EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2915-2929
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Spectrum and Energy of Identity Graph for Group
of Integers Modulo n, Zn

Mamika Ujianita Romdhini1,∗, Athirah Nawawi2, Faisal Al-Sharqi3,4, Salwa1

1 Department of Mathematics, Faculty of Mathematics and Natural Science,
University of Mataram, Mataram 83125, Indonesia
2 Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia,
43400 Serdang, Selangor, Malaysia
3 Department of Mathematics, Faculty of Education for Pure Sciences, University of Anbar,
Ramadi, Anbar, Iraq
4 College of Engineering, National University of Science and Technology, Dhi Qar, Iraq

Abstract. Groups and graphs are two concepts of algebraic mathematics. This paper focuses on
group structures that can be expressed in graphs known as identity graphs. We investigate the
energy of the identity graph for a group of integers modulo n, Zn, for odd and even n corresponding
to adjacency, Laplacian, and signless Laplacian matrices. It can be seen that the Laplacian and
signless Laplacian energies are always equal and are always an even integer. Meanwhile, the
adjacency energy is never an odd integer for n is odd.

2020 Mathematics Subject Classifications: 05C25, 15A18

Key Words and Phrases: Energy of a graph, Identity graph of a group, Zn

1. Introduction

Groups and graphs are two concepts of algebraic mathematics. A group is an algebraic
structure from a non-empty set with a binary operation and satisfies associative property,
there is an identity element and each element has an inverse. Furthermore, graph theory is
a discrete mathematics study that discusses vertices and edges. In this paper, we discuss
group structures that can be expressed in graphs, the name is identity graph.

Kandasamy and Smarandache in 2009 [5] described finite groups as graphs. They call
this the identity graph because the main key in constructing the graph is determined by
the group’s identity elements. The discussion on labeling of the identity graph can be
found in [10].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5375

Email addresses: mamika@unram.ac.id (M. U. Romdhini), athirah@upm.edu.my (A. Nawawi),
faisal.ghazi@uoanbar.edu.iq (F. Al-Sharqi), salwa@unram.ac.id (Salwa)

https://www.ejpam.com 2915 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2916

The graph energy concept was pioneered by Gutman in 1978 [4]. It should be noted
that the graph energy is never an odd integer [1, 9]. Moreover, several results on the energy
of a graph defined on groups can be found in [11, 13, 15]. They worked on non-commuting
graphs with Wiener-hosoya, closeness and degree subtraction matrices. Meanwhile, for
Sombor energy can be seen in [12]. Shi et al [17] found the energy of picture fuzzy graphs,
in line with the signless Laplacian energy [11] and Cayley of interval-valued fuzzy graphs
[2]. Kumari et al. [6] presented the quotient energy of the identity graph for Zp, for prime
number p and Romdhini et al. [14] showed the spectral properties of power graph for
dihedral groups. Meanwhile, the spectral discussion of the square power graph can be
seen in [18]. In addition, Shanthakumari et al. [16] described the Euclidean degree energy
and Lokesha et al. [8] investigated the skew energy of a graph.

Inspired by this, we work on the identity matrix. Our focus in this paper is a group
of integers modulo n, Zn =

{
0, 1, 2, . . . , n− 1

}
. We construct the identity graph based

on the group elements as vertices. We develop some graph matrices corresponding to this
graph concerning the adjacency, Laplacian, and signless Laplacian matrices. We formulate
the graph’s characteristic polynomial, spectrum, and energy, and analyze the relationship
between those energies. We also observe the energy values to draw interesting conclusions.

2. Preliminaries

In this part, we recall the fundamental definition and theorem that are useful for our
main results. We start with the definition of the identity graph.

Definition 1. [5] The identity graph of a group G, denoted by ΓG, is a graph whose vertex
set is the elements of the group and two distinct vertices u and v will be connected by an
edge if uv = e with every member of G\{e} is adjacent to e, where e is the identity element
of G.

Throughout this paper, we denote the identity graph for Zn as ΓZn . The next two
theorems are the description of ΓZn , for n is odd and even.

Theorem 1. [5] If Zn =
{
0, 1, 2, ..., n− 1

}
is a group of order n, n ≥ 3 with odd n, then

the identity graph of Zn contains n−1
2 of K3.

Theorem 2. [5] If Zn =
{
0, 1, 2, ..., n− 1

}
is a group of order n, n ≥ 2 with even n, then

the identity graph of Zn contains n−2
2 of K3 and a K2.

