriya A 

 

 

Definition 3.1. In a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 clockwise or 
anticlockwise and the spoke edges towards the central vertex. Call the resulting digraph 

as in-fan and denote it by𝑖𝐹𝑛⃗⃗  ⃗. 
 

Definition 3.2. In a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 clockwise or 
anticlockwise and the spoke edges away from the central vertex. Call the resulting 

digraph as out-fan and denote it by𝑜𝐹𝑛⃗⃗  ⃗. 
 

Definition 3.3. A fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, is said to be an alternating fan (𝐴𝐹𝑛⃗⃗  ⃗) if the edges of 
the path 𝑃𝑛 are oriented clockwise or anticlockwise and the spoke edges alternately. 
 

Definition 3.4. In a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the path edges alternately and the spoke 
edges towards the central vertex. Call the resulting digraph as alternating in-fan and 

denote it by𝐴𝑖𝐹𝑛⃗⃗  ⃗. 
 

Definition 3.5. In a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 alternately and the 
spoke edges away from the central vertex. Call the resulting digraph as alternating out-

fan and denote it by𝐴𝑜𝐹𝑛⃗⃗  ⃗. 
 

Definition 3.6. In a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 alternately and the 
spoke edges such that either 𝑑+(𝑣) > 0 or𝑑−(𝑣) > 0 ∀ 𝑣 ∈ 𝑉(𝑃𝑛). Call the resulting 

digraph as sole double alternating fan and denote it by𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗. 
 

Definition 3.7. In a fan𝐹𝑛 = 𝑃𝑛 + 𝐾1, orient the edges of the path 𝑃𝑛 alternately and the 
spoke edges such that neither 𝑑+(𝑣) > 0 nor𝑑−(𝑣) > 0 ∀ 𝑣 ∈ 𝑉(𝑃𝑛). Call the resulting 

digraph as di-double alternating fan and denote it by𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗. 
 

Theorem 3.8. In-fan (𝑖𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood prime labeling for  𝑛 ≥
2. 

Proof: Let 𝑛 ≥ 2 and let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗  ⃗ and 𝑢 be 
the apex vertex. 

Then 𝐴(𝑖𝐹𝑛⃗⃗  ⃗) = {𝑣𝑖𝑣𝑖+1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑣𝑖𝑢⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛} is the arc set. 

This digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. 

Define an injective function 𝑓:𝑉(𝑖𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1 and 𝑓(𝑣𝑖) = 𝛾𝑖+1 for1 ≤

𝑖 ≤ 𝑛. 

Here,𝑑−(𝑢) > 1. Further, the labels of the in-neighborhood vertices of 𝑢 are 
consecutive Gaussian integers in the spiral ordering and so they are relatively prime. 

𝑁−(𝑣1) = 𝜙 And𝑁
−(𝑣𝑖) = {𝑣𝑖−1} for 2 ≤ 𝑖 ≤ 𝑛. 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling.                                           
Next to prove 𝑓 is also a Gaussian out-nighborhood prime labeling. 

12



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

Now 𝑑+(𝑣𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛 − 1. Here, the out-neighborhood vertices of 𝑣𝑖(1 ≤ 𝑖 ≤
𝑛 − 1) contains the Gaussian integer 𝛾1 = 1 and 𝛾1 is relatively prime to all the 
Gaussian integers. 

Further, 𝑁+(𝑣𝑛) = {𝑢} and𝑁
+(𝑢) = 𝜙. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling.                                         
Which implies 𝑓 is a Gaussian twin neighborhood prime labeling. 

Hence, in-fan (𝑖𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood prime labeling for𝑛 ≥ 2. 

 

Theorem 3.9. Out-fan (𝑜𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood prime labeling for𝑛 ≥
2. 

Proof: Let 𝑛 ≥ 2 and let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗  ⃗ and 𝑢 be 
the apex vertex. 

Then 𝐴(𝑜𝐹𝑛⃗⃗  ⃗) = {𝑣𝑖𝑣𝑖+1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛} is the arc set. 

This digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. 

Define an injective function 𝑓:𝑉(𝑜𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1 and 𝑓(𝑣𝑖) = 𝛾𝑖+1 

for1 ≤ 𝑖 ≤ 𝑛. 

Now 𝑑−(𝑣𝑖) > 1 for 2 ≤ 𝑖 ≤ 𝑛. In the above labeling, the in-neighborhood vertices of 
𝑣𝑖 contains the Gaussian integer 𝛾1 = 1 which is relatively prime to all Gaussian 
integers. 

Further, 𝑁−(𝑣1) = {𝑢} and𝑁
−(𝑢) = 𝜙.  

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling.                                           
Next to prove 𝑓 is also a Gaussian out-nighborhood prime labeling. 

Now 𝑑+(𝑢) > 1 and the labels of the out-neighborhood vertices of 𝑢 are consecutive 
Gaussian integers in the spiral ordering and so they are relatively prime. 

Also, 𝑁+(𝑣𝑖) = {𝑣𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝑁
−(𝑣𝑛) = 𝜙 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling.                                         
Which implies 𝑓 is a Gaussian twin neighborhood prime labeling. 

Hence, out-fan (𝑜𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood prime labeling for 𝑛 ≥ 2. 

 

Theorem 3.10. Alternating fan (𝐴𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood prime 
labeling for 𝑛 ≥ 2. 

Proof: Let 𝑛 ≥ 2 and let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗  ⃗ and 𝑢 be 
the apex vertex.  

This digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. 

Case (i): 𝒏 is odd  

13



Palani K and Shunmugapriya A 

 

 

𝐴(𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖𝑣2𝑖+1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪

{𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} is the corresponding arc set. 

Define 𝑓:𝑉(𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤
𝑛+1

2
 and 

𝑓(𝑣2𝑖) = 𝛾(𝑛+1)
2

+𝑖+1
 for1 ≤ 𝑖 ≤

𝑛−1

2
. 

Clearly 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Now 𝑁−(𝑢) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤
𝑛+1

2
 and the labels of the in-neighborhood 

vertices of 𝑢 are consecutive Gaussian integers in the spiral ordering and hence are 
relatively prime. 

Further, 𝑁−(𝑣2𝑖) = {𝑢,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the label set of the in-neighbors of 

𝑣2𝑖 contains𝛾1 = 1. 

Also, 𝑁−(𝑣1) = 𝜙 and 𝑁
−(𝑣2𝑖−1) = {𝑣2𝑖−2} for 2 ≤ 𝑖 ≤

𝑛+1

2
 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is odd.                   

Next to prove 𝑓 is also a Gaussian out-neighborhood prime labeling. 

Now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for1 ≤ 𝑖 ≤
𝑛+1

2
. 

𝑁+(𝑢) = {𝑣2,𝑣4,…,𝑣2𝑖} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the out-neighborhood vertices of 𝑢 are 

labeled with the consecutive Gaussian integers in the spiral ordering and so by the result 

1.4(4), the labels are relatively prime. 

Now 𝑁+(𝑣2𝑖−1) = {𝑢,𝑣2𝑖} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and 𝑁+(𝑣𝑛) = {𝑢}. 

Also, the out-neighborhood vertices of 𝑣2𝑖−1(1 ≤ 𝑖 ≤
𝑛−1

2
) contains the Gaussian 

integer 𝛾1 = 1.  

Further, 𝑁+(𝑣2𝑖) = {𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is odd.                
𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is odd.   
Case (ii): 𝒏 is even 

𝐴(𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖𝑣2𝑖+1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} is the corresponding arc set. 

