EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 212-221
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Closeness Energy of Non-Commuting Graph for
Dihedral Groups

Mamika Ujianita Romdhini1,∗, Athirah Nawawi2, Faisal Al-Sharqi3, Ashraf Al-
Quran4

1 Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas
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 Basic Sciences Department, Preparatory Year Deanship, King Faisal University, Al-Ahsa,
Saudi Arabia

Abstract. This paper focuses on the non-commuting graph for dihedral groups of order 2n, D2n,
where n ≥ 3. We show the spectrum and energy of the graph corresponding to the closeness
matrix. The result is that the obtained energy is always twice its spectral radius and is never an
odd integer. Moreover, it is classified as hypoenergetic.

2020 Mathematics Subject Classifications: 05C25, 05C50, 15A18, 20D99

Key Words and Phrases: Closeness matrix, Energy of a graph, Non-commuting graph, Dihedral
Groups

1. Introduction

Let G be a group and Z(G) be a center of G. The non-commuting graph of G,
denoted as ΓG, has vertex set G\Z(G) and two distinct vertices vp, vq in ΓG are connected
by an edge whenever vpvq ̸= vqvp [1]. Many authors have studied non-commuting graphs
for various kinds of groups. According to Abdollahi [1], ΓG is always connected and its
diameter is always 2. Accordingly, (dpq), which is the shortest path between vp and vq,
is well defined in ΓG. This discussion continues by examining the isomorphic properties
of two non-commuting graphs related to the isomorphic properties of the corresponding
groups.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.5020

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

https://www.ejpam.com 212 © 2024 EJPAM All rights reserved.



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 212-221 213

The energy of ΓG, E(ΓG), is calculated by adding all the absolute values of its eigen-
values. This definition was pioneered by Gutman [6]. There is a classification of graphs
based on energy value [11]. Also, Sun et al. have shown that the clique path has the
maximum distance of eigenvalues and energy [23]. As has been shown in the literature,
the adjacency energy of a graph is never an odd integer and it is never its square root
either [4, 12].

A graph matrix based on the distance between two vertices was introduced by Indulal
and Gutman [7]. Readers may refer to [8] for information regarding degree product dis-
tance energy. In addition, Jog and Gurjar [9] discusses the degree sum exponent distance
of graphs. Accordingly, Romdhini et al. [16] investigated signless Laplacian energies of
interval-valued fuzzy graphs. In addition, Zheng and Zhou [24] presented the closeness
eigenvalues of graphs.

Throughout this work, the vertex set for ΓG is the non-abelian dihedral group of order
2n, where n ≥ 3, denoted by D2n =

〈
a, b : an = b2 = e, bab = a−1

〉
[3]. The center of

D2n is either Z (D2n) = {e} for n is odd, or
{
e, a

n
2

}
for n is even. The centralizer of

the element ai in D2n is CD2n(a
i) = {aj : 1 ≤ j ≤ n} and for the element aib is either

CD2n(a
ib) = {e, aib}, if n is odd or CD2n(a

ib) =
{
e, a

n
2 , aib, a

n
2
+ib

}
, if n is even.

Several authors have examined the energy of commuting and non-commuting graphs
involving D2n as the set of vertex. By considering the eigenvalues of the degree sum and
degree subtraction matrices, Romdhini and Nawawi [17, 19] and Romdhini et al. [22]
formulated the energy. In [18, 21], the sum of the degree exponent and the maximum and
minimum degree energies were presented for D2n. Therefore, the purpose of this paper is
to formulate the energy based on the closeness matrix for ΓG on D2n.

2. Preliminaries

In this part, we begin with the definition of the closeness matrix of a graph.

Definition 1. [24] Let dpq be the distance between vertex vp and vq. The closeness matrix
of order n× n associated with ΓG is given by C(ΓG) = [cpq] whose (p, q)-th entry is

cpq =

{
2−dpq , if vp ̸= vq
0, if vp = vq.

The closeness energy of ΓG can be written by

EC(ΓG) =
n∑

i=1

|λi| ,

where λ1, λ2, . . . , λn are eigenvalues of C(ΓG).

