EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 1098-1112
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Some Parameters of the Central Graphs of the
Identity Graphs of Finite Cyclic Groups

Clarence T. Alib1,∗, Daryl M. Magpantay2

1 College of Arts and Sciences, Camarines Sur Polytechnic Colleges, Nabua,
Camarines Sur, Philippines
2 College of Arts and Sciences, Batangas State University, Batangas City, Philippines

Abstract. The interplay of groups and graphs has been a subject of interest by mathematics
researchers nowadays. One particular instance is the identity graph of a group introduced by
Kandasamy [6]. Moreover, the concept of a central graph of any graph is widely used by many
graph theorist. The central graph of a graph G denoted by C(G) can be obtained by subdividing
the edge of G exactly once and joining all the nonadjacent vertices of G in C(G). In this paper,
we construct the central graph of the identity graph of finite cyclic group and investigate some of
its graph properties.

2020 Mathematics Subject Classifications: 05C07, 05C12, 05C25

Key Words and Phrases: Cyclic group, Identity graph of a group, Handshaking lemma, Central
graph of a graph, Distance, Eccentricities, Radius, Diameter, Center of a graph, Periphery of a
graph, Girth

1. Introduction

The interconnection between different fields of mathematics has been a subject of
interest by mathematics researchers nowadays. One particular instance is the interplay of
groups and graphs where the existence of the researches contribute to the productive area
of mathematics.

The collaboration of the two areas of mathematics mentioned gain attention to the
mathematical community because of its elegant results. Some of which are order divisor
graphs of finite groups [8], graphs and classes of finite groups [1], the power graph of a
finite group [2], commuting graphs of dihedral type groups [7] and many more.

In 2009, Kandasamy and Smarandache [6], wrote a short book entitled ”Groups as
Graphs”. They represented every finite group in the form of graph and choose to call
these graphs as identity graphs since the main role of obtaining the graph is played by

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4416

Email addresses: clarencealib@cspc.edu.ph (C. Alib),
darylmagpantay@g.batstate-u.edu.ph (D. Magpantay)

https://www.ejpam.com 1098 © 2022 EJPAM All rights reserved.



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1099

the identity element of the group. In this paper, we construct the central graph of the
identity graph of finite cyclic group and find its properties. Graph parameters like distance,
eccentricities, radius, diameter, girth, center, periphery are included.

2. Preliminaries

In this paper, all groups considered are finite cyclic groups.

Definition 2.1. A group is a nonempty set G together with a binary operation

(a, b) 7→ a · b : G × G → G

satisfying the following properties:

G1: (associativity) for all a, b, c ∈ G ,

(a · b) · c = a · (b · c);

G2: (existence of an identity element) there exists an element e ∈ G such that

a · e = a = e · a

for all a ∈ G ;
G3: (existence of inverse element) for each a ∈ G , there exists an a′ ∈ G such that

a · a′ = e = a′ · a

Note that the notation (a, b) 7→ a · b : G × G → G means that for any two elements
a, b ∈ G , a ·b also belongs to G . This is called closure property. This group G together
with the binary operation · is written as ( G , ·) but in this paper we will abbreviate ( G , ·)
to G . Also, we usually write ab for a · b and 1 for e; alternatively, we write a+ b for a · b
and 0 for e. In the first case, the group is said to be multiplicative, and in the second, it is
said to be additive. In some standard Group Theory books, · is usually written as * but
in this paper we use · as the binary operation for the group G .

Example 1. Let G = {1,−1}. G is a group under usual multiplication since G satisfices
all the properties of a group; that is, G is nonempty, closed under usual multiplication;
i.e. {(1× 1 = 1, 1×−1 = −1,−1×−1 = 1}. Since usual multiplication is associative G
is also associative. 1 ∈ G is the identity element of G and 1 and −1 are self-inverse
elements.

Definition 2.2. A group G is called cyclic if there exists a ∈ G such that

G = {an | n ∈ Z}.

Such an element a is called a generator of G . We may indicate that G is a cyclic group
generated by a by writing G = ⟨a⟩. We denote this group as Cn of order n.



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1100

Example 2. Consider the group Z6 = {0, 1, 2, 3, 4, 5} under usual addition modulo 6.
Note that 1 ∈ Z6 such that ⟨1⟩ = Z6. We can also verify that ⟨5⟩ = Z6, that is,

⟨5⟩ = {1 · 5 ≡ 5, 2 · 5 ≡ 4, 3 · 5 ≡ 3, 4 · 5 ≡ 2, 5 · 5 ≡ 1, 6 · 5 ≡ 0}
= {0 , 1, 2, 3, 4, 5} = Z6

Definition 2.3. [6] Given a group G with e as the identity element, define the identity
graph ΓG to have the vertex set G and the edge set E(ΓG ) satisfying two conditions:
(i) For every x, y ∈ G (x ̸= e, y ̸= e, x ̸= y), x and y are adjacent in ΓG if and only if

x · y = e ;

(ii) For each x ∈ G (x ̸= e), x and e are adjacent in ΓG .

Definition 2.4. [9] Given a group G , a line in the identity graph ΓG is an edge (x, e)
such that the degree of a vertex x ∈ G is one. The number of lines in the identity graph
ΓG is denoted by line(G ).

Definition 2.5. [9] A triangle in the identity graph ΓG is a subgraph which is isomorphic
to the cycle graph of length three. The number of triangles in the identity graph ΓG is
denoted by tri(G ).

Consider the identity graphs of cyclic groups below.

.................................... ....................................

....................................

.................................... .................................... ........................................................................

....................................
....................................

....................................

.................................... ....................................
ΓC5 :

e

g g4 g2 g3
..............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.. ....
................................

................................................................................................................
..........
.........
.........
.........
.........
.........
.........
.........
.......

....................................

....................................................................................................................

....................................

................................................................................................................

..................................................................................................................................... ....................................

....................................

................................................................
.................. ....................................

......................................................

....................................

............
...........
...........
...........
...........
...........
......

....................................

.......................................................................................................

....................................

..............
.............
.............
.............
....

....................................

................................................................................................ ....................................

....................................

............................................................... ....................................

.............................................................................................................