The construction of the graph matrices of ΓZn is based on the definition of the ad-
jacency, Laplacian, and signless Laplacian matrices. We refer these definition to ([3]) as
presented below:

Definition 2. ([3]) The adjacency matrix of order n× n associated with ΓZn is given by
A(ΓZn) = [aij ] whose (i, j)-th entry

aij =

{
1, if vi ̸= vj and they are adjacent
0, otherwise



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2917

Definition 3. ([3]) The n × n diagonal degree matrix of ΓZn is given by D(ΓZn) = [dij ]
whose (i, j)-th entry

dij =

{
dvi , if vi = vj
0, otherwise

where dvi is the vertex degree of vi.

Definition 4. ([3]) The n× n Laplacian matrix of ΓD2n is given by L(ΓZn) = D(ΓZn)−
A(ΓZn).

Definition 5. ([3]) The n × n signless Laplacian matrix of ΓZn is given by SL(ΓZn) =
D(ΓZn) +A(ΓZn).

The characteristic polynomial of A(ΓZn) is defined by

PA(ΓZn )
(λ) = |λIn −A(ΓZn)| , (1)

where In is an n × n identity matrix. Similarly, notation for other matrices can be used
in the same manner.

To formulate the determinant in Equation 1, we need row and column operations to
simplify the process. Let Ri be the i-th row and Ci be the i-th column of PA(ΓZn )

(λ).
Furthermore, the roots of PA(ΓZn )

(λ) = 0 are the eigenvalues of ΓZn . The graph energy
definition is based on the eigenvalues of ΓZn as presented below.

Definition 6. [4] The adjacency energy of ΓZn can be written by

EA(ΓZn) =

n∑
i=1

|λi| ,

where λ1, λ2, . . . , λn are eigenvalues of A(ΓZn).

The spectrum of ΓZn in accordance with the adjacency matrix is

SpecA(ΓZn) = {(λ1)
k1 , (λ2)

k2 , . . . , (λn)
kn},

where k1, k2, . . . , kn are the respective multiplicities of eigenvalus. The spectral radius of
ΓZn corresponding with the adjacency matrix is

ρA(ΓZn) = max{|λ| : λ ∈ SpecA(ΓZn)}.

The energy value of ΓZn is classified as hyperenergetic if the energy of ΓZn is greater
than 2(n− 1) [7].



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2918

3. Main Results

In this section, we begin with the analysis of the degree of every vertex in ΓZn . We
need this property for constructing the matrices of ΓZn .

Theorem 3. Let ΓZn be the identity graph on Zn. For n is odd, then

(i) the degree of 0 on ΓZn is deg(0) = n− 1, and

(ii) the degree of a on ΓZn is deg(a) = 2, for a ̸= 0.

Proof. From Theorem 1 for odd n, the identity graph of Zn contains n−1
2 of K3. Since

the identity of Zn is 0, then 0 is adjacent to all other vertices in Zn. This means that the
degree of 0 is equal to n − 1. Meanwhile, for a, where a ̸= 0, we know that the inverse
of a is n− a, since a+ (n− a) = 0. This implies that a and n− a are always adjacent.
Therefore, the degree of a is 2.

Theorem 4. Let ΓZn be the identity graph on Zn. For n is even, then

(i) the degree of 0 on ΓZn is deg(0) = n− 1,

(ii) the degree of n
2 on ΓZn is deg(n2 ) = 1, and

(iii) the degree of a on ΓZn is deg(a) = 2, for a ̸= 0, n2 .

Proof. Recall the fact from Theorem 2 that the identity graph of Zn contains n−2
2 of

K3 and a K2 for even n. By the same argument with the proofing part of Theorem 3 that
0 is the identity of Zn, then the degree of 0 is n− 1. Now, we concern with n

2 ∈ Zn. Since

the inverse of n
2 is itself, then n

2 is only adjacent to 0 which means the degree of n
2 is 1.

Meanwhile, for a ̸= 0, n2 , we have a+ (n− a) = 0. Consequently, a is adjacent to n− a
and also to 0 which we mentioned earlier. Therefore, the degree of a is 2.

Theorem 5. Let Mn×n be the matrix as follows:

M =



a c c . . . c c
c b 0 . . . 0 c
c 0 b . . . c 0
...

...
...

. . .
...

