Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

CLOSENESS LAPLACIAN AND CLOSENESS

SIGNLESS LAPLACIAN ENERGIES OF

NON-COMMUTING GRAPH FOR DIHEDRAL

GROUPS^∗

Mamika Ujianita Romdhini^†

Abdurahim Abdurahim^‡

Andika Ellena Sauﬁka Hakim Maharani^§

Athirah Nawawi^¶

Faisal Al-Sharqi^k

Ifan Hasnan Dani^∗∗

Communicated by G. Failla

10.56082/annalsarscimath.2026.2.145

DOI

Abstract

This research investigates the relationship between algebra and

graph theory, speciﬁcally how algebra facilitates graph theory. Asso-

ciating matrices with graphs introduces the concept of graph energies.

A new energy formula of non-commuting graphs for dihedral groups

^∗Accepted for publication on January 15, 2026

^†mamika@unram.ac.id, Department of Mathematics, Faculty of Mathematics and Nat-

ural Sciences, University of Mataram, Mataram 83125, Indonesia

^‡abdurahim@staff.unram.ac.id, Department of Mathematics, Faculty of Mathematics

and Natural Sciences, University of Mataram, Mataram 83125, Indonesia

^§a.ellena.saufika@staff.unram.ac.id, Department of Mathematics, Faculty of

Mathematics and Natural Sciences, University of Mataram, Mataram 83125, Indonesia

^¶athirah@upm.edu.my, Department of Mathematics and Statistics, Faculty of Science,

Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

^kfaisal.ghazi@uoanbar.edu.iq, Department of Mathematics, Faculty of Education

for Pure Sciences, University Of Anbar, Ramadi, Anbar, Iraq; College of Engineering, Na-

tional University of Science and Technology, Dhi Qar, Iraq; Department of Mathematics,

College of Education, Al-Ayen Iraqi University, An Nasiriyah, Iraq

^∗∗ifanhasnandani@gmail.com, Department of Mathematics, Faculty of Mathematics

and Natural Sciences, University of Mataram, Mataram 83125, Indonesia

145

Energy of non-commuting graph for dihedral groups

146

using closeness Laplacian and closeness signless Laplacian matrices is

investigated in this paper. It is found that both energies are always

equivalent and are categorized as hyperenergetic.

Keywords: energy of a graph, non-commuting graph, dihedral group,

closeness Laplacian matrix, closeness signless Laplacian matrix.

MSC: 05C25, 15A18.

1 Introduction

Algebraic graph theory is currently a prominent topic of research. This

study investigates the relationship between algebra and graph theory, focus-

ing on how algebra facilitates graph studies. This ﬁeld of study is signiﬁcant,

and it contributes to various other areas. Chemical graph theory is a branch

of mathematical chemistry that applies graph theory to the mathematical

modeling of chemical compounds mentioned by [16 ]. It also includes a dis-

cussion of graph energies presented by [5] by considering a chemical molecule

as a graph and estimating the π-electron energy. The eigenvalues of the ad-

jacency matrix denote the energy level of the electron in the molecule.

Neumann [8] introduced the concept of non-commuting graphs. How-

ever, [1 ] conducted extensive research on the features of non-commuting

graphs. Let G be a ﬁnite group and Z(G) be its center. The non-commuting

graph of G, represented by Γ_G, contains G excluding the center of G as its

vertex set, with two diﬀerent vertices v_pand v_qconnected by an edge when-

ever v_pv_q= v_qv_p.

The vertex set of this research is dihedral groups. For n ≥ 3, the

non-abelian dihedral group of order 2n is deﬁned as reﬂection and rota-

tion motions that return a regular n-gon to its initial state, with compo-

sition as the operation, represented by D_2n. The n rotations are aⁱ, and

the reﬂections are aⁱb, with 1 ≤ i ≤ n. Thus, D_2ncan be expressed as

ꢀ

ꢁ

a, b : aⁿ= b²= e, bab = a⁻¹[3]. For further study, we denote the non-

commuting graph for D_2nas Γ_D. Based on the center of D_2n, it is clear

2n

that Γ_D

has 2n − 1 and 2n − 2 vertices for odd and even n, respectively.

2n

The discussion on Γ_D

with several matrices has been done involving

2n

closeness matrix [12], Seidel Laplacian and Seidel signless Laplacian ma-

trices [13], Sombor matrix [14], and Wiener Hosoya matrix [15]. In 2022,

Zheng and Zhou introduced the deﬁnition of closeness Laplacian and close-

ness signless Laplacian matrices of a graph [17 ] as the extension of the close-

ness matrix of a graph [18]. On the other hand, distance-based matrices,

especially the distance Laplacian, have gained signiﬁcant attention for their

M.U. Romdhini, A. Abdurahim, A.E.S.H. Maharani, A. Nawawi, F.