...........
..........
..........
..........
..........
.......
....................................

....................................

ΓC7 :

e
g

g6g2

g5

g3 g4

....................................

.................................... ....................................

....................................

....................................

.................................... ....................................

....................................

....................................

...................................................................................... ....................................

..................................................................................................

..........
.........
.........
.........
.........
.........
.........
.........
.......
....................................

....................................

.......................................................................................................... ....................................

.........

........

........

........

........

........

........

.....
....................................

..............................
.............................

.............................
..................

....................................

....................................

..........
.........
.........
.........
.........
.........
.........
.........
.......
....................................

.........................................................................................

.............................................................................. ....................................

........................................................................................................................................

....................................

.........

........

........

........

........

........

........

.....

....................................

.......................
......................

......................
......................

.............. ....................................

....................................

ΓC9 :

e g

g8

g2g7

g3

g6

g4g5

....................................

.................................... ....................................

....................................

....................................

....................................

.................................... ....................................

....................................

....................................

....................................

............................................................................................ ....................................

......................................................................................................................

..........
.........
.........
.........
.........
.........
.........
.........
.........
.. ..
..................................

....................................

......................................................................................................... .............................................
........
........
........
........
........
.......
....................................

...............................................................................................................................
....................................

.................
................
................
................
................
..... ...
.................................

........................................................

....................................

..........
.........
.........
.........
.........
.........
.........
.........
.......

....................................

..................................................................................................................

....................................

...............
..............
..............
..............
.......

....................................

................................................................................. ....................................

....................................

.................
................
................
................
................
................
......

.................................... .........
........
........
........
........
........
....
....................................

....................................................................................... ....................................

....................................

ΓC11 :

e
g

g10

g2

g9g3
g8

g4

g7

g5 g6

Lemma 1. [9] For a group G of order n, we have

line(G ) + 2tri(G ) = n− 1.

Corollary 1. [4] If G is a cyclic group of odd order, then G has the identity graph
ΓG which is formed only by triangles with no lines.



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1101

....................................

.................................... ....................................

....................................

.................................... .................................... ........................................................................

....................................
....................................

....................................

.................................... ....................................

....................................ΓC6 :
e

g g5 g2 g4

g3

..............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.. ....
................................

................................................................................................................
..........
.........
.........
.........
.........
.........
.........
.........
.......

....................................

....................................................................................................................

....................................

................................................................................................................

..................................................................................................................................... ....................................

....................................
.........
........
........
........
........
........
........
.....

................................................................
.................. ....................................

......................................................

....................................

............
...........
...........
...........
...........
...........
......

....................................

.......................................................................................................

....................................

..............
.............
.............
.............
....

....................................

................................................................................................ ....................................

....................................

............................................................... ....................................

.............................................................................................................

...........
..........
..........
..........
..........
.......
....................................

....................................

................
...............
...............
...............
...............
...............
...............
...............
...............
...............
.

ΓC8 :

e
g

g7g2

g6

g3 g5 g4

....................................

.................................... ....................................

....................................

....................................

.................................... ....................................

....................................

....................................

....................................

...................................................................................... ....................................

..................................................................................................

..........
.........
.........
.........
.........
.........
.........
.........
.......
....................................

....................................

.......................................................................................................... ....................................

.........

........

........

........

........

........

........

.....
....................................

..............................
.............................

.............................
..................

....................................

....................................

..........
.........
.........
.........
.........
.........
.........
.........
.......
....................................

.........................................................................................

.............................................................................. ....................................

........................................................................................................................................

....................................

.........

........

........

........

........

........

........

.....

....................................

.......................
......................

......................
......................

.............. ....................................

....................................
.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
..

ΓC10 :

e g

g9

g2g8

g3

g7

g4g6 g5

....................................

.................................... ....................................

....................................

....................................

....................................

.................................... ....................................

....................................

....................................

....................................

....................................

............................................................................................ ....................................

......................................................................................................................

..........
.........
.........
.........
.........
.........
.........
.........
.........
.. ..
..................................

....................................

......................................................................................................... .............................................
........
........
........
........
........
.......
....................................

...............................................................................................................................
....................................

.................
................
................
................
................
..... ...
.................................

........................................................

....................................

..........
.........
.........
.........
.........
.........
.........
.........
.......

....................................

..................................................................................................................

....................................

...............
..............
..............
..............
.......

....................................

................................................................................. ....................................

....................................

.................
................
................
................
................
................
......

.................................... .........
........
........
........
........
........
....
....................................

....................................................................................... ....................................

....................................
.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
..

ΓC12 :

e
g

g11

g2

g10g3
g9

g4

g8

g5 g7 g6

Figure 1: The identity graphs of cyclic groups C5, C6 C7, C8 C9, C10, C11, and C12

.

Theorem 1. [6] If Cn is a cyclic group of order n (n is odd), then the identity graph ΓCn

of Cn is formed by n−1
2 triangles.

Theorem 2. [6] If Cn is a cyclic group of order n (n is even), then the identity graph
ΓCn of Cn is formed by n−2

2 triangles and a line.

Theorem 3. [9] For a cyclic group Cn of order n, we have

line(Cn) =

{
0, n = 2k + 1
1, n = 2k + 2

and

tri(Cn) =

{
n−1
2 , n = 2k + 1

n−2
2 , n = 2k + 2.

Theorem 4. [3], [5] Let Cn be the cyclic group of order n and ΓCn be the identity graph
associated with Cn. The size of ΓCn is

|E(ΓCn)| =

{
3n−3

2 , n is odd,
3n−4

2 , n is even.

Lemma 2. (Handshaking Lemma)
In a graph G, we have∑

x∈V (G) deg(x) = 2|E| where |E| is the total number of edges.



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1102

The distance d(u, v) between u, v ∈ V (G) is the length of a shortest u− v path in
the graph G. The eccentricity of a vertex u ∈ V (G) is e(u) = max{d(u, v) | u ∈ V (G)}.
The diameter of a graph G is diam = max{e(u) | u ∈ V (G)}. The radius of a graph G is
rad = min{e(u) | u ∈ V (G)}. If e(u) = diam(G), then u is a peripheral vertex. The set
of all such vertices make the periphery of G. If e(u) = rad(G), the vertex u is a central
vertex. The set of all such vertices make the center of G. The girth of a graph G denoted
by gir(G) is the length of the shortest cycle (if any) in G. These graph parameters will
be considered as the focus of the study.

