CONNECTED EDGE AND ENTIRE DOMINATION OF

INVOLUTORY ADDITION CAYLEY GRAPH

1E.Lavanya  2, *M.Siva Parvathi
1,2,3Department of Applied Mathematics, Sri Padmavati Mahila Visvavidyalayam, Tirupati, 

Andhra Pradesh, India

Corresponding Author: M.Siva Parvathi

Article History: Received 20.09.2024     Revised: 24.10.2024 Accepted: 30.11.2024

ABSTRACT

Graph theory is one of the most advanced branches of discrete mathematics with variety of 

applications to different branches of Science and Technology. For a positive integer 𝑛 > 1, 

the involutory addition Cayley graph 𝐺+(𝑍𝑛, 𝐼𝑣), is the graph whose vertex set is 𝑍𝑛 =

{0,1,2,3,… , 𝑛 − 1} and edge set 𝐸(𝐺𝑛) = {𝑥𝑦 /𝑥, 𝑦 𝜖 𝑍𝑛, 𝑥 + 𝑦 𝜖 𝐼𝑣}, where 𝐼𝑣 = {𝑥 𝜖 𝑍𝑛 ∶

𝑥2 ≡ 1(𝑚𝑜𝑑 𝑛)}  is the set of involutory elements of 𝑍𝑛.By taking Involutory Addition 

Cayley Graphs 𝐺+(𝑍𝑛 , 𝐼𝑣) the author’s evaluated graph related Connected edge domination 

numbers and Entire domination numbers. In this paper, Connected edge domination number, 

Entire domination number of the Involutory addition cayley graphs 𝐺+(𝑍𝑛 , 𝐼𝑣) were

discussed.

Keywords: Involutory addition cayley graphs, Connected edge dominating sets, Entire 

dominating sets, Connected edge Domination number, Entire domination number.

AMS Subject Classification: 05C40, 05C19.

1. INTRODUCTION

A graph 𝐺(𝑉, 𝐸) is a mathematical object that may be thought of as a collection of edges 

and a set of vertices that connect any or all of the vertices. In a graph 𝐺, two vertices are 

considered neighboring if an edge joins them; otherwise, the edge is considered non-adjacent. 

We indicate that a graph 𝐺 has 𝑉 (𝐺) vertices and 𝐸(𝐺) edges, accordingly. The cardinality 

of 𝑉 (𝐺) is the definition of the order of 𝐺. The cardinality of 𝐸(𝐺) is represented by |𝐸|, and 

that of 𝑉 (𝐺) by |𝑉 |. The number of edges that occur with a vertex 𝑣 in a graph 𝐺 is known 

as its degree, or deg(𝑣).

Involutory cayley graphs were introduced by Venkata Anusha et al. [8], who also looked into 

some of its characteristics. Involutory addition Cayley graphs were first shown by Shanmuga 

Priya et al. [7], who also examined several of their characteristics.In addition to studying the

characterization of the set of involutory elements of (𝑍𝑛,⨁,⨀), Prameela Rani et al. [3,4]

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investigated certain properties of the domatic number and dominance of the involutory cayley 

graph.  

Lavanya et al. [6] investigated the involutory addition cayley graphs edge domination number 

and total edge domination number. 

For a positive integer 𝑛 > 1, the involutory addition Cayley graph 𝐺+(𝑍𝑛, 𝐼𝑣),  is the graph 

whose vertex set is 𝑍𝑛 = {0,1,2,3,… , 𝑛 − 1}  and edge set 𝐸(𝐺𝑛) = {𝑥𝑦  /𝑥, 𝑦 𝜖 𝑍𝑛 , 𝑥 +

𝑦 𝜖 𝐼𝑣}, where 𝐼𝑣 = {𝑥 𝜖 𝑍𝑛 ∶ 𝑥
2 ≡ 1(𝑚𝑜𝑑 𝑛)}  is the set of involutory elements of 𝑍𝑛. 

The theory of domination in graphs was introduced in 1958 by Claude Berge[1] in which he 

used the concept ‘coefficient of external stability’ to refer the domination number of a graph. 