Define 𝑓:𝑉(𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by𝑓(𝑢) = 𝛾1; 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤
𝑛

2
  and 𝑓(𝑣2𝑖) =

𝛾𝑛
2
+𝑖+1

 for1 ≤ 𝑖 ≤
𝑛

2
. 

Clearly 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for1 ≤ 𝑖 ≤
𝑛

2
. 

Now 𝑁−(𝑢) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤
𝑛

2
 and the labels of the in-neighborhood 

vertices of 𝑢 are consecutive Gaussian integers in the spiral ordering and so are 
relatively prime. 

14



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

Further, 𝑁−(𝑣2𝑖) = {𝑢,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤
𝑛

2
 and the label set of in-neighbors of 𝑣2𝑖 

contains the Gaussian integer𝛾1 = 1.  

Also, 𝑁−(𝑣1) = 𝜙 and 𝑁
−(𝑣2𝑖−1) = {𝑣2𝑖−2} for 2 ≤ 𝑖 ≤

𝑛

2
 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is even.                 
Next to prove 𝑓 is also a Gaussian out-nighborhood prime labeling. 

Now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for1 ≤ 𝑖 ≤
𝑛

2
. 

𝑁+(𝑢) = {𝑣2,𝑣4,…,𝑣2𝑖} For1 ≤ 𝑖 ≤
𝑛

2
 and the out-neighborhood vertices of 𝑢 are 

labeled with the consecutive Gaussian integers in the spiral ordering and so they are 

relatively prime. 

Further, 𝑁+(𝑣2𝑖−1) = {𝑢,𝑣2𝑖} for 1 ≤ 𝑖 ≤
𝑛

2
 and the out-neighborhood vertices of 

𝑣2𝑖−1(1 ≤ 𝑖 ≤
𝑛

2
) contains the Gaussian integer𝛾1 = 1. 

Also, 𝑁+(𝑣2𝑖) = {𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−2

2
 and𝑁+(𝑣𝑛) = 𝜙. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is even.               
𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is even.   
Cases (i) and (ii) imply 𝑓 is a Gaussian twin neighborhood prime labeling. 

Hence, alternating fan (𝐴𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood prime labeling for𝑛 ≥
2. 

Theorem 3.11. Alternating in-fan (𝐴𝑖𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime 
digraph for𝑛 ≥ 2. 

Proof. Let 𝑛 ≥ 2 and let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗  ⃗ and 𝑢 be 
the apex vertex. 

Case (i): 𝒏 is odd 

𝐴(𝐴𝑖𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛+1

2
} ∪

{𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} is the corresponding arc set. 

Define an injective function 𝑓:𝑉(𝐴𝑖𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 

 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤
𝑛+1

2
 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1)

2
+𝑖+1

 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Here 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Clearly, the label set of  in-neighborhood vertices of 𝑢 contains consecutive Gaussian 
integers in the spiral ordering and so they are relatively prime.  

Now 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the labels of the in-neighborhood 

vertices of 𝑣2𝑖 are consecutive Gaussian integers in the spiral ordering  

Further, 𝑁−(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤
𝑛+1

2
 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is odd.                   
Next to prove 𝑓 is also a Gaussian out-neighborhood prime labeling. 

15



Palani K and Shunmugapriya A 

 

 

Now 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛+1

2
 and the labels of vertices in 𝑁+(𝑣2𝑖−1) contains 

the Gaussian integer 𝛾1 = 1 which is relatively prime to all the Gaussian integers. 

Further, 𝑁+(𝑣2𝑖) = {𝑢} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and 𝑁+(𝑢) = 𝜙. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is odd.                 
By (1) and (2), 𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is odd.   
Case (ii): 𝒏 is even  

𝐴(𝐴𝑖𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝐴𝑖𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by𝑓(𝑢) = 𝛾1; 

 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤
𝑛

2
 and 𝑓(𝑣2𝑖) = 𝛾𝑛

2
+𝑖+1

 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Here 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for1 ≤ 𝑖 ≤
𝑛−2

2
. 

Clearly, the label set of in-neighborhood vertices of 𝑢 contains consecutive Gaussian 
integers in the spiral ordering and so they are relatively prime.  

Now 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−2

2
 and the label set of in-neighborhood 

vertices of 𝑣2𝑖 are consecutive Gaussian integers in the spiral ordering. 

Also, 𝑁−(𝑣𝑛) = {𝑣𝑛−1} and 𝑁
−(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤

𝑛+1

2
. 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is even.                 

Now 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛

2
 and the out-neighborhood vertices of 𝑣2𝑖−1 contains 

the Gaussian integer 𝛾1 = 1 which is relatively prime to all the Gaussian integers. 

Further, 𝑁+(𝑣2𝑖) = {𝑢} for 1 ≤ 𝑖 ≤
𝑛

2
. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is even.               
By (3) and (4), 𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is even.   
Cases (i) and (ii) imply 𝑓 is a Gaussian twin neighborhood prime labeling. 

Thus, an alternating in-fan (𝐴𝑖𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime digraph for 
𝑛 ≥ 2. 

Theorem 3.12. Alternating out-fan (𝐴𝑜𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime 
digraph for 𝑛 ≥ 2. 

Proof: Let 𝑛 ≥ 2. 

Let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the directed path 𝑃𝑛⃗⃗  ⃗ and 𝑢 be the apex vertex. 

This digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. 

Case (i): 𝒏 is odd  

𝐴(𝐴𝑜𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑢𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛+1

2
} ∪

{𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} is the corresponding arc set. 

Define an injective function 𝑓:𝑉(𝐴𝑜𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 

16



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤
𝑛+1

2
 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1)

2
+𝑖+1

 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Now 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the label set of in-neighbors of 𝑣2𝑖 are the 

consecutive Gaussian integers in the spiral ordering. 

Further, 𝑁−(𝑣2𝑖−1) = {𝑢} for 1 ≤ 𝑖 ≤
𝑛+1

2
 and𝑁−(𝑢) = 𝜙. 

Therefore, 𝑓 is a Gaussian in- neighborhood prime labeling when 𝑛 is odd.                  
Next to prove 𝑓 is a Gaussian out-neighborhood prime labeling. 

Now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for2 ≤ 𝑖 ≤
𝑛−1

2
. 

Clearly, the label set of out-neighborhood vertices of 𝑢 contains all the vertices of the 

path 𝑃𝑛⃗⃗  ⃗ which are labeled with the consecutive Gaussian integers in the spiral ordering. 

Further, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤
𝑛−1

2
 and the labels of the out-

neighborhood vertices of 𝑣2𝑖−1 are consecutive Gaussian integers in the spiral ordering. 

Also, 𝑁+(𝑣1) = {𝑣2} ,  𝑁
+(𝑣𝑛) = {𝑣𝑛−1} and 𝑁

+(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is odd.                 
(1) and (2) imply 𝑓 is a Gaussian twin neighborhood prime labeling if 𝑛 is odd. 
Case (ii): 𝒏 is even  

𝐴(𝐴𝑜𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑢𝑣2𝑖−1⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} is the corresponding arc set. 

Define an injective function 𝑓:𝑉(𝐴𝑜𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 

 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤
𝑛

2
 and 𝑓(𝑣2𝑖) = 𝛾𝑛

2
+𝑖+1

 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Now 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛

2
 and the label set of the in-neighborhood vertices of 𝑣2𝑖 

are consecutive Gaussian integers in the spiral ordering and so they are relatively prime. 