3. Definition of Central Graph of ΓCn.

Consider the cyclic group Cn of order n (≥ 2). By definition 2.3, ΓCn is the identity
graph associated with the group Cn. In this section, central graph of ΓCn will be discussed.

Definition 3.1. [10] Let Cn be a finite cyclic group of order n (≥ 2) and ΓCn be the identity
graph of Cn. The central graph of ΓCn denoted by C(ΓCn) is obtained by subdividing the
edges of ΓCn exactly once and joining all the non-adjacent vertices of ΓCn in C(ΓCn).

Throughout this paper, we fix a notation for the vertex-set and the edge-set of C(ΓCn).
For any integer n ≥ 2, let ΓCn be the identity graph of cyclic group Cn and let V (ΓCn) =
{v0, v1, · · · , vn−1}. Consider its central graph C(ΓCn). The vertex-set and edge-set of
C(ΓCn) are V (C(ΓCn)) = V (ΓCn)

⋃
C, where C = {cij : (vi, vj) ∈ E(ΓCn)} and

E(C(ΓCn)) = {(vi, cij), (vj , cij) : (vi, vj) ∈ E(ΓCn)}
⋃

{(vi, vj) : (vi, vj) /∈ E(ΓCn)},
respectively.

We consider two cases:

• central graph of the identity graph of odd cyclic group;

• central graph of the identity graph of even cyclic group.

Definition 3.2. Let C(ΓCn) be the central graph of ΓCn for any odd integer n.
The vertex-set and edge-set of C(ΓCn) is given by

V (C(ΓCn)) = {v0, vi, c0i : 1 ≤ i ≤ n− 1}
⋃ {

c(2i−1)(2i) : 1 ≤ i ≤ n− 1

2

}
E(C(ΓCn)) = {(v0, c0i), (vi, c0i) : 1 ≤ i ≤ n− 1}⋃ {

(v2i−1, c(2i−1)(2i)), (v2i, c(2i−1)(2i)) : 1 ≤ i ≤ n− 1

2

}
⋃

{(vi, vj) : 1 ≤ i ≤ n− 3, i+ 2 ≤ j ≤ n− 1}⋃ {
(v2i, v2i+1) : 1 ≤ i ≤ n− 3

2

}
,

respectively.



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1103

.........

........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
.......
.........
.....................................................
..

.........................................................................................................
.....................................................
..

................................................................................................................................................................................................................................................................................................................................................................................................................................
.........
.....................................................
..

.........
.....................................................
..

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
....

.........
.....................................................
..

.....................................................................................
.........
.....................................................
..

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
.........

.........
.....................................................
..

.........
.....................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................

.........
.....................................................
..

...........
..........
..........
..........
..........
..........
..........
..........
....

.........
.....................................................
..

..............................................................................................................................................................................................................................................................................................................................................................
.........
.....................................................
..

.........
.....................................................
..

.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
. .........
.....................................................
..

.....................................................................
.........
.....................................................
..

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.

.........
.....................................................
..

.........
.....................................................
........................................................................................................................................................................................................................................................................................................................................

.........
.....................................................
..

.........
........
........
........
........
........
........
........
....

.........
.....................................................
..

................................................................................................................................................................................................................................................................................................... .........
.....................................................
..

.........
.....................................................
..

v0

v1

v2

v3

v4vn−4

vn−3

vn−2

vn−1

.................... ........................................ ....................

........................................

Figure 2: The identity graph of odd cyclic group

............ ............ ............ ....................................

....................................

....................................
....................................

....................................
.................................... ....................................

....................................

....................................
....................................

....................................

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

c01
c02
c03
c04c0(n−4)

c0(n−3)

c0(n−2)

c0(n−1)

v0

v1

v2

v3

v4vn−4

vn−3

vn−2

vn−1

c12

c34c(n−4)(n−3)

c(n−2)(n−1)
..........
.........
.........
.

......................................

.........

........

........

........

......

.............................................

............................................. ...................
..................

........

...................
..................

........
.........
........
........
........
......

............
...........
...........
....

.............................

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.....

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
.......

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
......

...................
..................

..................
..................

..................
..................

..............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.........

............
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
.

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..

..................................
.................................

.................................
.................................

.................................
.................................

....................

...................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

.............

................
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
............

.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
.

.............................................................................................................

...........................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................
.....................................

.....................................
.....................................

.....................................
.....................................

.....................................
.....................................

.....................................
.....................................

.....................

..........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
............

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
......

.............................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................

...................................................................
...................................................................

...................................................................
...................................................................

...................................................................
......

..............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
............

..................................................
.................................................

.................................................
.................................................

.................................................
.................................................

.................................................
.................................................

.................................................
.................................................

...............................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................

....................................................................................................................................................

.............................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................

....................................................................................................................................

..................
.................
.................
.......

.............
............
............
............
............
.......

...........
..........
..........
..........
..........
..........
..........
..

..........
.........
.........
.........
.........
.........
.........
.........
.........
..

.........
........
........
........
........
........
........
........
........
........
........
...........................................................................................

....................................................................................

.........................................................................

....................................................................

...........................................................

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.........

.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
......

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........

.........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
... ....................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................

.................... ........................................ ....................

........................................

Figure 3: The Central Graph of ΓCn (n is odd).

The degrees of each vertex in C(ΓCn) is summarized below:

deg(v0) = n− 1

deg(vi) = n− 1 ; 1 ≤ i ≤ n− 1

deg(c0i) = 2 ; 1 ≤ i ≤ n− 1

deg(c(2i−1)(2i)) = 2 ; 1 ≤ i ≤ n− 1

2
.

We will sum up all the degrees of vertices in C(ΓCn) and get∑
x∈V (C(ΓCn ))

deg(x) = (n− 1) + (n− 1)(n− 1) + 2(n− 1) + 2(
n− 1

2
)

= 4(n− 1) + n2 − 2n+ 1



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1104

= 4n− 4 + n2 − 2n+ 1

= n2 + 2n− 3.