In 1962.Dominating sets of edges were studied by Mitchell and Hedetniemi[3].Connected 

edge domination was introduced by Kulli and Sigarkanti[5].Entire domination was 

introduced by Kulli[6]. Kulli Sigarkanti and Soner established a relationship between the 

domination, edge domination and entire domination number. 

2 CONNECTED EDGE DOMINATION 

The theory of domination in graphs was introduced in 1958 by Claude Berge[1] in which 

he used the concept ‘coefficient of external stability’ to refer the domination number of a 

graph. In 1962, Oystein Ore[4] wrote another book on graph theory, in which he studied the 

concept of domination using the terms ‘dominating set’ and ‘domination number’ with 

notation 𝑑(𝐺) for the first time. Cockayne et al[2] discussed the review of results and 

applications concerning dominating sets in graphs.Dominating sets of edges were studied by 

Mitchell and Hedetniemi[3].Connected edge domination was introduced by Kulli and 

Sigarkanti[5]. 

An edge domination set 𝐹 of a graph 𝐺 is a connected edge domination set if the induced 

subgraph < 𝐹 > is connected. The connected edge domination number 𝛾𝑐
′(𝐺) of 𝐺 is the 

minimum cardinality of a connected edge dominating set.  

Theorem 2.1: For the involutory addition cayley graph 𝐺+(𝑍𝑛 , 𝐼𝑣), if n is even, 𝑛 > 2 then 

the connected edge domination number is  

𝛾𝑐
′(𝐺+(𝑍𝑛, 𝐼𝑣)) = 𝑛 − 2 

Proof: Consider a graph𝐺+(𝑍𝑛 , 𝐼𝑣) with vertex set 𝑍𝑛 = {0,1,2,3,… , 𝑛 − 1} where 𝐼𝑣 denotes 

the set of involutory elements in 𝑍𝑛. 

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Let 𝑛 be even and 𝑛 > 2. |𝐸| = 𝑛 or 2𝑛 

If |𝐸| = 𝑛, graph is a Hamilton cycle. If |𝐸| = 2𝑛, graph contains two Hamilton cycles, each 

contains `𝑛′ number of edges. From first Hamilton cycle, consider 𝐹 = {𝑒𝑖/𝑒𝑖 ≠ 𝑒1, 𝑒2}. 

Now every edge in 𝐸 − 𝐹 is adjacent to atleast one edge in 𝐹, 𝐹 is induced subgraph of 

𝐺+(𝑍𝑛 , 𝐼𝑣) and connected. Then 𝐹 becomes connected edge dominating set of 𝐺+(𝑍𝑛, 𝐼𝑣) and 

it is minimum. Therefore |𝐹| = |𝐸| − 2   (𝐸 = set of edges in one path) = 𝑛 − 2. 

Hence, 𝛾𝑐
′(𝐺+(𝑍𝑛 , 𝐼𝑣)) = 𝑛 − 2. 

Theorem 2.2: For the involutory addition cayley graph 𝐺+(𝑍𝑛 , 𝐼𝑣), if 𝑛 is odd, 𝑛 > 3 then 

the connected edge domination number is  

𝛾𝑐
′(𝐺+(𝑍𝑛, 𝐼𝑣)) = 𝑛 − 3 

Proof: Consider a graph𝐺+(𝑍𝑛 , 𝐼𝑣) with vertex set 𝑍𝑛 = {0,1,2,3,… , 𝑛 − 1} where 𝐼𝑣 denotes 

the set of involutory elements in 𝑍𝑛. 

Let 𝑛 be odd, 𝑛 > 3. |𝐸| = 𝑛 − 1 or 𝑛 − 2. 

If |𝐸| = 𝑛 − 1, graph is a path. If |𝐸| = 2𝑛 − 2, graph contains two paths and each path 

contains 𝑛 − 1 number of edges. From first path, consider 𝐹 = {𝑒𝑖/𝑒𝑖 ≠ 𝑒1, 𝑒𝑛}. 

Now every edge in 𝐸 − 𝐹 is adjacent to at least one edge in 𝐹, 𝐹 is induced subgraph of 

𝐺+(𝑍𝑛 , 𝐼𝑣) and connected. Then 𝐹 becomes connected edge dominating set of 𝐺+(𝑍𝑛, 𝐼𝑣) and 

it is minimum. Therefore |𝐹| = |𝐸| − 2   (𝐸 = set of edges in one path)= 𝑛 − 3. 