Further, 𝑁−(𝑢) = 𝜙 and 𝑁−(𝑣2𝑖−1) = {𝑢} for 1 ≤ 𝑖 ≤
𝑛

2
. 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is even.                
Next to prove 𝑓 is also a Gaussian out-neighborhood prime labeling.  

Here 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤
𝑛

2
. 

Clearly, the labels out-neighborhood vertices of 𝑢 contains consecutive Gaussian 
integers in the spiral ordering and so are relatively prime.  

Further, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤
𝑛

2
 and the labels of out-neighborhood 

vertices of 𝑣2𝑖−1 are consecutive Gaussian integers in the spiral ordering and so they are 
relatively prime. 

Also, 𝑁+(𝑣1) = {𝑣2} and 𝑁
+(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤

𝑛

2
. 

Therefore, 𝑓 is a Gaussian out- neighborhood prime labeling when 𝑛 is even.              
(3) and (4) imply 𝑓 is a Gaussian twin neighborhood prime labeling if 𝑛 is even. 

From the cases (i) and (ii), 𝑓 is a Gaussian twin neighborhood prime labeling.  

Thus, an alternating outfan (𝐴𝑜𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime digraph for 
𝑛 ≥ 2. 

17



Palani K and Shunmugapriya A 

 

 

Theorem 3.13. Sole double alternating fan (𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood 
prime digraph for 𝑛 ≥ 2. 

Proof: Let 𝑉(𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑢,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set where 𝑣𝑖 represent the ith 

vertex of the directed path 𝑃𝑛⃗⃗  ⃗ and 𝑢 is the apex vertex. 

This digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. 

Case (i): 𝒏 is odd  

𝐴(𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛+1

2
} ∪

{𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} is the corresponding arc set. 

Define an injective function 𝑓:𝑉(𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 

 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤
𝑛+1

2
 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1)

2
+𝑖+1

 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Now 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for1 ≤ 𝑖 ≤
𝑛−1

2
. 

The label set of in-neighborhood vertices of 𝑢 are consecutive Gaussian integers in the 
spiral ordering and so are relatively prime.  

Also, the in-neighborhood vertices of 𝑣2𝑖 contains the Gaussian integer 𝛾1 = 1 which is 
relatively prime to all the Gaussian integers. 

Further, 𝑁−(𝑣2𝑖−1) = {𝑢} for1 ≤ 𝑖 ≤
𝑛+1

2
. 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is odd.                   
Next to prove 𝑓 is also a Gaussian out-nighborhood prime labeling. 

Now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for1 ≤ 𝑖 ≤
𝑛+1

2
. 

The label set of out-neighborhood vertices of 𝑢 are consecutive Gaussian integers in the 
spiral ordering and so those are relatively prime.  

Also, the label set of out-neighborhood vertices of 𝑣2𝑖−1 contains the Gaussian integer 
𝛾1 = 1 which is relatively prime to all the Gaussian integers. 

𝑁+(𝑣2𝑖) = 𝜙 for1 ≤ 𝑖 ≤
𝑛−1

2
. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is odd.                
Case (ii): 𝒏 is even  

𝐴(𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 

 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤
𝑛

2
 and 𝑓(𝑣2𝑖) = 𝛾𝑛

2
+𝑖+1

 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Now 𝑑−(𝑢) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛

2
. 

The label set of of in-neighborhood vertices of 𝑢 are consecutive Gaussian integers in 
the spiral ordering and so those are relatively prime.  

18



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

Further, the label set of in-neighborhood vertices of 𝑣2𝑖 contains the Gaussian integer 
𝛾1 = 1 which is relatively prime to all the Gaussian integers. 

Also, 𝑁−(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is even.                 
Next to prove 𝑓 is also a Gaussian out-nighborhood prime labeling. 

Now 𝑑+(𝑢) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛

2
. 

The label set of out-neighborhood vertices of 𝑢 are consecutive Gaussian integers and 
so they are relatively prime. Then the out-neighborhood vertices of 𝑣2𝑖−1 contains the 
Gaussian integer𝛾1 = 1. 

Also, 𝑁+(𝑣2𝑖) = 𝜙 for1 ≤ 𝑖 ≤
𝑛

2
. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is even.               
Case (i) and (ii) imply 𝑓 is a Gaussian twin neighborhood prime labeling. 

Hence, the Sole-double alternating fan (𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime 
digraph for𝑛 ≥ 2. 

Theorem 3.14. Di-double alternating fan 𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗ is a Gaussian twin neighborhood 
prime digraph for𝑛 ≥ 2. 

Proof: Let 𝑛 ≥ 2 and let 𝑉(𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑢,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set where 𝑣𝑖 

represent the ith vertex of the directed path 𝑃𝑛⃗⃗  ⃗ and 𝑢 is the apex vertex. 

This digraph has 𝑛 + 1 vertices and 2𝑛 − 1 arcs. 

Case (i): 𝒏 is odd  

𝐴(𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗⃗⃗ ⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪

{𝑢𝑣2𝑖−1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤
𝑛+1

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 

 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for1 ≤ 𝑖 ≤
𝑛+1

2
 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1)

2
+𝑖+1

 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Now𝑑−(𝑢) > 1. The label set of in-neighborhood vertices of 𝑢 are consecutive 
Gaussian integers in the in the spiral ordering and so they are relatively prime.  

Also, 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the in-neighborhood vertices of 𝑣2𝑖 are labeled 

with the consecutive Gaussian integers. 

Further, 𝑁−(𝑣2𝑖−1) = {𝑢} for 1 ≤ 𝑖 ≤
𝑛+1

2
. 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is odd.                   
Now 𝑑+(𝑢) > 1 and the out-neighborhood vertices of 𝑢 are consecutive Gaussian 
integers in the labeling and so they are relatively prime.  

Also, 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤
𝑛−1

2
 and the label set of the out-neighborhood vertices 

of 𝑣2𝑖−1 contains the consecutive Gaussian integers in the spiral ordering. 

19



Palani K and Shunmugapriya A 

 

 

Further, 𝑁+(𝑣1) = {𝑣2} and 𝑁
+(𝑣𝑛) = {𝑣𝑛−1}.  

𝑁+(𝑣2𝑖) = {𝑢} for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is odd.                 
Case (ii): 𝒏 is even 

𝐴(𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑢𝑣2𝑖−1⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤
𝑛

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+1] by 𝑓(𝑢) = 𝛾1; 

 𝑓(𝑣2𝑖−1) = 𝛾𝑖+1 for 1 ≤ 𝑖 ≤
𝑛

2
 and 𝑓(𝑣2𝑖) = 𝛾𝑛

2
+𝑖+1

 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Now 𝑑−(𝑢) > 1 and labeling of the in-neighborhood vertices of 𝑢 are consecutive 
Gaussian integers in the spiral ordering and they are relatively prime.  

Also, 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛−2

2
 and the in-neighborhood vertices of 𝑣2𝑖 are labeled 

with the consecutive Gaussian integers and so are relatively prime.  

Further, 𝑁−(𝑣𝑛) = {𝑣𝑛−1} and 𝑁
−(𝑣2𝑖−1) = {𝑢} for 1 ≤ 𝑖 ≤

𝑛

2
. 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is even.                 
Next to prove 𝑓 is also a Gaussian out-nighborhood prime labeling. 

Now 𝑑+(𝑢) > 1 and the labeling of the out-neighborhood vertices of 𝑢 are consecutive 
Gaussian integers.  

Also, 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤
𝑛

2
 and the label set of the out-neighborhood vertices of 

𝑣2𝑖−1(2 ≤ 𝑖 ≤
𝑛

2
) are consecutive Gaussian integers in the spiral ordering and so are 

relatively prime. 