By Handsaking Lemma,

|E(C(ΓCn))| =
1

2
(

∑
x∈V (C(ΓCn ))

deg(x)) =
n2 + 2n− 3

2

From this result, we can characterize the order and size of C(ΓCn) (n is odd).

Proposition 1. Let Cn be a cyclic group of order n and ΓCn be the identity graph of Cn.

If n is odd, then |V (C(ΓCn))| = 5n−3
2 and |E(C(ΓCn))| = n2+2n−3

2 .

Proof. Let Cn be a cyclic group of order n (n is odd) and ΓCn be the identity graph
of Cn. By Theorem 4, the size of ΓCn is |E(ΓCn)| = 3n−3

2 . Thus, the order and size
of C(ΓCn) are

|V (C(ΓCn))| = n+

(
3n− 3

2

)
=

2n+ 3n− 3

2
=

5n− 3

2

and

|E(C(ΓCn))| =
n(n− 1)

2
+

(
3n− 3

2

)
=

n2 − n+ 3n− 3

2
=

n2 + 2n− 3

2
,

respectively.

Illustration 1. The identity graph of C9 and its central graph is in Figure 4.

ΓC9 : .........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

v0

v1

v2

v3

v4v5

v6

v7

v8
...................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..

.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
...

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
....

.........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
......................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................
..........................................................

................................................................................ ...............
..............
..............
..............
...........

..........
.........
.........
.........
.........
.........
...

C(ΓC9) : .........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
...........

.....................................................
..

.........
.....................................................
.. c12

c34c56

c78

v0

v1

v2

v3

v4v5

v6

v7

v8

.................................................................

...................................................................................................................................................................

.........................................................................................................................................................................................

............................................................................................................................................................................................................................................................................
................................................................................................

...........................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................
.................................

.................................
.................................

.................................
.................................

..........................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................... ...........
..........
..........
..........
..........
..........
....

..............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
..

....................
...................

...................
...................

...................
...................

...................
...................

...................
...................

...................
...................

...................
...................

.................

............................................
...........................................

...........................................
...........................................

...........................................
...........................................

...........................................
...........................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
.......

..............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
...........

.....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
.

.....................................
....................................

....................................
....................................

....................................
....................................

....................................
....................................

....................................
....................................

....................................
....................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................

............
...........
...........
........

...........
..........
..........
..........
..........
..

..........................................................................................................

.....................................................

..........................................

..........
.........
.........
.........
........

..............................................................................................................................................................................

.................................................................................................................................................................................................................

........................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
.

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
.

............
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
..........

...................
..................

..................
..................

..................
..................

..................
..................

..................
...........

.............................................
.........................................................

.................................................................

.............................................................................
........
........
........
........
........
........
........
...

...........
..........
..........
..........
..........
..........
....

............
...........
...........
...........
...........
.

...................
..................

........

............................

............................
............................

........................................................
............................

............................

............................
c01
c02

c03c04c05
c06
c07

c08

Figure 4: The Central Graph of ΓC9 .



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1105

Definition 3.3. Let C(ΓCn) be the central graph of ΓCn for any even integer n. The
vertex-set and edge-set of C(ΓCn) is given by

V (C(ΓCn)) = {v0, vi, c0i : 1 ≤ i ≤ n− 1}
⋃ {

c(2i−1)(2i) : 1 ≤ i ≤ n− 2

2

}
E(C(ΓCn)) = {(v0, c0i), (vi, c0i) : 1 ≤ i ≤ n− 1}⋃ {

(v2i−1, c(2i−1)(2i)), (v2i, c(2i−1)(2i)) : 1 ≤ i ≤ n− 2

2

}
⋃

{(vi, vj) : 1 ≤ i ≤ n− 3, i+ 2 ≤ j ≤ n− 1}⋃ {
(v2i, v2i+1) : 1 ≤ i ≤ n− 2

2

}
,

respectively.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
. ..........

.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..............................................................................................................................................................................................................................................................

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........

..........
.............................................
.

.............................................................

..........
.............................................
.

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........

..........
.............................................
.

..........
.............................................
.

.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
.......

..........
.............................................
.

............................................................
..........
.............................................
.

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
...............

..........
.............................................
.

..........
.............................................
.

.........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
.
..........
.............................................
.

..............................................................................................
.............................................
.

..........................................................................................................................................................................................................................................................................................................................................................................
..........
.............................................
.

..........
.............................................
.................................................................................................................................................................................................................................................................................................................................

..........
.............................................
.

..........
.........
.........
.........
.........
.........
.........
...

..........
.............................................
.

............................................................................................................................................................................................................................................................................................ ..........
.............................................
.

..........
.............................................
.

v0

vn−1

vn−2

vn−3

v6v5

v4

v3

v2

v1

....................
....................
....................

Figure 5: The Identity Graph of Even Cyclic Group

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
. ..........

.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

..........
.............................................
.

c01
c02c03c04c05 c06

c0(n−3)
c0(n−2)

c0(n−1)

c12

c34

c56

c(n−3)(n−2)

.........
........
........
........
........
........
........
......

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.

...............
..............
..............
.............
..............
..............
..............
..............
..............
..............
..............
....

............
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
....

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
....

.....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

...

................
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
...............
.............

.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
.

................................................................................................

..............................................................................................................................................................................................

.....................................................
....................................................

....................................................
....................................................

....................................................
....................................................

................

.........................
........................

........................
........................

........................
........................

........................
........................

........................
........................

........................
........................

........................
........................

........................
......................

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.

..........................................................................................................................................................................

................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................................................................................................

.....................................................
....................................................

....................................................
....................................................

....................................................
....................................................

....................................................
..............................................

...........................
..........................

..........................
..........................

..........................
..........................

..........................
..........................

..........................
..........................

..........................
..........................

..........................
..........................

..........................
..........................

..........................
.

.......................................................

.....................................................................................................................................

............................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................

................................................
................................................

................................................
................................................

................................................
................................................

................................................
.............................................................................................................................................................

.................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........
..........
.........

........................................

.........

........

........

........

......

...................................

............................................... ..............
.............
.............
.......