Hence 𝛾𝑐
′(𝐺+(𝑍𝑛, 𝐼𝑣)) = 𝑛 − 3 

3.   ENTIRE DOMINATION  

The term domination was first used by Ore[4]. Cockayne et al[2] discussed the review of 

results and applications concerning dominating sets in graphs. Dominating sets of edges were 

studied by Mitchell and Hedetniemi[3]. Entire domination was introduced by Kulli[6]. Kulli 

Singarkanti and Soner established a relationship between the domination, edge domination 

and entire domination number. The vertices and edges of a graph 𝐺 are called the elements of 

𝐺. Let 𝑥 and 𝑦 be two elements of 𝐺. We say 𝑥 dominates 𝑦 if 𝑥 = 𝑦 or if 𝑥 and 𝑦 are 

adjacent or incident. Thus a vertex 𝑣 of 𝐺 is said to dominate itself, all vertices adjacent to 𝑣 

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and all edges incident to it. Similarly, an edge 𝑒 of 𝐺 dominates itself, the two end vertices of 

𝑒 and all edges adjacent to 𝑒. 

A set 𝑆 of elements of 𝐺 is an entire dominating set if every element not in 𝑆 is either 

adjacent or incident to atleast one element in 𝑆. Equivalently, a set 𝑆 of elements of 𝐺 is an 

entire dominating set if each element in 𝐺 is dominated by some element in 𝑆. The entire 

domination number 𝛾𝑒𝑛(𝐺) is the cardinality of a smallest entire dominating set. 

 

Theorem 3.1: For the involutory addition cayley graph 𝐺+(𝑍𝑛 , 𝐼𝑣), if n is even |𝐼𝑣| = 2, 𝑛 >

2, then the Entire domination number is  

𝛾𝑒𝑛(𝐺
+(𝑍𝑛 , 𝐼𝑣)) = {

2𝑛

5
                   𝑖𝑓 2𝑛 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 5

2𝑛 − 𝑟

5
+ 1             𝑖𝑓 2𝑛 𝑛𝑜𝑡 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 5  

 

where 𝑟 is the remainder when 2𝑛 divided by 5. 

Proof: Consider a graph𝐺+(𝑍𝑛 , 𝐼𝑣) with vertex set 𝑍𝑛 = {0,1,2,3,… , 𝑛 − 1} where 𝐼𝑣 denotes 

the set of involutory elements in 𝑍𝑛 and 𝑛 is even, 𝑛 > 2, |𝑒| = 𝑛. 

Case 1: if 2𝑛 divisible by 5 (start Entire dominating set with edge) 

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Then consider 𝑆 = {𝑒𝑖, 𝑣𝑗/𝑖 = 0,5,10,… 𝑎𝑛𝑑 𝑗 = 3,8,13,… } 

Now every edge in 𝐺+(𝑍𝑛, 𝐼𝑣) adjacent to atleast one edge in 𝑆 or incident with atleast one 

vertex in 𝑆 also every vertex in 𝐺+(𝑍𝑛 , 𝐼𝑣) either incident with atleast one edge or adjacent to 

atleast one vertex in 𝑆. Therefore 𝑆 becomes entire domination set of 𝐺+(𝑍𝑛 , 𝐼𝑣) and it is 

minimum. 

|𝑆| =
|𝐸| + |𝑉|

5
=
𝑛 + 𝑛

5
=
2𝑛

5
 

Therefore 𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

2𝑛

5
 

Case 2: If 2𝑛 not divisible by 5 

Then consider 𝑆1 = {𝑒𝑖, 𝑣𝑗/𝑖 = 0,5,10,… 𝑎𝑛𝑑 𝑗 = 3,8,13,… } 

Take 𝑆 = 𝑆1 ∪ (𝐺 − 𝑆1). Now every edge in 𝐺+(𝑍𝑛, 𝐼𝑣) adjacent to atleast one edge in 𝑆 or 

incident with atleast one vertex in 𝑆 also every vertex in 𝐺+(𝑍𝑛 , 𝐼𝑣) either incident with 

atleast one edge or adjacent to atleast one vertex in 𝑆. Therefore 𝑆 becomes entire domination 

set of 𝐺+(𝑍𝑛 , 𝐼𝑣) and it is minimum. 