Also, 𝑁+(𝑣1) = {𝑣2} and 𝑁
+(𝑣2𝑖) = {𝑢} for 1 ≤ 𝑖 ≤

𝑛

2
. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is even.               
Cases (i) and (ii) imply 𝑓 is a Gaussian twin neighborhood prime labeling. 

Thus the di-double alternating fan (𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime 
digraph for𝑛 ≥ 2. 

 

4. Double fan digraphs 
In this section, some new digraphs are introduced by orienting double fan graphs 

in different possible ways and named accordingly. Also, the Gaussian twin 

neighborhood prime labeling is proved for those digraphs. 

 

Definition 4.1. In a double fan𝐷𝐹𝑛 = 𝑃𝑛 + 𝐾2̅̅ ̅, orient the edges of the common path 𝑃𝑛 
clockwise or anticlockwise and the spoke edges are towards the central vertex. Call the 

resulting digraph as double in-fan and denote it by𝐷𝑖𝐹𝑛⃗⃗  ⃗. 

20



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

Definition 4.2. In a double fan𝐷𝐹𝑛 = 𝑃𝑛 + 𝐾2̅̅ ̅, orient the edges of the common path 𝑃𝑛 
clockwise or anticlockwise and the spoke edges away from the central vertex. Call the 

resulting digraph as double out-fan and denote it by𝐷𝑜𝐹𝑛⃗⃗  ⃗. 
 

Definition 4.3. A double fan𝐷𝐹𝑛 = 𝑃𝑛 + 𝐾2̅̅ ̅, is said to be a double alternating fan 

(𝐷𝐴𝐹𝑛⃗⃗  ⃗) if the edges of the common path 𝑃𝑛 are oriented in clockwise or anticlockwise 
and the spoke edges alternately. 

 

Definition 4.4. In a double fan𝐷𝐹𝑛, orient the edges of the common path 𝑃𝑛 alternately 
and the spoke edges are towards the central vertices. Call the resulting digraph as 

double alternating in-fan and denote it by𝐷𝐴𝑜𝐹𝑛⃗⃗  ⃗. 

 

Definition 4.5. In a double fan𝐷𝐹𝑛, orient the edges of the common path 𝑃𝑛 alternately 
and the spoke edges are away the central vertices. Call the resulting digraph is called a 

double alternating out-fan and denote it by𝐷𝐴𝑖𝐹𝑛⃗⃗  ⃗. 

 

Definition 4.6. In a double fan 𝐷𝐹𝑛, orient the edges of the common path 𝑃𝑛  alternately 
and the spoke edges such that either 𝑑+(𝑣) = 0 or 𝑑−(𝑣) = 0 ∀ 𝑣 ∈ 𝑉(𝑃𝑛). The 
resulting digraph is called a double sole-double alternating fan and denoted it as 

𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗. 

 

Definition 4.7. In a double fan 𝐷𝐹𝑛, orient the edges of the common path 𝑃𝑛 alternately 
and the spoke edges such that neither 𝑑+(𝑣) = 0 nor 𝑑−(𝑣) = 0 ∀ 𝑣 ∈ 𝑉(𝑃𝑛). The 
resulting digraph is called a double di-double alternating fan and denote it by 

𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗. 

Theorem 4.8. Double in-fan (𝐷𝑖𝐹𝑛⃗⃗  ⃗ )is a Gaussian twin neighborhood prime digraph. 

Proof:  Let 𝑉(𝐷𝑖𝐹𝑛⃗⃗  ⃗) = {𝑢,𝑤,𝑣𝑖| 1 ≤ 𝑖 ≤ 𝑛} where 𝑢 and 𝑤 are the apex vertices and 

𝑣𝑖 represent the 𝑖𝑡ℎ vertex of the directed path 𝑃𝑛⃗⃗  ⃗. 

Then 𝐴(𝐷𝑖𝐹𝑛⃗⃗  ⃗) = {𝑣𝑖𝑣𝑖+1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ | 1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑣𝑖𝑢⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖𝑤⃗⃗⃗⃗⃗⃗ | 1 ≤ 𝑖 ≤ 𝑛} is the 
arc set. 

This digraph graph has 𝑛 + 2 vertices and 3𝑛 − 1 edges. 

Define an injective function 𝑓:𝑉(𝐷𝑖𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by(𝑢) = 𝛾1 , 𝑓(𝑤) = 𝛾2 and 𝑓(𝑣𝑖) =

𝛾𝑖+2      for 1 ≤ 𝑖 ≤ 𝑛. 
Here 𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1. 
By the definition of 𝑓, the in-neighborhood vertices of 𝑢 and 𝑤 are labeled by 
consecutive Gaussian integers 𝛾1 and 𝛾2 in the spiral ordering and so are relatively 
prime. 

Now, 𝑁−(𝑣1) = 𝜙 and 𝑁
−(𝑣𝑖) = {𝑣𝑖−1} for 2 ≤ 𝑖 ≤ 𝑛. 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling.                                            

21



Palani K and Shunmugapriya A 

 

 

Next to prove 𝑓 is also Gaussian out-neighborhood prime labeling. 
Now, 𝑑+(𝑣𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛. 
Further, 𝑁+(𝑣𝑖) = {𝑢,𝑣𝑖+1,𝑤} for 1 ≤ 𝑖 ≤ 𝑛 − 1 and the labels of vertices in 𝑁

+(𝑣𝑖) 
contains the Gaussian integer 𝛾1 = 1 which is relatively prime to all the Gaussian 
integers.  

Also, 𝑁+(𝑣𝑛) = {𝑢,𝑤} and labels of 𝑢 and 𝑤 are consecutive Gaussian integers. 
Further, 𝑁+(𝑢) = 𝑁+(𝑤) = 𝜙. 
Therefore, 𝑓 is a Gaussian out- neighborhood prime labeling.                                       
𝑓 is a Gaussian twin neighborhood prime labeling. 

Thus, double fan(𝐷𝑖𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime digraph. 
 

Theorem 4.9. Double out-fan(𝐷𝑜𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood prime 
labeling. 

Proof:  Let 𝑉(𝐷𝑜𝐹𝑛⃗⃗  ⃗) = {𝑢,𝑤,𝑣𝑖| 1 ≤ 𝑖 ≤ 𝑛} where 𝑢 and 𝑤 are the apex vertices and 

𝑣𝑖 represent the 𝑖𝑡ℎ vertex of the directed path 𝑃𝑛⃗⃗  ⃗. 

Then 𝐴(𝐷𝑜𝐹𝑛⃗⃗  ⃗) = {𝑣𝑖𝑣𝑖+1⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ | 1 ≤ 𝑖 ≤ 𝑛 − 1} ∪ {𝑢𝑣𝑖⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑤𝑣𝑖⃗⃗ ⃗⃗ ⃗⃗ | 1 ≤ 𝑖 ≤ 𝑛} is the 
arc set. 

This digraph has 𝑛 + 2 vertices and 3𝑛 − 1 edges. 

Define an injective function 𝑓:𝑉(𝐷𝑜𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by  

𝑓(𝑢) = 𝛾1 , 𝑓(𝑤) = 𝛾2 and 𝑓(𝑣𝑖) = 𝛾𝑖+2      for 1 ≤ 𝑖 ≤ 𝑛. 
Now, 𝑑−(𝑣𝑖) > 1 for 1 ≤ 𝑖 ≤ 𝑛. 
𝑁−(𝑣1) = {𝑢,𝑤} and 𝑁

−(𝑣𝑖) = {𝑢,𝑣𝑖−1,𝑤} for 2 ≤ 𝑖 ≤ 𝑛.   
Clearly, label set of vertices in 𝑁−(𝑣𝑖) contains the Gaussian integer 𝛾1 = 1 which is 
relatively prime to all the Gaussian integers. 