...................
..................

......

.......................................

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................

.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
........

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
...

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.......

.........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

.... .....................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................

............................................................................................................................................................................................

..................
.................
.................

.............
............
............
............
...........

...........
..........
..........
..........
..........
..........
......

..........
.........
.........
.........
.........
.........
.........
.........
...

.........

........

........

........

........

........

........

........

........

........

...................................................................................

.....................................................................

............................................................

........................................................................................
....................................
....................................
.................................... .................................... ....................................

....................................

....................................

....................................

v0

vn−1

vn−2

vn−3

v6v5

v4

v3

v2

v1

....................
....................
....................

........................
............

Figure 6: The Central Graph of ΓCn (n is even)



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1106

The degrees of each vertex in C(ΓCn) is summarized below:

deg(v0) = n− 1

deg(vi) = n− 1 ; 1 ≤ i ≤ n− 1

deg(c0i) = 2 ; 1 ≤ i ≤ n− 1

deg(c(2i−1)(2i)) = 2 ; 1 ≤ i ≤ n− 2

2
.

We will sum up all the degrees of vertices in C(ΓCn) and get∑
x∈V (C(ΓCn ))

deg(x) = (n− 1) + (n− 1)(n− 1) + 2(n− 1) + 2(
n− 2

2
)

= (n− 1) + (n− 2) + (2n− 2) + (n2 − 2n+ 1)

= n2 + 2n− 4.

By Handshaking Lemma,

|E(C(ΓCn))| =
1

2
(

∑
x∈V (C(ΓCn ))

deg(x)) =
n2 + 2n− 4

2

From this result, we can characterize the order and size of C(ΓCn) (n is even).

Proposition 2. Let Cn be a cyclic group of order n and ΓCn be the identity graph of Cn.

If n is even, then |V (C(ΓCn))| = 5n−4
2 and |E(C(ΓCn))| = n2+2n−4

2 .

Proof. Let Cn be a cyclic group of order n (n is even) and ΓCn be the identity graph
of Cn. By Theorem 4, the size of ΓCn is |E(ΓCn)| = 3n−4

2 . Thus, the order and size of
C(ΓCn) are

|V (C(ΓCn))| = n+

(
3n− 4

2

)
=

2n+ 3n− 4

2
=

5n− 4

2

and

|E(C(ΓCn))| =
n(n− 1)

2
+

(
3n− 4

2

)
=

n2 − n+ 3n− 4

2
=

n2 + 2n− 4

2
,

respectively.

4. Main Results

This section presents some graph parameters such as distance, eccentricities, radius,
diameter, center, periphery, and girth of C(ΓCn).



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1107

Proposition 3. Let ΓCn be the identity graph of finite cyclic group Cn (n ≥ 2) and C(ΓCn)
be the central graph of ΓCn. If u, v ∈ V (C(ΓCn)), then d(u, v) ≤ 3, where d(u, v) is the
distance between u and v.

Proof. For any integer n ≥ 2, let ΓCn be the identity graph of finite cyclic group Cn and
let V (ΓCn) = {v0, v1, · · · , vn−1}. Consider its central graph C(ΓCn). The vertex-set and
edge-set of C(ΓCn) are V (C(ΓCn)) = V (ΓCn)

⋃
C, where C = {cij : (vi, vj) ∈ E(ΓCn)}

and E(C(ΓCn)) = {(vi, cij), (vj , cij) : (vi, vj) ∈ E(ΓCn)}
⋃

{(vi, vj) : (vi, vj) /∈ E(ΓCn)},
respectively. Let x ∈ V (C(ΓCn)) for odd n.
If x = v0, d(x, u) ≤ 3 for any u ∈ V (C(ΓCn)).
If x ∈ {c0i : 1 ≤ i ≤ (n− 1)}, d(x, u) ≤ 3 for any u ∈ V (C(ΓCn)).
If x ∈ {vi : 1 ≤ i ≤ (n− 1)}, d(x, u) ≤ 3 for any u ∈ V (C(ΓCn)).
If x ∈ {c(2i−1)(2i) : 1 ≤ i ≤ n−1

2 }, d(x, u) ≤ 3 for any u ∈ V (C(ΓCn)).
The proof is analogous to the first case if n is even. Therefore, d(x, u) ≤ 3 for any
u, v ∈ V (C(ΓCn)).

Illustration 2. The central graph of ΓCn (n is odd) is given in Figure 7.

............ ............ ............ ....................................

....................................

....................................
....................................

....................................
.................................... ....................................

....................................

....................................
....................................

....................................

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

C(ΓCn) :
c01
c02
c03
c04c0(n−4)

c0(n−3)

c0(n−2)

c0(n−1)

v0

v1

v2

v3

v4vn−4

vn−3

vn−2

vn−1

c12

c34c(n−4)(n−3)

c(n−2)(n−1)

...........................................................................................................................................................................
...............................

........................................................................................................................................................................................................................... ......................
.....................

.....................
.....................

.....................
.....................

..............

......

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

.............
............

...........
............

............
...........

............
............

...........
............

............
...........

............
............

...........
............

.....

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
.......

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
........

...................................................................................................................................

....................................................................... .............
............

...........
............

............
...........

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........

..........
......
.........
......
.........
......
.........
......
.........
......
.........
......
.........
......
.........
......
.........
......
.........
......
.........
......
.....

........
.......
......
.......
.......
......
.......
.......
......
.......
.......
......
.......
.......
......
.......
.......
......
.......
.......
......
.......
.......
......
.......
.......
......
.......
.......
......
.......
.......
....

.............................................................................................................................................

.................................................................................................................................................................................................

....................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................................

......

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

............................................................

....................................................................................................................................................................................................

.............
............

......

..........
.........
.........
.........
........

..........
.........
.........
.

......................................

.........

........

........

........

......

............................................. ...................
..................

........
.........
........
........
........
......

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.....

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
.......

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
......

............................................................................................................................................................................................................................................................................................................................................................................
.....................................

.....................................
.....................................

.....................................
.....................................

.....................................
.....................................

.....................................
.....................................

.....................

..........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
.........................

.........................
............

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
......

........................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................

...................................................................
...................................................................