...
c 0 c . . . b 0
c c 0 . . . 0 b


,

where n is odd, and real numbers a, b, c. The characteristic polynomial of M is

PM (λ) =
(
λ2 − (a+ b+ c)λ+ a(b+ c)− c2(n− 1)

)
(λ− b− c)

n−3
2 (λ− b+ c)

n−1
2 .



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2919

Proof. Let n be an odd number. The characteristic polynomial of M is given by

PM (λ) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

λ− a −c −c . . . −c −c
−c λ− b 0 . . . 0 −c
−c 0 λ− b . . . −c 0
...

...
...

. . .
...

...
−c 0 −c . . . λ− b 0
−c −c 0 . . . 0 λ− b

∣∣∣∣∣∣∣∣∣∣∣∣∣
,

where the numbers a, b, c are real. We need to simplify the above determinant by applying
row and column operations.

(i) Rn+1
2

+i −→ Rn+1
2

+i −Rn+3
2

−i, for i = 1, 2, . . . , n−1
2 .

Then we have

PM (λ) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

λ− a −c −c . . . −c −c
−c λ− b 0 . . . 0 −c
−c 0 λ− b . . . −c 0
...

...
...

. . .
...

...
0 0 −λ+ b− c . . . λ− b+ c 0
0 −λ+ b− c 0 . . . 0 λ− b+ c

∣∣∣∣∣∣∣∣∣∣∣∣∣
.

(ii) Cn+3
2

−i −→ Cn+3
2

−i + Cn+1
2

+i, for i = 1, 2, . . . , n−1
2 .

Consequently,

PM (λ) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

λ− a −2c −2c . . . −c −c
−c λ− b− c 0 . . . 0 −c
−c 0 λ− b− c . . . −c 0
...

...
...

. . .
...

...
0 0 0 . . . λ− b+ c 0
0 0 0 . . . 0 λ− b+ c

∣∣∣∣∣∣∣∣∣∣∣∣∣
.

(iii) C1 −→ C1 +
c

λ−b−cC2 +
c

λ−b−cC3 + . . .+ c
λ−b−cCn+1

2
.

Hence we can write

PM (λ) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

(λ−a)(λ−b−c)−c2(n−1)
λ−b−c −2c −2c . . . −c −c

0 λ− b− c 0 . . . 0 −c
0 0 λ− b− c . . . −c 0
...

...
...

. . .
...

...
0 0 0 . . . λ− b+ c 0
0 0 0 . . . 0 λ− b+ c

∣∣∣∣∣∣∣∣∣∣∣∣∣
.



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2920

It follows that

PM (λ) =
(
λ2 − (a+ b+ c)λ+ a(b+ c)− c2(n− 1)

)
(λ− b− c)

n−3
2 (λ− b+ c)

n−1
2 ,

due to it is an upper triangular matrix.

Theorem 6. Let n is even and Mn×n be the matrix as follows:

M =



a d d . . . d d d . . . d d
d b 0 . . . 0 0 0 . . . 0 d
d 0 b . . . 0 0 0 . . . d 0
...

...
...

. . .
...

...
...

...
...

...
d 0 0 . . . b 0 d . . . 0 0
d 0 0 . . . 0 c 0 . . . 0 0
d 0 0 . . . d 0 b . . . 0 0
...

...
...

. . .
...

...
...

...
...

...
d 0 d . . . 0 0 0 . . . b 0
d d 0 . . . 0 0 0 . . . 0 b


,

where a, b, c, d are real numbers. The characteristic polynomial of M is

PM (λ) =
(
λ3 − (a+ b+ c+ d)λ2 + ((a+ c)(b+ d) + ac− d2(n− 1))λ+ (b+ d)(d2 − ac) + cd2(n− 2)

)
(λ− b− d)

n
2
−2(λ− b+ d)

n
2
−1..

Proof. Let n is even, and a, b, c, d are real numbers. The characteristic polynomial of
M is given by

PM (λ) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

λ− a −d −d . . . −d −d −d . . . −d −d
−d λ− b 0 . . . 0 0 0 . . . 0 −d
−d 0 λ− b . . . 0 0 0 . . . −d 0
...

...
...

. . .
...

...
...

. . .
...

...
−d 0 0 . . . λ− b 0 −d . . . 0 0
−d 0 0 . . . 0 λ− c 0 . . . 0 0
−d 0 0 . . . −d 0 λ− b . . . 0 0
...