Al-Sharqi, I.H. Dani

147

ability to capture both structural and spectral properties of graphs [11].

Notably, their characteristic polynomials and spectra have been analyzed

for algebraically deﬁned graphs, including power graphs [10]. These ﬁnd-

ings motivated us to construct the matrices for Γ_D

and to formulate the

2n

spectrum and energy of Γ_D

.

2n

2 Preliminaries

This section focuses on the basic deﬁnitions and theorems from the previous

literature. Let d_pqbe the distance between vertex p and q. We denote

P

−d_pq

c(p) =

2

, for adjacent vertices p and q. Hence, we have

q∈V (Γ_D_2n)\{p}

two deﬁnitions below.

Deﬁnition 1. [17] The closeness Laplacian matrix of Γ_D

is given by

2n

CL(Γ_D) = [l_pq] whose (p, q)-entry is

2n

ꢂ

−2^−d

,

if p = q

pq

l_pq

=

c(p),

if p = q.

Deﬁnition 2. [17] The closeness signless Laplacian matrix of Γ_D

is given

2n

by CSL(Γ_D) = [s_pq] whose (p, q)-entry is

2n

ꢂ

2^−d

,

if p = q

pq

s_pq

=

c(p),

if p = q.

Deﬁnition 1 and Deﬁnition 2 require the distance between any two ver-

tices in a graph. Therefore, we write the following theorem to construct the

matrices in the next section.

Theorem 1. [12] The distance between two distinct vertices p, q in Γ_D

is

2n

ꢂ

2,

if v_p, v_q∈ G₁

1. for odd n, d_pq

=

, and

1,

otherwise,



2,

if v_p, v_q∈ G₁





n

o

n

2

2. for even n, d_pq

=

2, v_p∈ G₂, v_q∈ a ⁺ⁱb





1,

otherwise,

where G₁is the set of rotations (excluding identity), and G₂is the set of

reﬂections.

Energy of non-commuting graph for dihedral groups

148

The spectrum of Γ_D

It is deﬁned as λ^k₁¹, λ^k₂², . . . , λ^k_m^m, where λ₁, λ₂, . . . , λ_mare eigenvalues of

is denoted by Spec_CL(Γ_D) or Spec_CSL(Γ_D).

2n

n

o

CL(Γ_D) or CSL(Γ_D), and k₁, k₂, . . . , k_mare their respective multiplici-

2n

ties. Moreover, the spectral radius of Γ_D

as max {λ|λ ∈ Spec(Γ_D)}.

i=1

2n

P

m

Furthermore, the energy of Γ_D

is deﬁned as

|λ_i| [5]. Γ_{D n}can be

2n

2

categorized as hyperenergetic if the energy is greater than 4(n − 1) for odd

n, or it is greater than 4(n − 1) − 2 for even n [7].

3 Main results

3.1

Characteristic polynomial

Let J_nbe the matrix of size n×n with all entries are 1, and I_nbe an identity

matrix of size n × n. In this subsection, we prove two beneﬁcial theorems to

formulate the characteristic polynomial of matrices. In the process of proof,

we use row and column operations with the following notation:

1. R_i: the i−th row;

2. R_i⁰: the new i−th row obtained from a row operation;

3. C_i: the i−th column;

4. C_i⁰: the new i−th column obtained from a column operation.

Theorem 2. Suppose s, t, u, and v are real numbers. The characteristic

polynomial of a (2n − 1) × (2n − 1) matrix,

ꢃ

ꢄ

(s + v)I_n−1− vJ_n−1

−tJ_(n−1)×n

M =

−tJ_n×(n−1)

(u + t)I_n− tJ_n

can be simpliﬁed into an expression as

ꢅ

P_M(λ) = (λ − s − v)ⁿ⁻²(λ − u + t(n − 1))(λ − s + (n − 2)v)

ꢆ

+(1 − n)nt²(λ − t − u)ⁿ⁻¹

.

Proof. For real numbers s, t, u and v, the characteristic polynomial of M is

ꢇ

(λ − s − v)I_n−1+ vJ_n−1

tJ_(n−1)×n

P_M(λ) =

.

tJ_n×(n−1)

(λ − u − t)I_n+ tJ_n)

M.U. Romdhini, A. Abdurahim, A.E.S.H. Maharani, A. Nawawi, F.