...................................................................
...................................................................

...................................................................
......

..............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
.............................

.............................
............

..................................................
.................................................

.................................................
.................................................

.................................................
.................................................

.................................................
.................................................

.................................................
.................................................

...............................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................................................................................

.............................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................

....................................................................................................................................

..................
.................
.................
.......

.............
............
............
............
............
.......

...........
..........
..........
..........
..........
..........
..........
..

..........
.........
.........
.........
.........
.........
.........
.........
.........
.. ....................................................................................

.........................................................................

....................................................................

...........................................................

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.........

.............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............
............

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
......

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
......... .................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................. ............
...........
...........
....

.............................

......................................................................................................................................................................................................................

.................... ........................................ ....................

........................................

.............................................................................................................................................................................................................................................................................................................................................

.......................

......................

......................

......................

......................

.....................

.....................

.....................

.....................

....................

....................

...................

...................

..................

..................

.................

.................

...............

...............

..............

..............

...........

...........

.........

.........

..

..

.............................................................................................................................................................................................................................................................................................................................................

.......................

......................

......................

......................

......................

.....................

.....................

.....................

.....................

....................

....................

...................

...................

..................

..................

.................

.................

...............

...............

..............

..............

...........

...........

.........

.........

..

..

...................................................................................................................................................................................................................
.............
............
............
...........
...........
..........
..........
........
........
......
......

.............................................................................................................................................................................................................................................................................................................................................

.......................

......................

......................

......................

......................

.....................

.....................

.....................

.....................

....................

....................

...................

...................

..................

..................

.................

.................

...............

...............

..............

..............

...........

...........

.........

.........

..

..

.............................................................................................................................................................................................................................................................................................................................................

.......................

......................

......................

......................

......................

.....................

.....................

.....................

.....................

....................

....................

...................

...................

..................

..................

.................

.................

...............

...............

..............

..............

...........

...........

.........

.........

..

..

...................................................................................................................................................................................................................
.............
............
............
...........
...........
..........
..........
........
........
......
......

.............................................................................................................................................................................................................................................................................................................................................

.......................

......................

......................

......................

......................

.....................

.....................

.....................

.....................

....................

....................

...................

...................

..................

..................

.................

.................

...............

...............

..............

..............

...........

...........

.........

.........

..

..

.............................................................................................................................................................................................................................................................................................................................................

.......................

......................

......................

......................

......................

.....................

.....................

.....................

.....................

....................

....................

...................

...................

..................

..................

.................

.................

...............

...............

..............

..............

...........

...........

.........

.........

..

..

Figure 7: Central Graph of ΓCn , n is odd

Figure 7 shows that any arbitrary vertex x ∈ V (C(ΓCn)) has the maximum distance
of 3. The same argument for the central graph of the identity graph of Cn (n is even).
The only difference is that d(vn−1, x) ≤ 2, for any other x ∈ V (C(ΓCn)).

The next proposition determines the eccentricities of the vertices of C(ΓCn).

Proposition 4. Let C(ΓCn) (n ≥ 2) be the central graph of ΓCn. The eccentricities of the
vertices of C(ΓCn) are as follows:
i. If n is odd, then e(u) = 3 for all u ∈ V (C(ΓCn)).
ii. If n is even, then e(u) is either 2 or 3, u ∈ V (C(ΓCn)).



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1108

Proof. Let C(ΓCn) (n ≥ 2) be the central graph of ΓCn .
(i). If n is odd, (see Figure 3), every vertex u ∈ V (C(ΓCn)) has a maximum distance of
3, that is, e(v0) = 3, e(c0i) = 3, 1 ≤ i ≤ (n− 1), e(vi) = 3, 1 ≤ i ≤ (n− 1),
e(c(2i−1)(2i)) = 3, 1 ≤ i ≤ n−1

2 . Thus, for any u ∈ V (C(ΓCn)), e(u) = 3.
(ii). If n is even, (see Figure 6), e(v0) = 3, e(c0i) = 3 (1 ≤ i ≤ (n− 1)),
e(vi) = 3 (1 ≤ i ≤ (n− 2)), e(c(2i−1)(2i)) = 3 (1 ≤ i ≤ n−2

2 ), e(v(n−1)) = 2.
Therefore, e(u) is either 2 or 3..

Illustration 3. Consider the eccentricities of the vertices of C(ΓC4) and C(ΓC5) in
Figure 8. Notice that all the vertices of C(ΓC4) have eccentricities 3 except the vertex v3
which is 2. On the other hand, all the vertices of C(ΓC5) have eccentricities 3.

C(ΓC4) :

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........

........

........

........

........

........

....

.........

........

........

........

........

........

....

............
...........
...........
...........
...........
......

............
...........
...........
...........
...........
......

................................................................................................ ................................................................................................

..............................................................

..............................................................

.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
..

.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.. 2

3

33 3

3
3

3

v3

c03

v0

c01

v1c12v2

c02

C(ΓC5) :

v0

v4

v3 v2

v1

c04

c03 c02

c01

c34 c12

3

3

33

3 3

3

33

3

3

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
...........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
...........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
...........

.....................................................
..

..........................................................................

.........
.....................................................
..

..........................................................................

.........
.....................................................
..

...........
..........
..........
..........
..........
........

.........
.....................................................
..

...........
..........
..........
..........
..........
........

.........
.....................................................
..

..........................................................................................
.........
.....................................................
..

..........................................................................................
.........
.....................................................
..

.........
.....................................................
..

......................
.....................

.....................
.......... .........

.....................................................
..

......................
.....................

.....................
.......... .........

.....................................................
..

...........................................................

.........
.....................................................
..

...........................................................

.........
.....................................................
..

..........
.........
.........
.........
.........
.........
.........
.........
.........
........

.........
.....................................................
..

..........
.........
.........
.........
.........
.........
.........
.........
.........
........

.........
.....................................................
..

.........
.....................................................
..

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

.................

Figure 8: The Central Graphs of C(ΓC4) and ΓC5 .

The radius and diameter of C(ΓCn) will be discussed in the next proposition.