...
...

. . .
...

...
...

. . .
...

...
−d 0 −d . . . 0 0 0 . . . λ− b 0
−d −d 0 . . . 0 0 0 . . . 0 λ− b

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.

We need to simplify the above determinant by applying row and column operations.

(i) Rn
2
+1+i −→ Rn

2
+1+i −Rn

2
+1−i, for i = 1, 2, . . . , n2 − 1.



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2921

Then we obtain

PM (λ) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

λ− a −d −d . . . −d −d −d . . . −d −d
−d λ− b 0 . . . 0 0 0 . . . 0 −d
−d 0 λ− b . . . 0 0 0 . . . −d 0
...

...
...

. . .
...

...
...

. . .
...

...
−d 0 0 . . . λ− b 0 −d . . . 0 0
−d 0 0 . . . 0 λ− c 0 . . . 0 0
0 0 0 . . . −λ+ b− d 0 λ− b+ d . . . 0 0
...

...
...

. . .
...

...
...

. . .
...

...
0 0 −λ+ b− d . . . 0 0 0 . . . λ− b+ d 0
0 −λ+ b− d 0 . . . 0 0 0 . . . 0 λ− b+ d

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.

(ii) Cn
2
+1−i −→ Cn

2
+1−i + Cn

2
+1+i, for i = 1, 2, . . . , n2 − 1.

Consequently, we have

PM (λ) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

λ− a −2d −2d . . . −2d −d −d . . . −d −d
−d λ− b− d 0 . . . 0 0 0 . . . 0 −d
−d 0 λ− b− d . . . 0 0 0 . . . −d 0
...

...
...

. . .
...

...
...

. . .
...

...
−d 0 0 . . . λ− b− d 0 −d . . . 0 0
−d 0 0 . . . 0 λ− c 0 . . . 0 0
0 0 0 . . . 0 0 λ− b+ d . . . 0 0
...

...
...

. . .
...

...
...

. . .
...

...
0 0 0 . . . 0 0 0 . . . λ− b+ d 0
0 0 0 . . . 0 0 0 . . . 0 λ− b+ d

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.

(iii) C1 −→ C1 +
d

λ−b−dC2 +
d

λ−b−dC3 + . . .+ d
λ−b−dCn

2
−1 +

d
λ−b−dCn

2
.

Then, we can state that PM (λ) is∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(λ−a)(λ−b−d)−d2(n−2)
λ−b−d

−2d −2d . . . −2d 0 −d . . . −d −d

0 λ− b− d 0 . . . 0 0 0 . . . 0 −d
0 0 λ− b− d . . . 0 0 0 . . . −d 0
...

...
...

. . .
...

...
...

. . .
...

...
0 0 0 . . . λ− b− d 0 −d . . . 0 0
−d 0 0 . . . 0 λ− c 0 . . . 0 0
0 0 0 . . . 0 0 λ− b+ d . . . 0 0
...

...
...

. . .
...

...
...

. . .
...

...
0 0 0 . . . 0 0 0 . . . λ− b+ d 0
0 0 0 . . . 0 0 0 . . . 0 λ− b+ d

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.

(iv) C1 −→ C1 +
d

λ−cCn
2
+1.

It follows that PM (λ) is
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(λ−a)(λ−c)(λ−b−d)−d2((n−2)(n−c)+(λ−b−d)
(λ−b−d)(λ−c)

−2d −2d . . . −2d 0 −d . . . −d −d

0 λ − b − d 0 . . . 0 0 0 . . . 0 −d
0 0 λ − b − d . . . 0 0 0 . . . −d 0

.

.

.

.

.

.

.

.

.
.
. .

.

.

.

.

.

.

.

.

.
.
. .

.

.

.

.

.

.
0 0 0 . . . λ − b − d 0 −d . . . 0 0
0 0 0 . . . 0 λ − c 0 . . . 0 0
0 0 0 . . . 0 0 λ − b + d . . . 0 0

.

.

.

.

.

.

.

.

.
.
.
.

.

.

.

.

.

.

.

.

.
.
.
.

.

.

.

.

.

.
0 0 0 . . . 0 0 0 . . . λ − b + d 0
0 0 0 . . . 0 0 0 . . . 0 λ − b + d

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2922

The matrix form is upper triangular, consequently, we derive the following formula:

PM (λ) =
(
λ3 − (a+ b+ c+ d)λ2 + ((a+ c)(b+ d) + ac− d2(n− 1))λ+ (b+ d)(d2 − ac) + cd2(n− 2)

)
(λ− b− d)

n
2
−2(λ− b+ d)

n
2
−1.