The spectral radius of ΓG corresponding with closeness matrix is

ρC(ΓG) = max{|λ| : λ ∈ SpecC(ΓG)}.



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 212-221 214

We know that ΓG has 2n − 1 and 2n − 2 vertices for odd and even n, respectively,
then ΓG corresponding to the C−matrix can be classified as hypoenergetic graph if the
C−energy complies with the statement below:

[11]EC(ΓG) <

{
2n− 1, for odd n
2n− 2, for even n,

The following theorem is useful to construct the closeness matrix of ΓG. We define
G1 = {ai : 1 ≤ i ≤ n}\Z (D2n) and G2 = {aib : 1 ≤ i ≤ n}.
Theorem 1. [10] For a non-commuting graph for G, ΓG,

(i) if G = G1, then ΓG
∼= K̄m, where m = |G1|,

(ii) if G = G2, then ΓG
∼=

{
Kn, if n is odd
Kn − n

2K2, if n is even.

where n
2K2 denotes n

2 copies of K2.

Lemma 1. [5] The adjacency spectrum of Kn is {(n− 1)(1), (−1)(n−1)}.
In order to simplify the determinant in the characteristic polynomial of ΓG, we need

the following three results.

Lemma 2. [13] If a, b, c and d are real numbers, then the determinant of the (n1 + n2)×
(n1 + n2) matrix of the form∣∣∣∣ (λ+ a)In1 − aJn1 −cJn1×n2

−dJn2×n1 (λ+ b)In2 − bJn2

∣∣∣∣
can be simplified in an expression as

(λ+ a)n1−1(λ+ b)n2−1 ((λ− (n1 − 1) a) (λ− (n2 − 1) b)− n1n2cd) ,

where 1 ≤ n1, n2 ≤ n and n1 + n2 = n.

Theorem 2. [20] If s, t are real numbers, then the characteristic polynomial of an n× n
matrix

M =

[
t(J − I)n

2
t(J − I)n

2
+ sIn

2

t(J − I)n
2
+ sIn

2
t(J − I)n

2

]
is

PM (λ) = (λ− s+ 2t)
n
2
−1 (λ− s− (n− 2) t) (λ+ s)

n
2 .

Theorem 3. [20] If r, s, t, u are real numbers, then the characteristic polynomial of an
(2n− 2)× (2n− 2) matrix

M =

 r(J − I)n−2 tJ(n−2)×n
2

tJ(n−2)×n
2

tJn
2
×(n−2) u(J − I)n

2
u(J − I)n

2
+ sIn

2

tJn
2
×(n−2) u(J − I)n

2
+ sIn

2
u(J − I)n

2


is

PM (λ) = (λ+ r)n−3 (λ− s+ 2u)
n
2
−1 (λ+ s)

n
2(

λ2 − (s+ (n− 2)u+ r(n− 3))λ+ r(n− 3) (s+ (n− 2)u)− n(n− 2)t2
)
.



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 212-221 215

3. Main Results

In this section, we begin with the distance between two distinct vertices in ΓG.

Theorem 4. Let ΓG be the non-commuting graph on G = G1 ∪ G2. For two distinct
vertices vp, vq ∈ V (ΓG), then the distance between vp and vq

(i) for the odd n, dpq =

{
2, if vp, vq ∈ G1

1, otherwise,
, and

(ii) for the even n, dpq =


2, if vp, vq ∈ G1

2, vp ∈ G2, vq ∈
{
a

n
2
+ib

}
1, otherwise.

Proof. For odd n case, since CD2n(a
i) = {aj : 1 ≤ j ≤ n}, then the vertex ai, for

1, 2, . . . , n − 1, is not adjacent to all vertices of G1, however, it always has an edge with
all members of G2. Thus, it is proven that dpq = 1, where vp belongs to G1 and vq ∈ G2,
or vice versa. Suppose now two distinct vertices ap, aq ∈ G1 with p ̸= q, meaning from ap

there are two vertices that must be passed to arrive at the terminal vertex vq, they are
one of aib and vq itself. From this fact, we then get dpq = 2.