Proposition 5. Let C(ΓCn) be the central graph of ΓCn for any integer n ≥ 3.
i. If n is odd, then rad(C(ΓCn)) = diam(C(ΓCn)).
ii. If n is even, then rad(C(ΓCn)) = 2 and diam(C(ΓCn)) = 3.

Proof. Let C(ΓCn) be the central graph of ΓCn for any integer n ≥ 3.
(i). By Proposition 4-(i), for any u ∈ V (C(ΓCn)), e(u) = 3. Hence, rad(C(ΓCn)) =
3 = diam(C(ΓCn)).
(ii). It is straightforward to prove case ii by using Proposition 4-(ii) since min {e(u) : ∃u ∈
V (C(ΓCn))} = 2 and max {e(u) : ∃u ∈ V (C(ΓCn))} = 3}. Hence, rad(C(ΓCn)) = 2
and diam(C(ΓCn)) = 3.

Illustration 4. In Figure 8, rad(C(ΓC5)) = diam(C(ΓC5)) = 3 and rad(C(ΓC4)) = 2
and diam(C(ΓC4)) = 3.

In the next two propositions below, the center of C(ΓCn) is given.

Proposition 6. For any even integer n ≥ 4, let C(ΓCn) be the central graph of ΓCn. The
center of C(ΓCn) denoted by Cen(C(ΓCn)) is a complete graph K1.



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1109

Proof. By Proposition 5-(ii), rad(C(ΓCn)) = 2. And, by Proposition 4-(ii), the only
vertex of (C(ΓCn)) that has of eccentricity 2 is the vertex vn−1. Hence, vn−1 is the only
central vertex of C(ΓCn). Thus, Cen(C(ΓCn)) is K1 (consists of a single vertex vn−1).

Illustration 5. Let C(ΓC8) be the central graph of ΓC8 given below.

................................................

................................................

................................................

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

..........
.........
.........
.........
.........
.........
.........
.........
.........
....

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
.......

.........................................................................................................................
........................

........................
.......................

........................
........................

........................
........................

..

..............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.......

......................................................................................................................................................................................................

.....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
....................

....................
..

........................................................
.......................................................

.......................................................
.......................................................

.......................................................
....................................................

.............................................................................................................................................

..........................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................
..............................

..............................
..............................

..............................
..............................

..............................
..............................

..............................
..............................

..............................
..............................

......................................................................................................................................................

................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................................................................................................

...............................................................................................................................................................................................................................................................................................................................................................................................
..................................
..................................

.................................. .................................. ..............
.............
..........

..............
.............
..........

.................
................
................
................
................
................
................
................
................
.........

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
.......

..........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
........

.........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

..... ..............................................................................................................................................................................................................

............................................................................................................................................................................................

..........................................................................................................................................................

.................
................
......

...........
..........
..........
..........
.........

..........
.........
.........
.........
.........
.......

.........

........

........

........

........

........

............................................................

..................................................

.......................................C(ΓC8) :
c07
c06

c05c04c03
c02

c01

c56
c34

c12

v0

v7

v6

v5v4
v3

v2

v1

....................................

.................................... .................................... .................................... ....................................
....................................

....................................

....................................

v7

Cen(C(ΓC8)) :

Figure 9: The Central Graph of ΓC8 and its center.

The center of ΓC8 is a subgraph induced by a vertex v7.

If n is odd, the center of the graph C(ΓCn) is given in the next proposition.

Proposition 7. Let C(ΓCn) be the central graph of ΓCn. If n is odd (≥ 3), then C(ΓCn)
is a self-centered graph.

Proof. Let Cn be a cyclic group of odd order n ≥ 3. We can associate an identity
graph (see Figure 2) and its central graph (Figure 3). Proposition 4-(i), tells us that for
any u ∈ V (C(ΓCn)), e(u) = 3. Hence, rad(C(ΓCn)) = 3 = e(u). Thus, all u ∈ V (C(ΓCn))
are central vertices of C(ΓCn). Therefore a subgraph induced by the central vertices of
C(ΓCn) is C(Γn) itself.

C(ΓC7) : .........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

c01
c02

c03c04
c05

c06

c12

c34

c56

...................
..................

................

...........
..........
..........
..........
..........
....

.........
........
........
........
........
........
...........................................................

.......................................................

....................................................

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.....

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..

.........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
....... ........................................................................................................................................................................................................

...........................................................................................................................................................................................................

..................................................................................................................................................................................................
............................................

............................................

.............................................. ..............................................

...........
..........
..........
..........
...
...........
..........
..........
..........
.....................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................

........................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................
..................................

..................................
..................................

..................................
..................................

..................................
..................................

..................................
..................................

..................................
..................................

..................

.......................................
......................................

......................................
......................................

......................................
......................................

...

...................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

...........

.....................
....................

....................
....................

....................
.......

..............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
....

v0

v1

v2

v3v4

v5

v6

3

3

3

33

3

3

3
3

3
3

33

3

3

3

Cen(C(ΓC7)) : .........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
.. .........

.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

.........
.....................................................
..

c01
c02

c03c04
c05

c06

c12

c34

c56

...................
..................

................

...........
..........
..........
..........
..........
....

.........
........
........
........
........
........
...........................................................

.......................................................

....................................................

..................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.....

...........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..

.........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
....... ........................................................................................................................................................................................................

...........................................................................................................................................................................................................

..................................................................................................................................................................................................
............................................

............................................

.............................................. ..............................................

...........
..........
..........
..........
...
...........
..........
..........
..........
.....................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................

........................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................
..................................

..................................
..................................

..................................
..................................

..................................
..................................

..................................
..................................

..................................
..................................

..................

.......................................
......................................

......................................
......................................

......................................
......................................

...

...................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

..................
..................

...........

.....................
....................

....................
....................

....................
.......

..............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
....

v0

v1

v2

v3v4

v5

v6

Figure 10: The Central Graph of ΓC7 .



C. Alib, D. Magpantay / Eur. J. Pure Appl. Math, 15 (3) (2022), 1098-1112 1110

Illustration 6. For the central graph of ΓC7 of Figure 10, the eccentricities of all the
vertices of C(ΓC7) are all the same. Thus, all the vertices are central vertices of C(ΓC7).
Therefore, the center of C(ΓC7) is C(ΓC7) itself.