Also, 𝑁−(𝑢) = 𝑁−(𝑤) = 𝜙. 
Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling.                                           
Next to prove 𝑓 is also Gaussian out-neighborhood prime labeling.  
Here 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1. 
By the definition of 𝑓, the set of out-neighborhood vertices of 𝑢 and 𝑤 are labeled by 
the consecutive Gaussian integers in the spiral ordering and which are relatively prime. 

Further, 𝑁+(𝑣𝑖) = {𝑣𝑖+1} for 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝑁
+(𝑣𝑛) = 𝜙. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling.                                          
𝑓 is a Gaussian twin neighborhood prime labeling. 

Hence, double out-fan (𝐷𝑜𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood prime labeling. 
 

Theorem 4.10. Double alternating fan (𝐷𝐴𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime 
digraph. 

Proof: Let 𝑣1,𝑣2,…,𝑣𝑛 be the vertices of the path 𝑃𝑛⃗⃗  ⃗ and 𝑢,𝑤 be the apex vertices. 

Let 𝑉(𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set. 

This digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. 
Case (i): 𝑛 is even 

𝐴(𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖𝑣2𝑖+1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛

2
} ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} is the arc set. 

22



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

Define an injective function 𝑓:𝑉(𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1;𝑓(𝑤) = 𝛾2 and  

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 and 𝑓(𝑣2𝑖) = 𝛾𝑛
2
+𝑖+2

 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Here, 𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Clearly, the label set of in-neighborhood vertices of 𝑢 and 𝑤 are consecutive Gaussian 
integers in the spiral ordering.  

Further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛

2
. By the definition of 𝑓, the vertex 𝑢 is 

labeled as 𝛾1 = 1, which is relatively prime to all the Gaussian integers. 

Also, 𝑁−(𝑣1) = 𝜙 and 𝑁
−(𝑣2𝑖−1) = {𝑣2𝑖−2} for 2 ≤ 𝑖 ≤

𝑛

2
. 

Therefore,  𝑓 is  Gaussian in-neighborhood prime labeling when 𝑛 is even.                  
Next to prove 𝑓 is also Gaussian out-neighborhood prime labeling. 

Now 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Clearly, the label set of out-neighborhood vertices of 𝑢 and 𝑤 are labeled by 
consecutive Gaussian integers in the spiral ordering.  

Further, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖,𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛

2
 and by the definition of 𝑓, the label of 𝑢 

is 𝛾1 = 1, which is relatively prime to all the Gaussian integers. 

Also, 𝑁+(𝑣2𝑖) = {𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−2

2
 and 𝑁+(𝑣𝑛) = 𝜙. 

Therefore,  𝑓 is a Gaussian out neighborhood prime labeling when 𝑛 is even.               
𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is even. 
Case (ii): 𝑛 is odd. 

Then, 𝐴(𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖𝑣2𝑖+1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛−1

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1;𝑓(𝑤) = 𝛾2 and  

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1
2
)+𝑖+2

 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Here,𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Clearly, the label set of in-neighborhood vertices of 𝑢 and 𝑤 are consecutive Gaussian 
integers in the spiral ordering.  

Further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and by the definition of𝑓, the vertex 𝑢 

has the label 𝛾1 = 1, which is relatively prime to all the Gaussian integers. 

Also,  𝑁−(𝑣1) = 𝜙 and 𝑁
−(𝑣2𝑖−1) = {𝑣2𝑖−2} for 2 ≤ 𝑖 ≤

𝑛+1

2
. 

Therefore,  𝑓 is a Gaussian in- neighborhood prime labeling when 𝑛 is odd.                 
Next to prove 𝑓 is also a Gaussian out-nighborhood prime labeling. 

Now𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛+1

2
. 

By the definition of𝑓, the label set of out-neighborhood vertices of 𝑢 and 𝑤 consecutive 
Gaussian integers in the spiral ordering.  

Also, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖,𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the label of 𝑢 is 𝛾1 = 1 is relatively 

prime to the Gaussian integers. 

23



Palani K and Shunmugapriya A 

 

 

𝑁+(𝑣2𝑖) = {𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Further,𝑁+(𝑣𝑛) = {𝑢,𝑤}. The vertices 𝑢 and 𝑣 are labeled by 𝛾1 and 𝛾2 respectively. 
Since 𝛾1 and 𝛾2  are consecutive Gaussian integers in the spiral ordering and so they are 
relatively prime. 

Therefore, 𝑓 is a Gaussian out- neighborhood prime labeling when 𝑛 is odd.                
𝑓 is a Gaussian twin neighborhood prime labeling. 
From both the cases, 𝑓 is a Gaussian twin neighborhood prime labeling. 

Hence, double alternating fan (𝐷𝐴𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime digraph. 
 

Theorem 4.11. Double alternating in-fan (𝐷𝐴𝑖𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood 
prime labeling. 

Proof. Let 𝑉(𝐷𝐴𝑖𝐹𝑛⃗⃗  ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} where 𝑢,𝑤the apex vertices are and 𝑣𝑖 

represent the 𝑖th vertex of the common path𝑃𝑛⃗⃗  ⃗. 
This digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. 
Case (i): 𝑛 is odd. 

𝐴(𝐷𝐴𝑖𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑣2𝑖𝑤⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛−1

2
} is the 

corresponding arc set. 

Define an injective function 𝑓:𝑉(𝐷𝐴𝑖𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by(𝑢) = 𝛾1, 𝑓(𝑤) = 𝛾2 , 

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2       for 1 ≤ 𝑖 ≤
𝑛+1

2
 and  𝑓(𝑣2𝑖) = 𝛾(𝑛+1

2
)+𝑖+2

 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Here, 𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣1,𝑣2,…,𝑣𝑛}. The vertices 𝑣1,𝑣2,…,𝑣𝑛 are labeled with the 
consecutive Gaussian integers in the spiral ordering and so they are relatively prime.  

Further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the labels of 𝑁−(𝑣2𝑖) are 

consecutive Gaussian integers in the spiral ordering. 

Also, 𝑁−(𝑣2𝑖−1) =  𝜙 for1 ≤ 𝑖 ≤
𝑛+1

2
.       

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is odd.                   
Next to prove 𝑓 is also Gaussian out-neighborhood prime labeling. 

Here 𝑑+(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛−1

2
 and 𝑑+(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤

𝑛−1

2
 

Further, 𝑁+(𝑣1) = {𝑣2,𝑢,𝑤} and 𝑁
+(𝑣2𝑖−1) = {𝑢,𝑤,𝑣2𝑖−2,𝑣2𝑖} for2 ≤ 𝑖 ≤

𝑛−1

2
. 

and the label set of out-neighborhood vertices 𝑣2𝑖−1(1 ≤ 𝑖 ≤
𝑛−1

2
) contains the 

Gaussian integer 𝛾1 = 1 which is relatively prime to all the Gaussian integers. 