3.1. Adjacency Matrix

In this part, the goal is to provide the energy formula of ΓZn associated with the
adjacency matrix. We begin with the case for n is odd.

Theorem 7. Let ΓZn be the identity graph on Zn. The adjacency energy of ΓZn for odd
n is

EA(ΓZn) = n− 2 +
√
4n− 3.

Proof. According to Theorems 1 and 3, we can construct an n × n adjacency matrix
of ΓZn as follows:

A(ΓZn) =

0 1 2 . . . n− 2 n− 1



0 0 1 1 . . . 1 1
1 1 0 0 . . . 0 1
2 1 0 0 . . . 1 0
...

...
...

...
. . .

...
...

n− 2 1 0 1 . . . 0 0
n− 1 1 1 0 . . . 0 0

(2)

Following the principle of Theorem 5 with a = b = 0 and c = 1, then we derive

PA(ΓZn )
(λ) =

(
λ2 − λ− (n− 1)

)
(λ− 1)

n−3
2 (λ+ 1)

n−1
2 .

The roots of PA(ΓZn )
(λ) = 0 are λ1 = 1 of multiplicity n−3

2 , λ2 = −1 of multiplicity n−1
2 ,

and λ3,4 =
1
2 ±

√
4n−3
2 of multiplicity 1, respectively. Consequently, the spectrum of ΓZn is

SpecA(ΓZn) =

{(
1

2
+

√
4n− 3

2

)1

, (1)
n−3
2 , (−1)

n−1
2 ,

(
1

2
−

√
4n− 3

2

)1
}
.

It is clear that the spectral radius of ΓZn is

ρA(ΓZn) =
1

2
+

√
4n− 3

2
.

Therefore, the adjacency energy of ΓZn is as follows:

EA(ΓZn) =

(
n− 3

2

)
|1|+

(
n− 1

2

)
| − 1|+

∣∣∣∣12 ±
√
4n− 3

2

∣∣∣∣
= n− 2 +

√
4n− 3.



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2923

Theorem 8. Let ΓZn be the identity graph on Zn. The characteristic polynomial of A(ΓZn)
for even n is

PA(ΓZn )
(λ) =

(
λ3 − λ2 − (n− 1)λ+ 1

)
(λ− 1)

n
2
−2(λ+ 1)

n
2
−1.

Proof. According to Theorems 2 and 4, we can construct an n × n adjacency matrix
of ΓZn as follows:

A(ΓZn) =

0 1 2 . . . n
2 − 1 n

2
n
2 + 1 . . . n− 2 n− 1



0 0 1 1 . . . 1 1 1 . . . 1 1
1 1 0 0 . . . 0 0 0 . . . 0 1
2 1 0 0 . . . 0 0 0 . . . 1 0
...

...
...

...
. . .

...
...

...
...

...
...

n
2 − 1 1 0 0 . . . 0 0 1 . . . 0 0

n
2 1 0 0 . . . 0 0 0 . . . 0 0

n
2 + 1 1 0 0 . . . 1 0 0 . . . 0 0
...

...
...

...
. . .

...
...

...
...

...
...

n− 2 1 0 1 . . . 0 0 0 . . . 0 0
n− 1 1 1 0 . . . 0 0 0 . . . 0 0

. (3)

According to Theorem 6 with a = b = c = 0, d = 1, consequently, we derive the following
formula:

PA(ΓZn )
(λ) =

(
λ3 − λ2 − (n− 1)λ+ 1

)
(λ− 1)

n
2
−2(λ+ 1)

n
2
−1.

3.2. Laplacian Matrix

This part focuses on the Laplacian matrix of ΓZn , for odd and even n, followed by
calculating the energy.

Theorem 9. Let ΓZn be the identity graph on Zn. The Laplacian energy of ΓZn for odd
n is

EL(ΓZn) = 3(n− 1).

Proof. Based on Theorem 3, we have n × n degree matrix of ΓZn as diag(n −
1, 2, 2, ..., 2, 2). According to Definition 4 and Equation 2, we can construct an n × n
Laplacian matrix of ΓZn as follows:

L(ΓZn) =D(ΓZn)−A(ΓZn) (4)



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2924

=

0 1 2 . . . n− 2 n− 1



0 n− 1 −1 −1 . . . −1 −1
1 −1 2 0 . . . 0 −1
2 −1 0 2 . . . −1 0
...