Proposition 8. All vertices of C(ΓCn) ( odd integer n ≥ 3) are peripheral vertices.

Proof. Proposition 5-(i) tells us that for all v ∈ V (C(ΓCn)), diam(C(ΓCn)) = 3 = e(v).
Therefore, by definition of peripheral vertex of a graph, all vertices v ∈ V (C(ΓCn)) are
peripheral vertices.

Proposition 9. Let C(ΓCn) be the central graph of ΓCn (even integer n ≥ 4). The set of
vertices A = {vi : 0 ≤ i ≤ (n−2)}

⋃
{c0i : 1 ≤ i ≤ (n−1)}

⋃
{c(2i−1)(2i) : 1 ≤ i ≤ n−2

2 }
are peripheral vertices of C(ΓCn).

Proof. The proof follows from Propositions 4-(ii) and 5-(ii), respectively.

Proposition 10. For any odd integer n ≥ 3, Per(C(ΓCn)) = C(ΓCn).

Proof. By Proposition 7, if n is odd, V (C(ΓCn)) is the set of peripheral vertices of
C(ΓCn). Therefore, Per(C(ΓCn)) = C(ΓCn).

Proposition 11. Let C(ΓCn) (even integer n ≥ 4) with the vertex set V (C(ΓCn)) =
{v0, vi, c0i : 1 ≤ i ≤ (n−1), c(2i−1)(2i) : 1 ≤ i ≤ n−2

2 }. Take K = {vn−1}. The subgraph
C(ΓCn)\K of C(ΓCn) is the periphery of C(ΓCn).

Proof. The proof follows from Proposition 4-(ii) and Proposition 8.

Proposition 12. The graph C(ΓCn) is an eccentric graph if and only if n is odd.

Proof. Let C(ΓCn) be the central graph of ΓCn where n is odd. Then, C(ΓCn) is an
eccentric graph since every vertex of C(ΓCn) is a peripheral vertex and so is an eccentric
vertex of the other in C(ΓCn). Conversely, let C(ΓCn) be an eccentric graph. Thus,
every vertex of C(ΓCn) is an eccentric vertex of the other. There are only two cases for n.
For the case that n is even, vn−1 ∈ V (C(ΓCn)) is not an eccentric vertex of any other
vertex of C(ΓCn) and thus C(ΓCn) for even n is not an eccentric graph. So n must be
odd since in this case every vertex is an eccentric vertex of the other.

Proposition 13. The girth of C(ΓCn) is

gir(C(ΓCn)) =


6, if n = 3
4, if n = 4, 5
3, if n > 5.



REFERENCES 1111

Proof. Let ΓCn be the identity graph associated with a cyclic group Cn of order n. For
n = 3, C(ΓC3) is isomorphic to a cycle graph of length 6. Thus, gir(C(ΓC3)) = 6. For n =
4, let V (C(ΓC4)) = {v0, vi, c0i, c12 : 1 ≤ i ≤ 3} and E(C(ΓC4)) = {(v0, c0i), (vi, c0i) : 1 ≤
i ≤ 3}

⋃
{(v1, c12), (v1, v3), (c12, v2), (v2, v3)}. Clearly, the cycle {v1, c12, v2, v3, v1} is

the smallest cycle of C(ΓC4). Hence, gir(C(ΓC4)) = 4. For n = 5, let V (C(ΓC5)) =
{v0, vi, c0i, c12 : 1 ≤ i ≤ 4} and E(C(ΓC5)) = {(v0, c0i), (vi, c0i) : 1 ≤ i ≤ 4}

⋃
{(v1, c12),

(v1, v3), (v1, v4), (c12, v2), (v2, v3), (v2, v4), (v3, c34), (c34, (v4))}. Clearly, the cycle {v1, c12,
v2, v3, v1} is one of the smallest cycles of C(ΓC4). Thus, gir(C(ΓC5)) = 4. For n ≥ 6,
the cycle {v1, v3, vn−1, v1} is always in C(ΓCn). In fact, there are many cycle graph of
lenght 3 in C(C(ΓCn)). Thus, gir(C(ΓCn)) = 3.

5. Summary and Conclusion

In this paper, we investigated the structures and some properties of the central graphs
of the identity graphs of finite cyclic groups.

Acknowledgements

The authors would like to thank the referees for their thoughtful comments and efforts
towards improving their manuscript. Also, they would like to thank the Department
of Science and Technology Science Education Institute (DOST-SEI)-Philippines through
the Science and Technology Regional Alliance of Universities for National Development
(STRAND) for the grant to make this paper possible.

References

[1] A. Ballester-Bolinches and J. Cossey. Graphs and classes of finite groups. Note di
Matematica, 33(1):89–94, 2013.

[2] P. Cameron and S. Ghosh. The power graph of a finite group. Discrete Mathematics,
311(13):1220–1222, 2011.

[3] Yalçın Nazmiye Feyza and Kırğıl Yakup. Identity Graphs of Finite Cyclic Groups.
Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(1):149–158, 2020.

[4] A. Godase. Unit Graph of Some Finite Group Zn, Cn, Dn. pages 122–130, 2015.

[5] M. Herawati and P. Henryanti. Identity Graphs of Finite Cyclic Groups. International
Journal of Applied Sciences and Smart Technologies, 3(1), 2021.

[6] W. Kandasamy and F Smarandache. Groups as Graphs.
https://arxiv.org/abs/0906.5144, 2009.



REFERENCES 1112

[7] Z. Raza and S. Faizi. Commuting graphs of dihedral type groups. Applied Mathe-
matics ENotes, 13:221–227, 2013.

[8] S. Rehman, A. Baig, M. Imran, and Z. Khan. Order divisor graphs of finite groups.
Analele Universitatii ”Ovidius” Constanta-Seria Matematica, 26(3):29–40, 2018.

[9] W. Somnuek. Counting Lines and Triangle in a Unit Graph. Current Applied Science
and Technology, 17(1), 2017.

[10] J. Vernold. Harmonious coloring of total graphs, n-leaf, central graphs, circumdetic
graphs. Ph.D Thesis Bharathia University Coimbatore India, 2007.