While for the even n case, the centralizer of aib in D2n is {e, aib} implies that for
1 ≤ i ≤ n, vertex aib is connected with all other elements of G1 ∪ G2. Therefore, for
vp, vq ∈ G2, it is shown that dpq = 1. Now when n is even, as a result of CD2n(a

ib) =
{e, a

n
2 , aib, a

n
2
+ib} for all 1 ≤ i ≤ n, then vertices aib and a

n
2
+1b are always disconnected

in ΓG. Hence, for vp ∈ G2 and vq ∈
{
a

n
2
+ib

}
, dpq = 2. This also applies vice versa when

vq ∈ G2 and vp ∈
{
a

n
2
+ib

}
. However, when one of vp and vq is not in

{
a

n
2
+ib

}
, then

dpq = 1.

The closeness energy of the non-commuting graph on G, for G = G1 or G = G2 is
presented in the Theorem below:

Theorem 5. Let ΓG be the non-commuting graph on G.

(i) If G = G1, then EC(ΓG) is undefined, and

(ii) If G = G2, then EC(ΓG) =

{
n− 1, if n is odd
n− 3

2 , if n is even.
.

Proof.

(i) For G = G1 case, by Theorem 1 (1), ΓG
∼= K̄m, where m = |G1|. Then ΓG consists

of m isolated vertices which implies the distance of every pair vertices of G1 is
undefined.

(ii) For the second case when G = G2, we first proceed if n is odd. Again, by Theorem
1 (2), ΓG

∼= Kn. Then every pair of vertices are at distance 1. Now the closeness



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 212-221 216

matrix of ΓG is C(ΓG) = cpq, with (p, q)−entry if vp ̸= vq is 2−1, and zero if vp = vq.
Hence,

C(ΓG) =


0 1

2
1
2 . . . 1

2
1
2 0 1

2 . . . 1
2

1
2

1
2 0 . . . 1

2
...

...
...

. . .
...

1
2

1
2

1
2 . . . 0

 =
1

2
A(Kn).

In other words, C(ΓG) is the product of
1
2 and the adjacency matrix ofKn. Therefore,

from Lemma 1, the closeness energy of ΓG will be n− 1.

Meanwhile for the even n, by Theorem 1, ΓG
∼= Kn− n

2K2, then the distance between

every pair aib and a
n
2
+i for all 1 ≤ i ≤ n is 2, and 1, otherwise. Thus, C(ΓG) = cpq

and for vp ̸= vq,

cij =


1
4 , if vp = aib, vq = a

n
2
+ib, 1 ≤ i ≤ n

1
2 , if vp = aib, vq ̸= a

n
2
+ib, 1 ≤ i ≤ n

0, otherwise.

Now we can construct C(ΓG) as follows:

C(ΓG) =



0 . . . 1
2

1
4 . . . 1

2
...

...
...

...
. . .

...
1
2 . . . 0 1

2 . . . 1
4

1
4 . . . 1

2 0 . . . 1
2

...
...

...
...

. . .
...

1
2 . . . 1

4
1
2 . . . 0


=

[
1
2(J − I)n

2

1
4(J − I)n

2
+ 1

4In
2

1
4(J − I)n

2
+ 1

4In
2

1
2(J − I)n

2

]
.

In this case, we have four block matrices of C(ΓG):

C(ΓG) =

[
A B
B A

]
,

where A is a matrix of order n
2 with zero diagonal entries and all of the non-diagonal

entries as 1
2 and B is the matrix of order n

2 with diagonal entries are 1
4 and the

non-diagonal entries are 1
2 . By Theorem 2 with s = 1

4 and t = 1
2 , Equation 2 is

PC(ΓG)(λ) =

(
λ+

3

4

)n
2
−1(

λ+
3

4
− 1

2
n

)(
λ+

1

4

)n
2

.

Therefore, using the roots of Equation 2, the closeness energy of ΓG is

EC(ΓG) =
(n
2

) ∣∣∣∣−1

4

∣∣∣∣+ (n
2
− 1

) ∣∣∣∣−3

4

∣∣∣∣+ ∣∣∣∣12n− 3

4

∣∣∣∣ = n− 3

2
.