Al-Sharqi, I.H. Dani

149

Step 1: Replace R_1+iby R₁⁰_+i= R_1+i− R₁, for every 1 ≤ i ≤ n − 2, and

R_n+iby R_n⁰_+i= R_n+i− R_n, for every 1 ≤ i ≤ n − 1 Then we have P_M(λ)

as follows:

P_M(λ)

ꢇ

λ − s

vJ_1×(n−2)

t

tJ_1×(n−1)

ꢇ

(v + s − λ)J_(n−2)×1(λ − s − v)I_n−2

0_(n−2)×1

λ − u

0_n−1

=

.

t

tJ_1×(n−2)

0_{(n−1)×(n−2)}

tJ_1×(n−1)

ꢇ

0_(n−1)×1

(t + u − λ)J_(n−1)×1(λ − u − t)I_n−1

Step 2: Replace C₁with C⁰= C₁+ C₂+ . . . + C_n−1and replace C_n⁰

C_n+ C_n+1+ . . . + C_2n−1, then we have

=

1

P_M(λ)

ꢇ

λ − s + (n − 2)v

0_(n−2)×1

vJ_1×(n−2)

(λ − s − v)I_n−2

tJ_1×(n−2)

tn

tJ_1×(n−1)

ꢇ

0_(n−2)×1

0_n−1

=

.

t(n − 1)

λ − u + t(n − 1)

tJ_1×(n−1)

ꢇ

0_(n−1)×1

0_{(n−1)×(n−2)}

0_(n−1)×1

(λ − u − t)I_n−1

Step 3: Replace R_nby R⁰= R_n+ _{λ−s+(n−2)v}R₁and following by

t(1−n)

n

(s − λ) + tv

R⁰= R_n+

R₂+

R₃

n

(λ − s + (n − 2)v)(λ − s − v)

(s − λ) + tv

+ . . . +

R_n−1

,

(λ − s + (n − 2)v)(λ − s − v)

then we can write P_M(λ) as

ꢇ

λ − s + (n − 2)v

vJ_1×(n−2)

tn

0_(n−2)×1

ꢇ

0_(n−2)×1

(λ − s − v)I_n−2

ꢇ

(λ − u + t(n − 1))(λ − s + (n − 2)v) + (1 − n)nt²

0

0_1×(n−2)

λ − s + (n − 2)v

ꢇ

0_(n−1)×1

0_{(n−1)×(n−2)}

0_(n−1)×1

ꢇ

tJ_1×(n−1)

ꢇ

0_n−1

ꢇ

(1 − n)t²+ (λ − s + (n − 2)v)t

J_1×(n−1)

λ − s + (n − 2)v

ꢇ

(λ − u − t)I_n−1

Energy of non-commuting graph for dihedral groups

150

It is a diagonal matrix which implies that

ꢅ

P_M(λ) = (λ − s − v)ⁿ⁻²(λ − u + t(n − 1))(λ − s + (n − 2)v)

ꢆ

+(1 − n)nt²(λ − t − u)ⁿ⁻¹

.

Theorem 3. Suppose r, s, t and u are real numbers. The characteristic

polynomial of a (2n − 2) × (2n − 2) matrix





n

2

n

2

(r + u) I_n−2− uJ_n−2

−tJ_(n−2)×









n

−tJ

n

2

n

2

(s + t) I − tJ

(−u + t)I − tJ

M =

₂×(n−2)

2

n

−tJ

n

2

n

2

n

2

n

2

(−u + t)I − tJ

(s + t) I − tJ

₂×(n−2)

is

n

2

P_M(λ) =(λ − r − u)ⁿ⁻³(λ + u − s − 2t)

(λ − s − u)

₂−1

ꢅ

λ²+ ((n − 2)(u + t) − r − s)λ

+ (u(n − 3) − r)(u − s + (n − 2)t) − n(n − 2)t²

ꢆ

.

Proof. The determinant below is the characteristic polynomial of M

P_M(λ)

ꢇ

n

2

n

2

(λ − r − u) I_n−2+ uJ_n−2

tJ_(n−2)×

n

2

n

2

tJ

(λ − s − t) I + tJ

(u − t)I + tJ

=

.

₂×(n−2)

2

n

2

n

2

n

2

n

2

tJ

(u − t)I + tJ

(λ − s − t) I + tJ

₂×(n−2)

ⁿ₂+i

The ﬁrst stage we operate row operations by replacing R_n−2+

with

R_n⁰₋₂₊_n

= R_n−2+

− R_n−2+i, for every 1 ≤ i ≤ ⁿ₂, and we have

n

₂+i

P_M(λ)

ꢇ

n

2

n

2

(λ − r − u) I_n−2+ uJ_n−2

tJ_(n−2)×

n

2

n

2

tJ

(λ − s − t) I + tJ

(u − t)I + tJ

=

.

₂×(n−2)

2

n

2

n

2

0

(−λ + s + u)I

(λ − s − u) I

₂×(n−2)

0

n

2

Then, we consider column operations C_n−2+i= C_n−2+i+ C_n−_{2+ +i}, for

n

2

every 1 ≤ i ≤

− 1,

P_M(λ)

M.U. Romdhini, A. Abdurahim, A.E.S.H. Maharani, A. Nawawi, F.