Consider 𝑟 is the remainder when 2𝑛 divisible by 5 

|𝑆| =
|𝐸| + |𝑉| − 𝑟

5
+ 1 =

𝑛 + 𝑛 − 𝑟

5
+ 1 =

2𝑛 − 𝑟

5
+ 1. 

Therefore 𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

2𝑛−𝑟

5
+ 1 

Theorem 3.2: For the involutory addition cayley graph 𝐺+(𝑍𝑛 , 𝐼𝑣), if 𝑛 is odd |𝐼𝑣| = 2,  

𝑛 > 2 then the entire domination number  

𝛾𝑒𝑛(𝐺
+(𝑍𝑛 , 𝐼𝑣)) = {

2𝑛 − 1

5
                   𝑖𝑓 2𝑛 − 1 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 5

2𝑛 − 𝑟 + 4

5
             𝑖𝑓 2𝑛 − 1 𝑛𝑜𝑡 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 5  

 

where 𝑟 is the remainder when 2𝑛 − 1 divided by 5. 

Proof: Consider a graph𝐺+(𝑍𝑛 , 𝐼𝑣) with vertex set 𝑍𝑛 = {0,1,2,3,… , 𝑛 − 1} where 𝐼𝑣 denotes 

the set of involutory elements in 𝑍𝑛 and 𝑛 is odd, 𝑛 > 2, |𝑒| = 𝑛 − 1 

Case 1: If 2𝑛 − 1 divisible by 5. Consider 𝑆 = {𝑒𝑖, 𝑣𝑗/𝑖 = 0,5,10,… 𝑎𝑛𝑑 𝑗 = 3,8,13,… }. 

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Now every edge in 𝐺+(𝑍𝑛, 𝐼𝑣) adjacent to atleast one edge in 𝑆 or incident with atleast one 

vertex in 𝑆 also every vertex in 𝐺+(𝑍𝑛 , 𝐼𝑣) either incident with atleast one edge or adjacent to 

atleast one vertex in 𝑆. Therefore 𝑆 becomes entire domination set of 𝐺+(𝑍𝑛 , 𝐼𝑣) and it is 

minimum. 

|𝑆| =
|𝐸| + |𝑉|

5
=
𝑛 + 𝑛 − 1

5
=
2𝑛 − 1

5
. 

Therefore 𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

2𝑛−1

5
 

Case 2: If 2𝑛 − 1 not divisible by 5. Then consider 𝑆1 = {𝑒𝑖, 𝑣𝑗/𝑖 = 0,5,10,… 𝑎𝑛𝑑 𝑗 =

3,8,13,… }. Take 𝑆 = 𝑆1 ∪ (𝐺 − 𝑆1). Now every edge in 𝐺+(𝑍𝑛 , 𝐼𝑣) adjacent to atleast one 

edge in 𝑆 or incident with atleast one vertex in 𝑆 also every vertex in 𝐺+(𝑍𝑛, 𝐼𝑣) either 

incident with atleast one edge or adjacent to atleast one vertex in 𝑆 

Therefore 𝑆 becomes entire domination set of 𝐺+(𝑍𝑛 , 𝐼𝑣) and it is minimum. 

|𝑆| =
|𝐸| + |𝑉|

5
+ 1 =

(𝑛 + 𝑛 − 1) − 𝑟

5
+ 1 =

2𝑛 − 𝑟 − 1

5
+ 1 =

2𝑛 − 𝑟 + 4

5
. 

Therefore 𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

2𝑛−𝑟+4

5
 

Theorem 3.3: For the involutory addition cayley graph 𝐺+(𝑍𝑛 , 𝐼𝑣), if n is even and𝑛 >

2, |𝐼𝑣| = 4, 𝑛 = 2
𝑞, 𝑞 = 3,4,5,…  then the entire domination number is  

𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

𝑛

2
. 

Proof: Consider a graph𝐺+(𝑍𝑛 , 𝐼𝑣) with vertex set 𝑍𝑛 = {0,1,2,3,… , 𝑛 − 1} where 𝐼𝑣 denotes 

the set of involutory elements in 𝑍𝑛 and 𝑛 is even, 𝑛 > 2, |𝑒| = 2𝑛. 