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 212-221 217

Theorem 6. The characteristic polynomial of ΓG, where G = G1 ∪G2, is

(i) for n is odd: PC(ΓG)(λ) = (λ+ 2)n−2 (λ+ 1)n−1
(
λ2 − (3n− 5)λ+ (n− 1)(n− 4)

)
,

(ii) for n is even: PD(ΓG)(λ) = λ
n
2
−1 (λ+ 2)n−3+n

2
(
λ2 − 3 (n− 2)λ+ n(n− 4)

)
.

Proof.

(i) When n is odd and G = G1 ∪G2, by Theorem 4, we have the distance of every pair
of vertices. Since Z(D2n) = {e}, consequently, ΓG has 2n − 1 vertices. They are
n− 1 vertices of ai, for 1 ≤ i ≤ n− 1, and n vertices of aib, 1 ≤ i ≤ n. Hence, from
Definition 1, C(ΓG) is a (2n− 1)× (2n− 1) matrix as the following:

C(ΓG) =



0 . . . 1
4

1
2 . . . 1

2
...

. . .
...

...
. . .

...
1
4 . . . 0 1

2 . . . 1
2

1
2 . . . 1

2 0 . . . 1
2

...
. . .

...
...

. . .
...

1
2 . . . 1

2
1
2 . . . 0


=

[
1
4(J − I)n−1

1
2J(n−1)×n

1
2J(n−1)×n

1
2(J − I)n

]
.

Now the characteristic polynomial of Equation 1 is

PC(ΓG)(λ) = |λI2n−1 − C(ΓG)| =
∣∣∣∣ (λ+ 2)In−1 − 2Jn−1 −J(n−1)×n

−Jn×(n−1) (λ+ 1)In − Jn)

∣∣∣∣ .
Using Lemma 2, with a = 1

4 , b = 1
2 , c = d = 1

2 , and n1 = n − 1, n2 = n, then we
obtain the formula of PC(ΓG)(λ),

PC(ΓG)(λ) =

(
λ+

1

4

)n−2(
λ+

1

2

)n−1(
λ2 + 1−

(
3

4
n

)
λ− 1

8
(n+ 2)(n− 1)

)
(ii) Now for the even n case and G = G1 ∪G2, we know that Z(D2n) = {e, a

n
2 }. Then,

the cardinality of the vertex set of ΓG is 2n− 2 with detail n− 2 vertices of ai, for
1 ≤ i < n

2 ,
n
2 < i < n, and n vertices of aib, for 1 ≤ i ≤ n. Following the result of

Theorem 4 and by Definition 1, then C(ΓG) is a (2n − 2) × (2n − 2) matrix as the
following:

C(ΓG) =



0 . . . 1
4

1
2 . . . 1

2
1
2 . . . 1

2
...

. . .
...

...
. . .

...
...

. . .
...

1
4 . . . 0 1

2 . . . 1
2

1
2 . . . 1

2
1
2 . . . 1

2 0 . . . 1
2

1
4 . . . 1

2
...

. . .
...

...
. . .

...
...

. . .
...

1
2 . . . 1

2
1
2 . . . 0 1

2 . . . 1
4

1
2 . . . 1

2
1
4 . . . 1

2 0 . . . 1
2

...
. . .

...
...

. . .
...

...
. . .

...
1
2 . . . 1

2
1
2 . . . 1

4
1
2 . . . 0





M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 212-221 218

Now we provide nine block matrices of C(ΓG) as follows:

C(ΓG) =


1
4(J − I)n−2

1
2J(n−2)×n

2

1
2J(n−2)×n

2
1
2Jn

2
×(n−2)

1
2(J − I)n

2

1
2(J − I)n

2
+ 1

4In
2

1
2Jn

2
×(n−2)

1
2(J − I)n

2
+ 1

4In
2

1
2(J − I)n

2

 .

By Theorem 3 with r = s = 1
4 and t = u = 1

2 , we then obtain

PC(ΓG)(λ) =

(
λ+

1

4

) 3n−6
2

(
λ+

3

4

)n
2
−1(

λ2 − 3

4
(n− 2)λ− 1

16

(
2n2 + n− 9

))
.