...
...

...
. . .

...
...

n− 2 −1 0 −1 . . . 2 0
n− 1 −1 −1 0 . . . 0 2

. (5)

According to Theorem 5 with a = n− 1, b = 2, and c = −1, then we obtain

PL(ΓZn )
(λ) = λ(λ− n)(λ− 1)

n−3
2 (λ− 3)

n−1
2 .

The roots of PL(ΓZn )
(λ) = 0 are λ1 = 0 of multiplicity 1, λ2 = n of multiplicity 1, λ3 = 1

of multiplicity n−3
2 , and λ4 = 3 of multiplicity n−1

2 . Consequently, the spectrum of ΓZn is

SpecL(ΓZn) =
{
(n)1 , (1)

n−3
2 , (3)

n−1
2 , (0)1

}
.

It is clear that the spectral radius of ΓZn is

ρL(ΓZn) = n.

Therefore, the Laplacian energy of ΓZn is as follows:

EL(ΓZn) = (1) |n|+
(
n− 1

2

)
|3|+

(
n− 3

2

)
|1|+ (1) |0|

= 3(n− 1).

Theorem 10. Let ΓZn be the identity graph on Zn. The Laplacian energy of ΓZn for even
n is

EL(ΓZn) = 3n− 4.

Proof. From Theorem 2, we have the degree of every vertex in ΓZn for even n. Then the
diagonal matrix of ΓZn is D(ΓZn) = diag(n − 1, 2, 2, ..., 2, 1, 2, ..., 2). Based on Definition
4 and Equation 3, we can construct an n× n Laplacian matrix of ΓZn as follows:

L(ΓZn
) =D(ΓZn

)−A(ΓZn
) (6)



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2925

=

0 1 2 . . . n
2 − 1 n

2
n
2 + 1 . . . n− 2 n− 1



0 n− 1 −1 −1 . . . −1 −1 −1 . . . −1 −1
1 −1 2 0 . . . 0 0 0 . . . 0 −1
2 −1 0 2 . . . 0 0 0 . . . −1 0
...

...
...

...
. . .

...
...

...
...

...
...

n
2 − 1 −1 0 0 . . . 2 0 −1 . . . 0 0

n
2 −1 0 0 . . . 0 1 0 . . . 0 0

n
2 + 1 −1 0 0 . . . −1 0 2 . . . 0 0
...

...
...

...
. . .

...
...

...
...

...
...

n− 2 −1 0 −1 . . . 0 0 0 . . . 2 0
n− 1 −1 −1 0 . . . 0 0 0 . . . 0 2

. (7)

Following the guideline in Theorem 6 with a = n− 1, b = 2, c = 1, and d = −1, then we
can write the following expression:

PL(ΓZn )
(λ) = λ(λ− n)(λ− 1)

n
2
−1(λ− 3)

n
2
−1.

The roots of PL(ΓZn )
(λ) = 0 are λ1 = 0 of multiplicity 1, λ2 = n of multiplicity 1, λ3 = 1

of multiplicity n
2 − 1, and λ4 = 3 of multiplicity n

2 − 1. Consequently, the spectrum of ΓZn

is

SpecL(ΓZn) =
{
(n)1 , (3)

n
2
−1, (1)

n
2
−1, (0)1

}
.

It is clear that the spectral radius of ΓZn is

ρL(ΓZn) = n.

Therefore, the Laplacian energy of ΓZn is as follows:

EL(ΓZn) = (1) |n|+
(n
2
− 1
)
|3|+

(n
2
− 1
)
|1|+ (1) |0|

= 3n− 4.

3.3. Signless Laplacian Matrix

Next, we show the energy of ΓZn with respect to the signless Laplacian matrix, for odd
and even n.

Theorem 11. Let ΓZn be the identity graph on Zn. The signless Laplacian energy of ΓZn

for odd n is

ESL(ΓZn) = 3(n− 1).



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2926

Proof. Based on Theorem 3, we have n × n degree matrix of ΓZn as diag(n −
1, 2, 2, ..., 2, 2). According to Definition 5 and Equation 2, we can construct an n × n
signless Laplacian matrix of ΓZn as follows:

SL(ΓZn) =D(ΓZn) +A(ΓZn) (8)

=

0 1 2 . . . n− 2 n− 1



0 n− 1 1 1 . . . 1 1
1 1 2 0 . . . 0 1
2 1 0 2 . . . 1 0
...