Also, 𝑁+(𝑣2𝑖) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the vertices 𝑢 and 𝑤 are labeled by the 

Gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖 respectively.  
Now, 𝑁+(𝑢) = 𝑁+(𝑤) = 𝜙.   
Therefore, 𝑓 is  a Gaussian out-neighborhood prime labeling when 𝑛 is odd.                
𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is odd 
Case (ii): 𝑛 is even 

24



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

𝐴(𝐷𝐴𝑖𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛

2
} ∪ {𝑣2𝑖𝑤⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝐷𝐴𝑖𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1;𝑓(𝑤) = 𝛾2 and  

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 and𝑓(𝑣2𝑖) = 𝛾𝑛
2
+𝑖+2

 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Here, 𝑑−(𝑢) > 1, 𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛−2

2
. 

By the definition of𝑓, the label set of the in-neighborhood vertices of 𝑢 and 𝑤 are 
consecutive Gaussian integers in the spiral ordering and so they are relatively prime. 

Further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−2

2
 and the vertices in 𝑁−(𝑣2𝑖) are 

labeled by consecutive Gaussian integers in the spiral ordering. 

Also, 𝑁−(𝑣𝑛) = {𝑣𝑛−1} and 𝑁
−(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤

𝑛

2
.     

Therefore, 𝑓 admits a Gaussian in-neighborhood prime labeling when 𝑛 is even.         

Now 𝑑+(𝑣2𝑖−1) > 1 and 𝑑
+(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤

𝑛

2
. 

Here, 𝑁+(𝑢) = 𝑁+(𝑤) = 𝜙.      

Also, 𝑁+(𝑣1) = {𝑣2,𝑢,𝑤} and 𝑁
+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖,𝑢,𝑤} for 2 ≤ 𝑖 ≤

𝑛

2
 and the 

labels of 𝑁+(𝑣2𝑖−1)(1 ≤ 𝑖 ≤
𝑛

2
) contains the Gaussian integer 𝛾1 = 1 which is 

relatively prime to all the Gaussian integers. 

𝑁+(𝑣2𝑖) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛

2
 and the vertices 𝑢 and 𝑤 are labelled by the 

consecutive Gaussian integers 𝛾1 =  1 and 𝛾2 =  1 +  𝑖 respectively. So the labels of 
vertices in 𝑁+(𝑣2𝑖) are relatively prime. 
Therefore, 𝑓 is a Gaussian out- neighborhood prime labeling when 𝑛 is even.              
𝑓 is a Gaussian twin neighborhood prime labeling. 

Hence, double alternating in-fan (𝐷𝐴𝑖𝐹𝑛⃗⃗  ⃗) admits Gaussian twin neighborhood prime 
labeling.  

 

Theorem 4.12. Double alternating out-fan (𝐷𝐴𝑜𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood 
prime digraph. 

Proof. Let 𝑉(𝐷𝐴𝑜𝐹𝑛⃗⃗  ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} where 𝑢,𝑤 are the apex vertices and 𝑣𝑖 

represent the 𝑖th vertex of the common path 𝑃𝑛⃗⃗  ⃗. 
This digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. 
Case (i): 𝑛 is odd. 

𝐴(𝐷𝐴𝑜𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑢𝑣2𝑖−1⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑤𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛−1

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝐷𝐴𝑜𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by (𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2 , 

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2       for 1 ≤ 𝑖 ≤
𝑛+1

2
 and 𝑓(𝑣2𝑖) = 𝛾(𝑛+1

2
)+𝑖+2

 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Here, 𝑑−(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛+1

2
 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤

𝑛−1

2
. 

The vertices 𝑢 and 𝑤 has no in-neighbors. 
That is, 𝑁−(𝑢) = 𝑁−(𝑤) = 𝜙.       

25



Palani K and Shunmugapriya A 

 

 

Now 𝑁−(𝑣2𝑖−1) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛+1

2
 and the vertices 𝑢 and 𝑤 are labelled by the 

consecutive Gaussian integers 𝛾1 =  1 and 𝛾2 =  1 +  𝑖 respectively. So the labels of 
vertices in 𝑁−(𝑣2𝑖) are relatively prime. 

Further, 𝑁−(𝑣2𝑖) = {𝑢,𝑤,𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the label set of in-

neighborhood vertices 𝑣2𝑖 contains the Gaussian integer 𝛾1 = 1 which is relatively 
prime to all the Gaussian integers. 

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is odd.                   
Next to prove 𝑓 is also Gaussian out-neighborhood prime labeling. 

Now, 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤
𝑛−1

2
. 

The out-neighborhood vertices of 𝑢 and 𝑤 are labeled by consecutive Gaussian integers 
in the spiral ordering and so they are relatively prime. 

Also, 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤
𝑛−1

2
 and the vertices 𝑣2𝑖−2 and 𝑣2𝑖 are 

labeled by 𝛾
(
𝑛+1

2
)+𝑖+1

 and 𝛾
(
𝑛+1

2
)+𝑖+2

 which are consecutive Gaussian integers in the 

spiral ordering. 

𝑁+(𝑣1) = {𝑣2} and 𝑁
+(𝑣𝑛) = {𝑣𝑛−1}. 

Then, 𝑁+(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤
𝑛−1

2
.     

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is odd.                 
𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is odd 
Case (ii): 𝑛 is even 

𝐴(𝐷𝐴𝑜𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑢𝑣2𝑖−1⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑤𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} is the corresponding arc 

set. 

Define an injective function 𝑓:𝑉(𝐷𝐴𝑜𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by (𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2 , 

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2       for 1 ≤ 𝑖 ≤
𝑛

2
 and  𝑓(𝑣2𝑖) = 𝛾(𝑛

2
)+𝑖+2

 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Here, 𝑑−(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛

2
 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤

𝑛

2
. 

Further, 𝑁−(𝑢) = 𝑁−(𝑤) = 𝜙.       

Also,  𝑁−(𝑣2𝑖−1) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛

2
 and the vertices 𝑢 and 𝑤 are labeled with the 

consecutive Gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖 respectively.  

Here, 𝑁−(𝑣2𝑖) = {𝑢,𝑤,𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−2

2
 and 𝑁−(𝑣𝑛) = {𝑢,𝑤,𝑣𝑛−1}.  

Further, the label of the in-neighborhood vertices of 𝑣2𝑖(1 ≤ 𝑖 ≤
𝑛

2
) contains the 

Gaussian integer  𝛾1 = 1 which is relatively prime to all the Gaussian integers. 
Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is even.                 
Next to prove 𝑓 is also a Gaussian out-neighborhood prime labeling. 

Now, 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤
𝑛

2
. 

Clearly, the label of the out-neighborhood vertices of 𝑢 and 𝑤 are labeled by 
consecutive Gaussian integers in the spiral ordering and so are relatively prime. 

Also, 𝑁+(𝑣1) = {𝑣2}. 

26



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} For2 ≤ 𝑖 ≤
𝑛

2
. The vertices 𝑣2𝑖−2 and 𝑣2𝑖 are labeled by 

𝛾𝑛
2
+𝑖+1

 and 𝛾𝑛
2
+𝑖+2

 which are consecutive Gaussian integers in the labeling in spiral 

ordering.  

Also, 𝑁+(𝑣2𝑖) = 𝜙 for1 ≤ 𝑖 ≤
𝑛

2
.     

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is even.               
Therefore, 𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is even. 

Thus, double alternating out-fan (𝐷𝐴𝑜𝐹𝑛⃗⃗  ⃗) is a Gaussian twin neighborhood prime 
digraph. 

 

Theorem 4.13. Double sole double alternating fan (𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) admits Gaussian twin 
neighborhood prime labeling. 

Proof: Let 𝑉(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set where 𝑣𝑖 represent the 

ith vertex of the common path 𝑃𝑛⃗⃗  ⃗ and 𝑢,𝑤 be the apex vertices. 
This digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. 
Case (i) 𝑛 is odd. 