Theorem 7. The C−spectral radius for ΓG, where G = G1 ∪G2, is

(i) for n is odd: ρC(ΓG) =
1
8

(
3n− 4 +

√
n(17n− 16)

)
,

(ii) for n is even: ρC(ΓG) =
1
8

(
3n− 6 +

√
n(17n− 32)

)
.

Proof.

(i) According to Theorem 6 (1), for the odd n case gives four eigenvalues. They are
λ1 = −1

4 of multiplicity (n − 2), λ2 = −1
2 of multiplicity (n − 1), and λ3,4 =

1
8

(
3n− 4±

√
n(17n− 16)

)
. Hence, the spectrum of ΓG as the following:

SpecC(ΓG) =

{(
1

8

(
3n− 4 +

√
n(17n− 16)

))1

,

(
−1

4

)n−2

,

(
−1

2

)n−1

,

(
1

8

(
3n− 4−

√
n(17n− 16)

))1
}
.

We take the maximum absolute eigenvalues and get the spectral radius of ΓG as the
desired result.

(ii) For n is even and following Theorem 6 (2) implies that ΓG has four eigenvalues.
They are λ1 = −1

4 of multiplicity n − 3 + n
2 , λ2 = −3

4 of multiplicity n
2 − 1 and

λ3,4 =
1
8

(
3n− 6±

√
n(17n− 32)

)
. Hence, the spectrum of ΓG as the following:

SpecC(ΓG) =

{(
1

8

(
3n− 6 +

√
n(17n− 32)

))1

,

(
−1

4

)n−3+n
2

,

(
−3

4

)n
2
−1

,

(
1

8

(
3n− 6−

√
n(17n− 32)

))1
}
.

The maximum of |λi|, i = 1, 2, 3, 4 is the C−spectral radius of ΓG, and we complete
the proof.



M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 212-221 219

Theorem 8. The C−energy for ΓG, where G = G1 ∪G2, is

(i) for n is odd: EC(ΓG) =
1
4

(
3n− 4 +

√
n(17n− 16)

)
(ii) for n is even: EC(ΓG) =

1
4

(
3n− 6 +

√
n(17n− 32)

)
.

Proof.

(i) By Theorem 7 (1), for the odd n, the C−energy of ΓG can be calculated as follows:

EC(ΓG) = (n− 2)

∣∣∣∣−1

4

∣∣∣∣+ (n− 1)

∣∣∣∣−1

2

∣∣∣∣+ ∣∣∣∣18 (
3n− 4±

√
n(17n− 16)

)∣∣∣∣
=

1

4

(
3n− 4 +

√
n(17n− 16)

)
.

(ii) For even n, by Theorem 7 (2), then the C−energy of ΓG is

EC(ΓG) =

(
3n− 6

2

) ∣∣∣∣−1

4

∣∣∣∣+ (n
2
− 1

) ∣∣∣∣−3

4

∣∣∣∣+ ∣∣∣∣18 (
3n− 6±

√
n(17n− 32)

)∣∣∣∣
=

1

4

(
3n− 6 +

√
n(17n− 32)

)
.

As a result of Theorem 8, in the following, we obtain the classification of the closeness
energy of ΓG for D2n, where G = G1 ∪G2.

Corollary 1. ΓG associated with the closeness matrix is hypoenergetic.

Moreover, based on the energies in Theorem 8, we can conclude the following fact:

Corollary 2. C−energy for ΓG is never an odd integer.

The statements in Corollary 2 comply with the well-known facts from [4] and [12].
Furthermore, the comparison between energy in Theorem 8 and its spectral radius in
Theorem 7 can be determined in the following statement:

Corollary 3. C−energy for ΓG is always twice its spectral radius.

As a future view of this research, we recommend combining them with [2], which is
essentially an extension of the graph matrix based on Q-NSS matrix. In addition, this
work can be extended to the neutrosophic soft rings and neutrosophic soft field [14, 15].

Acknowledgements

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



REFERENCES 220

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