𝐺+(𝑍𝑛 , 𝐼𝑣) contains two Hamiltonian cycles. From first Hamiltonian cycle. 

Then consider  𝑆 = {𝑒𝑖𝑗/0 ≤ 𝑖 ≤
𝑛

2
− 1 𝑎𝑛𝑑  

𝑛

2
≤ 𝑗 ≤ 𝑛 − 1}. Now every edge in 𝐺+(𝑍𝑛, 𝐼𝑣) 

adjacent to atleast one edge in 𝑆 or incident with atleast one vertex in 𝑆 also every vertex in 

𝐺+(𝑍𝑛 , 𝐼𝑣) either incident with atleast one edge or adjacent to atleast one vertex in 𝑆. 

Therefore 𝑆 becomes entire domination set of 𝐺+(𝑍𝑛 , 𝐼𝑣) and it is minimum, |𝑆| =
|𝑉|

2
=

𝑛

2
 

Therefore 𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

𝑛

2
. 

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Theorem 3.4: For the involutory addition cayley graph 𝐺+(𝑍𝑛 , 𝐼𝑣), if n is even and 𝑛 >

2, |𝐼𝑣| = 4, 𝑛 = 4𝑝, 𝑝 > 2, 𝑝 𝑖𝑠 𝑝𝑟𝑖𝑚𝑒  then the entire domination number is  

𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

{
 
 

 
 

𝑛

2
                                        𝑓𝑜𝑟 𝑝 = 3        

2𝑛

5
             𝑖𝑓 2𝑛 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 5(𝑝 > 3)

2𝑛 − 𝑟 + 5

5
   𝑖𝑓 2𝑛 𝑛𝑜𝑡 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 5(𝑝 > 3)

 

where 𝑟 is the remainder when 2𝑛 divided by 5. 

Proof: Consider a graph𝐺+(𝑍𝑛 , 𝐼𝑣) with vertex set 𝑍𝑛 = {0,1,2,3,… , 𝑛 − 1} where 𝐼𝑣 denotes 

the set of involutory elements in 𝑍𝑛 and 𝑛 is even, 𝑛 > 2, |𝐼𝑣| = 4, 𝑛 = 4𝑝,  𝑝 > 2, 𝑝 is 

prime|𝑒| = 2𝑛. 𝐺+(𝑍𝑛, 𝐼𝑣) contains two Hamiltonian cycles, each Hamiltonian cycle contains 

𝑛 number of elements. 

Case 1: From first Hamiltonian cycle, for 𝑝 = 3. Then consider  𝑆 = {
𝑒𝑖𝑗

0
≤ 𝑖 ≤

𝑛

2
−

1 𝑎𝑛𝑑  
𝑛

2
≤ 𝑗 ≤ 𝑛 − 1}. Now every edge in 𝐺+(𝑍𝑛, 𝐼𝑣) adjacent to atleast one edge in 𝑆 or 

incident with atleast one vertex in 𝑆 also every vertex in 𝐺+(𝑍𝑛 , 𝐼𝑣) either incident with 

atleast one edge or adjacent to atleast one vertex in 𝑆. Therefore 𝑆 becomes entire domination 

set of 𝐺+(𝑍𝑛 , 𝐼𝑣) and it is minimum, |𝑆| =
|𝑉|

2
=

𝑛

2
. Therefore 𝛾𝑒𝑛(𝐺

+(𝑍𝑛, 𝐼𝑣)) =
𝑛

2
 

Case 2: From first Hamiltonian cycle, for 𝑝 > 3, If 2𝑛 divisible by 5. Then consider  𝑆 =

{𝑒𝑖, 𝑣𝑗/𝑖 = 0,5,10,… 𝑎𝑛𝑑 𝑗 = 3,8,13,… }. Now every edge in 𝐺+(𝑍𝑛 , 𝐼𝑣) adjacent to atleast 

one edge in 𝑆 or incident with atleast one vertex in 𝑆 also every vertex in 𝐺+(𝑍𝑛 , 𝐼𝑣) either 

incident with atleast one edge or adjacent to atleast one vertex in 𝑆. Therefore 𝑆 becomes 

entire domination set of 𝐺+(𝑍𝑛 , 𝐼𝑣) and it is minimum, 

|𝑆| =

|𝐸|

2
+ |𝑉|

5
=

2𝑛

2
+ 𝑛

5
=
𝑛 + 𝑛

5
=
2𝑛

5
. 