...
...

...
. . .

...
...

n− 2 1 0 1 . . . 2 0
n− 1 1 1 0 . . . 0 2

(9)

From Theorem 5 with a = n− 1, b = 2 and c = 1, we can simplify PSL(ΓZn )
(λ) as follows:

PSL(ΓZn )
(λ) = (λ2 − (2 + n)λ+ 2(n− 1))(λ− 3)

n−3
2 (λ− 1)

n−1
2 .

The roots of PSL(ΓZn )
(λ) = 0 are λ1 = 3 of multiplicity n−3

2 , λ2 = 1 of multiplicity n−1
2 ,

λ3,4 =
2+n
2 ±

√
n2−4n+12

2 of multiplicity 1, respectively. Consequently, the spectrum of ΓZn

is

SpecSL(ΓZn
) =


(
2 + n

2
+

√
n2 − 4n+ 12

2

)1

, (3)
n−3
2 , (1)

n−1
2 ,

(
2 + n

2
−

√
n2 − 4n+ 12

2

)1
 .

It is clear that the spectral radius of ΓZn is

ρSL(ΓZn) =
2 + n

2
+

√
n2 − 4n+ 12

2
.

Therefore, the signless Laplacian energy of ΓZn is as follows:

ESL(ΓZn) =

(
n− 3

2

)
|3|+

(
n− 1

2

)
|1|+

∣∣∣∣∣2 + n

2
±

√
n2 − 4n+ 12

2

∣∣∣∣∣
= 3(n− 1).

Theorem 12. Let ΓZn be the identity graph on Zn. The signless Laplacian energy of ΓZn

for even n is

ESL(ΓZn) = 3n− 4.

Proof. Since the diagonal matrix of ΓZn is D(ΓZn) = diag(n − 1, 2, 2, ..., 2, 1, 2, ..., 2).
Based on Definition 5 and Equation 3, we can construct an n×n signless Laplacian matrix
of ΓZn as follows:



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2915-2929 2927

SL(ΓZn) =D(ΓZn) +A(ΓZn) (10)

=

0 1 2 . . . n
2 − 1 n

2
n
2 + 1 . . . n− 2 n− 1



0 n− 1 1 1 . . . 1 1 1 . . . 1 1
1 1 2 0 . . . 0 0 0 . . . 0 1
2 1 0 2 . . . 0 0 0 . . . 1 0
...

...
...

...
. . .

...
...

...
...

...
...

n
2 − 1 1 0 0 . . . 2 0 1 . . . 0 0

n
2 1 0 0 . . . 0 1 0 . . . 0 0

n
2 + 1 1 0 0 . . . 1 0 2 . . . 0 0
...

...
...

...
. . .

...
...

...
...

...
...

n− 2 1 0 1 . . . 0 0 0 . . . 2 0
n− 1 1 1 0 . . . 0 0 0 . . . 0 2

. (11)

Again, by Theorem 6 with a = n− 1, b = 2, c = 1, and d = 1, we have

PSL(ΓZn )
(λ) =(λ3 − (n+ 3)λ2 + 3nλ− 2(n− 2))(λ− 3)

n
2
−2(λ− 1)

n
2
−1

=(λ− 2)(λ2 − (n+ 1)λ+ n− 2)(λ− 3)
n
2
−2(λ− 1)

n
2
−1.

The roots of PSL(ΓZn )
(λ) = 0 are λ1 = 2 of multiplicity 1, λ2 = 3 of multiplicity n

2 − 2,

λ3 = 1 of multiplicity n
2 − 1, and λ4,5 = n+1

2 ±
√
n2−2n+9

2 of multiplicity 1, respectively.
Consequently, the spectrum of ΓZn is

SpecSL(ΓZn) =


(
n+ 1

2
+

√
n2 − 2n+ 9

2

)1

, (3)
n
2 −2, (2)1, (1)

n
2 −1,

(
n+ 1

2
−

√
n2 − 2n+ 9

2

)1
 .

It is clear that the spectral radius of ΓZn is

ρSL(ΓZn) =
n+ 1

2
+

√
n2 − 2n+ 9

2
.