 𝐴(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛−1

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by (𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2 , 

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2 For 1 ≤ 𝑖 ≤
𝑛+1

2
 and  𝑓(𝑣2𝑖) = 𝛾(𝑛+1

2
)+𝑖+2

 for1 ≤ 𝑖 ≤
𝑛−1

2
. 

Here, 𝑑−(𝑢) > 1,𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for1 ≤ 𝑖 ≤
𝑛+1

2
. 

Further, 𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤
𝑛+1

2
 and the labels of 

vertices in 𝑁−(𝑢) and 𝑁−(𝑤) are consecutive Gaussian integers in the spiral ordering 
and hence are relatively prime. 

Also 𝑁−(𝑣2𝑖) = {𝑢,𝑤,𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the label of the vertex 𝑢 is 

𝛾1 = 1 which is relatively prime to all the Gaussian integers. 

Here, 𝑁−(𝑣2𝑖−1) = 𝜙 for1 ≤ 𝑖 ≤
𝑛+1

2
.   

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is odd.                  
Next to prove 𝑓 is also Gaussian out-neighborhood prime labeling. 

Here,  𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1  for 1 ≤ 𝑖 ≤
𝑛+1

2
. 

Now 𝑁+(𝑢) = 𝑁+(𝑤) = {𝑣2,𝑣4,…,𝑣2𝑖} for1 ≤ 𝑖 ≤
𝑛−1

2
. 

Clearly, the label set of out-neighborhood vertices of 𝑢 and 𝑤 are labeled by the 
consecutive Gaussian integers in the spiral ordering and so they are relatively prime. 

Also, 𝑁+(𝑣1) = {𝑢,𝑤,𝑣2},   𝑁
+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖,𝑢,𝑤} for 2 ≤ 𝑖 ≤

𝑛−3

2
 and  

𝑁+(𝑣𝑛) = {𝑢,𝑤,𝑣𝑛−1}. 

Further, the label of vertices in 𝑁+(𝑣2𝑖−1) for 1 ≤ 𝑖 ≤
𝑛−1

2
 contains the Gaussian 

integer 𝛾1 = 1 which is relatively prime to all the Gaussian integers. 

Also, 𝑁+(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤
𝑛−1

2
.  

27



Palani K and Shunmugapriya A 

 

 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is odd.                 
Therefore, 𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is odd. 
Case (ii): 𝑛 is even 

𝐴(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑣2𝑖−1𝑢⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑢𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖−1𝑤⃗⃗⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪ {𝑤𝑣2𝑖⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2, 

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2       for 1 ≤ 𝑖 ≤
𝑛

2
 and  𝑓(𝑣2𝑖) = 𝛾(𝑛

2
)+𝑖+2

     for 1 ≤ 𝑖 ≤
𝑛

2
. 

Here, 𝑑−(𝑢) > 1,𝑑−(𝑤) > 1 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤ 𝑖 ≤
𝑛

2
. 

Now 𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤
𝑛

2
 and the label set of vertices in 

𝑁−(𝑢) and 𝑁−(𝑤) are consecutive Gaussian integers in the spiral ordering.  

Further, 𝑁−(𝑣2𝑖) = {𝑢,𝑤,𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−2

2
  and 𝑁−(𝑣𝑛) = {𝑢,𝑤,𝑣𝑛−1}.  

By the definition of 𝑓, the label of the vertex 𝑢 is 𝛾1 which is relatively prime to all the 
Gaussian integers. 

𝑁−(𝑣2𝑖−1) = 𝜙 for 1 ≤ 𝑖 ≤
𝑛

2
.    

Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is even.                 
Next to prove 𝑓 is also a Gaussian out-nighborhood prime labeling. 

Here 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1 and 𝑑+(𝑣2𝑖−1) > 1  for 1 ≤ 𝑖 ≤
𝑛

2
. 

Now 𝑁+(𝑢) = 𝑁+(𝑤) = {𝑣2,𝑣4,…,𝑣2𝑖} for 1 ≤ 𝑖 ≤
𝑛

2
 and the out-neighborhood 

vertices of 𝑢 and 𝑤 are labeled by the consecutive Gaussian integers𝛾
(
𝑛

2
)+3

,𝛾
(
𝑛

2
)+4

, …, 

𝛾
(
𝑛

2
)+𝑖+2

 and so they are relatively prime. 

Further, 𝑁+(𝑣1) = {𝑢,𝑤,𝑣2} and 𝑁
+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖,𝑢,𝑤} for 2 ≤ 𝑖 ≤

𝑛

2
.  

and the labels of vertices in 𝑁+(𝑣1) and 𝑁
+(𝑣2𝑖−1) for 2 ≤ 𝑖 ≤

𝑛

2
 contains the Gaussian 

integer 𝛾1 = 1 which is relatively prime to all the Gaussian integers. 

Also, 𝑁+(𝑣2𝑖) = 𝜙 for 1 ≤ 𝑖 ≤
𝑛

2
.  

Therefore, 𝑓 is a Gaussian out- neighborhood prime labeling when 𝑛 even.                 
From both the cases, 𝑓 is a Gaussian twin neighborhood prime labeling.  

Hence double sole double alternating fan (𝐷𝑆𝐷𝐴𝐹𝑛⃗⃗  ⃗) admits a Gaussian twin 
neighborhood prime labeling. 

 

Theorem 4.14. Double di-double alternating fan 𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗ is a Gaussian twin 
neighborhood prime digraph. 

Proof: Let 𝑉(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑢,𝑤,𝑣𝑖|1 ≤ 𝑖 ≤ 𝑛} be the vertex set where 𝑣𝑖 represent the 

ith vertex of the common path 𝑃𝑛⃗⃗  ⃗ and 𝑢,𝑤 be the apex vertices. 
This digraph has 𝑛 + 2 vertices and 3𝑛 − 1 arcs. 
Case (i): 𝑛 is odd 

𝐴(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛−1

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑢𝑣2𝑖−1⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛−1

2
} ∪ {𝑤𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛+1

2
} ∪ {𝑣2𝑖𝑤⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛−1

2
} is the arc set. 

28



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

Define an injective function 𝑓:𝑉(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by 𝑓(𝑢) = 𝛾1;𝑓(𝑤) = 𝛾2 and  

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2       for 1 ≤ 𝑖 ≤
𝑛+1

2
 and  𝑓(𝑣2𝑖) = 𝛾(𝑛+1

2
)+𝑖+2

 for 1 ≤ 𝑖 ≤
𝑛−1

2
. 

Here, 𝑑−(𝑢) > 1,𝑑−(𝑤) > 1, 𝑑−(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛+1

2
 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤

𝑖 ≤
𝑛−1

2
. 

Now 𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣2,𝑣4,…,𝑣2𝑖} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the label set of in-

neighborhood vertices of 𝑢 and 𝑤 are consecutive Gaussian integers in the spiral 
ordering and so those are relatively prime.   

Also, 𝑁−(𝑣2𝑖−1) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛+1

2
 and the vertices 𝑢 and 𝑤 are labeled with 

consecutive Gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖. Since the consecutive Gaussian 
integers in the spiral ordering are relatively prime. 

Further, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−1

2
 and the labels of the vertices in 

𝑁−(𝑣2𝑖) are consecutive Gaussian integers and so are relatively prime. 
Therefore, 𝑓 is a Gaussian in-neighborhood prime labeling when 𝑛 is odd.                  

Here𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1, 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤
𝑛−1

2
 and 𝑑+(𝑣2𝑖) > 1 for 1 ≤

𝑖 ≤
𝑛−1

2
. 