Therefore 𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

2𝑛

5
 

Case 3: From first Hamiltonian cycle, for 𝑝 > 3, If 2𝑛 not divisible by 5. Then consider 

𝑆1 = {𝑒𝑖, 𝑣𝑗/𝑖 = 0,5,10,… 𝑎𝑛𝑑 𝑗 = 3,8,13,… } 

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Take 𝑆 = 𝑆1 ∪ (𝐺 − 𝑆1). Now every edge in 𝐺+(𝑍𝑛, 𝐼𝑣) adjacent to atleast one edge in 𝑆 or 

incident with atleast one vertex in 𝑆 also every vertex in 𝐺+(𝑍𝑛 , 𝐼𝑣) either incident with 

atleast one edge or adjacent to atleast one vertex in 𝑆. Therefore 𝑆 becomes entire domination 

set of 𝐺+(𝑍𝑛 , 𝐼𝑣) and it is minimum,  

|𝑆| =
(
|𝐸|

2
+ |𝑉|) − 𝑟

5
+ 1 =

(
2𝑛

2
+ 𝑛) − 𝑟

5
+ 1 =

(𝑛 + 𝑛) − 𝑟

5
+ 1 =

2𝑛 − 𝑟 + 5

5
. 

 

Therefore 𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

2𝑛−𝑟+5

5
 

Theorem 3.5: For the involutory addition cayley graph 𝐺+(𝑍𝑛 , 𝐼𝑣), if n is odd and 𝑛 >

2, |𝐼𝑣| = 4, 𝑛 = 3𝑝, 𝑝 > 3  then the entire domination number is  

𝛾𝑒𝑛(𝐺
+(𝑍𝑛 , 𝐼𝑣)) =

𝑛 + 1

2
. 

Proof: Consider a graph𝐺+(𝑍𝑛 , 𝐼𝑣) with vertex set 𝑍𝑛 = {0,1,2,3,… , 𝑛 − 1} where 𝐼𝑣 denotes 

the set of involutory elements in 𝑍𝑛 and 𝑛 is odd, 𝑛 > 2,|𝐼𝑣| = 4, 𝑛 = 3𝑝, 𝑝 > 3, 𝑝 is 

prime|𝑒| = 2𝑛 − 2. 𝐺+(𝑍𝑛 , 𝐼𝑣) contains two paths and each path contains 𝑛 − 1 number of 

edges. From first Hamiltonian cycle, consider  𝑆 = {𝑒𝑖𝑗,
𝑣𝑘

0
≤ 𝑖 ≤

𝑛−3

2
 ,

𝑛+1

2
≤ 𝑗 ≤ 𝑛 −

1 𝑎𝑛𝑑 𝑘 =
𝑛−1

2
}. Now every edge in 𝐺+(𝑍𝑛, 𝐼𝑣) adjacent to atleast one edge in 𝑆 or incident 

with atleast one vertex in 𝑆 also every vertex in 𝐺+(𝑍𝑛, 𝐼𝑣) either incident with atleast one 

edge or adjacent to atleast one vertex in 𝑆. Therefore 𝑆 becomes entire domination set of 

𝐺+(𝑍𝑛 , 𝐼𝑣) and it is minimum. 

|𝑆| =
|𝑉| − 1

2
+ 1 =

𝑛 − 1

2
+ 1 =

𝑛 + 1

2
. 

Therefore 𝛾𝑒𝑛(𝐺
+(𝑍𝑛, 𝐼𝑣)) =

𝑛+1

2
 

CONCLUSION 

Using involutory addition cayley graphs, it is interesting to find the Connected edge 

domination number and Entire domination number of involutory addition cayley graphs and 

the authors have also studied this aspect. 

 

Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 31 No. 7s (2024)

https://internationalpubls.com 788



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Communications on Applied Nonlinear Analysis
ISSN: 1074-133X
Vol 31 No. 7s (2024)

https://internationalpubls.com 789


	Article History: Received 20.09.2024     Revised: 24.10.2024 Accepted: 30.11.2024