Therefore, the signless Laplacian energy of ΓZn is as follows:

ESL(ΓZn) =
(n
2
− 2
)
|3|+ (1) |2|+

(n
2
− 1
)
|1|+

∣∣∣∣∣n+ 1

2
±

√
n2 − 2n+ 9

2

∣∣∣∣∣
= 3n− 4.

4. Discussion

From the results of the previous section, we can conclude several interesting statements.



REFERENCES 2928

Corollary 1. The Laplacian energy of ΓZn is always similar to the signless Laplacian
energy of ΓZn.

Corollary 2. The energy of ΓZn is always an even integer associated with the Laplacian
and signless Laplacian matrices.

Corollary 3. The energy of ΓZn for odd n is never an odd integer associated with the
adjacency matrix.

Corollary 4. ΓZn is hyperenergetic associated with the Laplacian and signless Laplacian
matrices.

Acknowledgements

We wish to express our gratitude to University of Mataram, Indonesia, for providing
partial funding assistance.

References

[1] R B Bapat and S Pati. Energy of a Graph Is Never an Odd Integer. Bulletin of
Kerala Mathematics Association, 1:129–132, 2004.

[2] R A Borzooei and H Rashmanlou. Cayley interval-valued fuzzy graphs. UPB Scientific
Bulletin, Series A: Applied Mathematics and Physics, 78(3):83–94, 2016.

[3] A E Brouwer and W H Haemers. Spectra of graphs. Springer, New York, 2011.

[4] I Gutman. The Energy of Graph. Ber. Math.-Stat. Sekt. Forschungsz. Graz, 103:1–2,
1978.

[5] W B V Kandasamy and F Smarandache. Groups as Graphs. Editura CuArt, Romania,
2009.

[6] L M Kumari, L Pandiselvi, and K Palani. Quotient energy of zero divisor graphs and
identity graph. Baghdad Sci. J., 20(1(SI)):277–282, 2023.

[7] X Li, Y Shi, and I Gutman. Graph Energy. Springer, New York, 2012.

[8] V Lokesha, , Y Shanthakumari, and Y Zeba. Energy and Skew Energy of a Modified
Graph. Creat. Math. Inform., 30(1):41–48, 2021.

[9] S Pirzada and I Gutman. Energy of a Graph Is Never the Square Root of an Odd
Integer. Applicable Analysis and Discrete Mathematics, 2:118–121, 2008.

[10] N M Rilwan and A S Hussain. Different kinds of cordial labeling on identity graphs.
Eur. Chem. Bull., 12(10):12374–12381, 2023.



REFERENCES 2929

[11] M U Romdhini and A Nawawi. Degree subtraction energy of commuting and non-
commuting graphs for dihedral groups. International Journal of Mathematics and
Computer Science, 18(3):497–508, 2023.

[12] M U Romdhini and A. Nawawi. On the spectral radius and Sombor energy of the
non-commuting graph for dihedral groups. Mal. J. Fund. Appl. Sci., 20(1):65–73,
2024.

[13] M U Romdhini, A. Nawawi, F Al-Sharqi, and A Al-Quran. Closeness energy of
non-commuting graph for dihedral groups. Eur. J. Pure Appl. Math., 17(1):212–221,
2024.

[14] M U Romdhini, A. Nawawi, F Al-Sharqi, and A Al-Quran. Spectral Properties of
Power Graph of Dihedral Groups. Eur. J. Pure Appl. Math., 17(2):591–603, 2024.

[15] M U Romdhini, A. Nawawi, F Al-Sharqi, A Al-Quran, and S R Kamali. Wiener-
Hosoya energy of non-commuting graph for dihedral groups. Asia Pac. J. Math.,
11(9):1–9, 2024.

[16] Y Shanthakumari, M Smitha, and V Lokesha. Euclidean Degree Energy Graphs.
Montes Taurus J. Pure Appl. Math., 3(1):89–105, 2021.

[17] X Shi, S Kosari, A A Talebi, S H Sadati, and H Rashmanlou. Investigation of the
main energies of picture fuzzy graph and its applications. Int. J. Comput. Intell.
Syst., 15(1):31, 2022.

[18] A Siwach, V Bhatia, A Sehgal, and P Rana. Characteristic polynomial of maximum
and minimum matrix of square power graph of dihedral group of order 2n with odd
natural number n. Int. J. Stat. Appl. Math., 9(3):57–64, 2024.