Now 𝑁+(𝑢) = 𝑁+(𝑤) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤
𝑛+1

2
. 

By the definition of𝑓, the out-neighborhood vertices of 𝑢 and 𝑤 are labeled by the 
consecutive Gaussian integers𝛾3, 𝛾4, …, 𝛾𝑖+2. Since the consecutive Gaussian integers 
in the spiral ordering are relatively prime. 

𝑁+(𝑣1) = {𝑣2} and 𝑁
+(𝑣𝑛) = {𝑣𝑛−1}.     

Now 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤
𝑛−1

2
. The vertices 𝑣2𝑖−2and 𝑣2𝑖 are labeled 

by the consecutive Gaussian integers and which are relatively prime. 

Also, 𝑁+(𝑣2𝑖) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛−1

2
. Since the label of the vertex 𝑢 is 𝛾1 = 1 which 

is relatively prime to all the Gaussian integers. 

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is odd.                 
(I) and (II) imply, 𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is odd. 
Case (ii): 𝑛 is even 

𝐴(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) = {𝑣2𝑖−1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖+1𝑣2𝑖⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ |1 ≤ 𝑖 ≤

𝑛−2

2
} ∪ {𝑣2𝑖𝑢⃗⃗⃗⃗⃗⃗⃗⃗ |1 ≤ 𝑖 ≤

𝑛

2
} ∪

{𝑢𝑣2𝑖−1⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤
𝑛

2
} ∪ {𝑣2𝑖𝑤⃗⃗⃗⃗⃗⃗⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} ∪ {𝑤𝑣2𝑖−1⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗  ⃗|1 ≤ 𝑖 ≤

𝑛

2
} is the arc set. 

Define an injective function 𝑓:𝑉(𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗) → [𝛾𝑛+2] by (𝑢) = 𝛾1,𝑓(𝑤) = 𝛾2 , 

𝑓(𝑣2𝑖−1) = 𝛾𝑖+2       for 1 ≤ 𝑖 ≤
𝑛

2
 and  𝑓(𝑣2𝑖) = 𝛾(𝑛

2
)+𝑖+2

     for 1 ≤ 𝑖 ≤
𝑛

2
. 

Here, 𝑑−(𝑢) > 1,𝑑−(𝑤) > 1, 𝑑−(𝑣2𝑖−1) > 1 for 1 ≤ 𝑖 ≤
𝑛

2
 and 𝑑−(𝑣2𝑖) > 1 for 1 ≤

𝑖 ≤
𝑛−2

2
. 

Now 𝑁−(𝑢) = 𝑁−(𝑤) = {𝑣2,𝑣4,…,𝑣2𝑖} for 1 ≤ 𝑖 ≤
𝑛

2
. Further, the labels of vertices in 

𝑁−(𝑢) and 𝑁−(𝑤) are consecutive Gaussian integers in the spiral ordering and so those 
are relatively prime. 

29



Palani K and Shunmugapriya A 

 

 

Also, 𝑁−(𝑣2𝑖) = {𝑣2𝑖−1,𝑣2𝑖+1} for 1 ≤ 𝑖 ≤
𝑛−2

2
 and the vertices in 𝑁−(𝑣2𝑖) are labeled 

with the consecutive Gaussian integers and so those are relatively prime. 

𝑁−(𝑣𝑛) = {𝑣𝑛−1}.  

𝑁−(𝑣2𝑖−1) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛

2
.   Since the vertices 𝑢 and 𝑤 are labeled with the 

consecutive Gaussian integers 𝛾1 = 1 and 𝛾2 = 1 + 𝑖 respectively. Then 𝛾1 and 𝛾2 are 
relatively prime.  

Therefore, 𝑓 is a Gaussian in- neighborhood prime labeling when 𝑛 is even.                
Next to prove 𝑓 is also Gaussian out-neighborhood prime labeling. 

Here 𝑑+(𝑢) > 1, 𝑑+(𝑤) > 1, 𝑑+(𝑣2𝑖−1) > 1 for 2 ≤ 𝑖 ≤
𝑛

2
 and 𝑑+(𝑣2𝑖) > 1   for 1 ≤

𝑖 ≤
𝑛

2
. 

Now 𝑁+(𝑢) = 𝑁+(𝑤) = {𝑣1,𝑣3,…,𝑣2𝑖−1} for 1 ≤ 𝑖 ≤
𝑛

2
. 

By the definition of 𝑓, the label set of the out-neighborhood vertices of 𝑢 and 𝑤 are 
labeled by the consecutive Gaussian integers 𝛾3, 𝛾4, …, 𝛾𝑖+2 and so those are relatively 
prime. 

Also, 𝑁+(𝑣1) = {𝑣2}.     

Now 𝑁+(𝑣2𝑖−1) = {𝑣2𝑖−2,𝑣2𝑖} for 2 ≤ 𝑖 ≤
𝑛

2
 and the vertices 𝑣2𝑖−2 and 𝑣2𝑖 are labeled 

by the consecutive Gaussian integers 𝛾𝑛
2
+𝑖+1

 and 𝛾𝑛
2
+𝑖+2

.  

𝑁+(𝑣2𝑖) = {𝑢,𝑤} for 1 ≤ 𝑖 ≤
𝑛

2
.  Since the label of the vertex 𝑢 is 𝛾1 = 1 which is 

relatively prime to all the consecutive Gaussian integers.  

Therefore, 𝑓 is a Gaussian out-neighborhood prime labeling when 𝑛 is even.             
(III) and (IV) imply 𝑓 is a Gaussian twin neighborhood prime labeling when 𝑛 is even. 
From cases (i) and (ii), 𝑓 is a Gaussian twin neighborhood prime labeling. 

Hence, double di-double alternating fan 𝐷𝐷𝐷𝐴𝐹𝑛⃗⃗  ⃗ is a Gaussian twin neighborhood 
prime digraph. 

 

5 Conclusions 
In this article, we established Gaussian twin neighborhood prime labeling in fan 

and double fan digraphs. In this way, we extend our thoughts to different types of 

labeling in digraphs. 

 

 

References 

[1] Alex Rosa, On certain valuation of the vertices of a graph, In theory of graphs,      

Gordon and Breach, 1967. 

[2] Gallian J A, A Dynamic Survey of Graph Labeling, The Electronic Journal of         

Combinatorics, #DS6, 2019. 

[3] Harary F, Graph Theory, Addison - Wesley Publishing Company, Reading, Mass,       

1972. 

30



Gaussian twin neighborhood prime labeling of fan digraphs 

 

 

[4] Klee S, Lehmann H and Park A, Prime labeling of families of trees with Gaussian      

Integers, AKCE International Journal of Graphs and Combinatorics, 13(2): 165-176, 

2016. 

[5] Lehmann H and Park A, Prime labeling of small trees with Gaussian integers, Rose      

Hulman Undergraduate Mathematics Journal, 17(1): 72-97, 2016. 

[6] Palani K and Shunmugapriya A, Gaussian twin neighborhood prime labeling on        

Wheel digraphs, Industrial Engineering Journal, 15(11): 226-243, 2022. 

[7] Rajesh Kumar T J & Mathew Varkey T K, Gaussian prime labeling of some classes      

of graphs and cycles, Annals of Pure and Applied Mathematics, 16 (1): 133-140,      

2018. 

[8] Rokad A H, Product cordial labeling of double wheel and double fan related graphs,       

Kragujevac Journal of Mathematics, 43(1): 7-13, 2019. 

 

 

31