Al-Sharqi, I.H. Dani

151

ꢇ

n

2

n

2

(λ − r − u) I_n−2+ uJ_n−2

2tJ_(n−2)×

tJ_(n−2)×

n

2

n

2

tJ

(λ + u − s − 2t) I + 2tJ

(u − t)I + tJ

=

.

₂×(n−2)

2

n

2

0

(λ − s − u) I

₂×(n−2)

2

0

n

Following by R_n−1+i= R_n−1+i−R_n−1, for every 1 ≤ i ≤ ₂−1, and replacing

C_n−1with C_n⁰₋₁= C_n−1+ C_n+ C_n+1+ . . . + C

, consequently, P_M(λ)

n

n−2+ ₂

can be written as

ꢇ

λ − r

(n−3)×1

uJ_1×(n−3)

(λ − r − u) I_n−3+ uJ_n−3

tJ_1×(n−3)

nt

2tJ_1×

2tJ_(n−3)×

2tJ_1×

n

2

−1

(

)

uJ

ntJ_(n−3)×1

λ + u − s − t(n − 2)

n

2

−1

(

n

)

ꢇ

t

−1

(

)

2

n

0

(λ + u − s − 2t)I

n

2

n

2

n

2

₂−1

−1 ×1

0

−1 ×(n−3)

−1 ×1

0

(

)

(

)

(

)

0_1×(n−3)

0_1×

0

n

2

−1

(

)

n

0

n

2

n

2

n

2

₂−1

−1 ×1

−1 ×(n−3)

−1 ×1

)

(

)

(

)

ꢇ

t

tJ_1×

(

n

2

−1

)

tJ_(n−3)×1

tJ_(n−3)×

n

2

−1

(

)

ꢇ

u

tJ_1×

(

n

2

−1

)

n

(t − u)J

(u − t)I

n

2

₂−1

−1 ×1

(

)

λ − s − u

0_1×

n

2

−1

(

)

n

0

(λ − s − u)I

n

2

₂−1

−1 ×1

(

)

The next step is operating C⁰= C_i− C_n−2, for 1 ≤ i ≤ n − 3 and following

i

by R_n⁰₋₂= R_n−2+R₁+R₂+. . .+R_n−3. Hence, we derive P_M(λ) as follows:

ꢇ

(λ − r − u)I_n−3

0_1×(n−3)

uJ_(n−3)×1

ntJ_(n−3)×1

n(n − 2)t

2tJ_(n−3)×

(

n

2

−1

)

λ − r + (n − 3)u

2(n − 2)tJ_1×

n

2

−1

(

)

ꢇ

0_1×(n−3)

t

λ + u − s − t(n − 2)

2tJ_1×

n

2

−1

(

)

n

0

(λ + u − s − 2t)I

n

2

n

2

₂−1

−1 ×1

0

−1 ×1

(

)

(

)

0_1×(n−3)

0

0_1×

0

n

2

−1

(

)

n

0

n

2

n

2

₂−1

−1 ×1

)

(

)

ꢇ

tJ_(n−3)×

(

n

2

n

2

−1

(

)

(n − 2)t

(n − 2)tJ_1×

n

2

−1

(

)

ꢇ

u

tJ_1×

n

2

−1

₂−1

(

)

n

(t − u)J

(u − t)I

n

2

−1 ×1

(

)

λ − s − u

0_1×

n

2

−1

(

)

n

0

(λ − s − u)I

n

2

₂−1

−1 ×1

(

)

Energy of non-commuting graph for dihedral groups

152

Then we can simplify P_M(λ) as

n

2

P_M(λ) =(λ − r − u)ⁿ⁻³(λ + u − s − 2t)

(λ − s − u)

₂−1

ꢅ

λ²+ ((n − 2)(u + t) − r − s)λ

+ (u(n − 3) − r)(u − s + (n − 2)t) − n(n − 2)t²

ꢆ

.

3.2

Closeness Laplacian energy

We begin this subsection by formulating the characteristic polynomial of

Γ_Dassociated with the closeness Laplacian matrix, CL(Γ_D).

2n

Theorem 4. The characteristic polynomial of Γ_D

associated with CL(Γ_D

)

2n

is

ꢈ

ꢉ

ꢈ

ꢉ

n−2

n

1

4

1

2

1. for odd n: P_CL(Γ

₎(λ) = λ λ − (3n − 1)

λ − n +

,

D

2n

2. for even n:

P_CL(Γ

₎(λ)

D

2n

n

2

ꢃ

ꢄ

ꢃ

ꢄ ꢃ

ꢄ

n−3

n

3

λ − n +

4

1

2

1

₂−1

=

(λ − n)

λ − n +

λ²− nλ − n .

2

Proof.

1. By Theorem 1 for odd n, we have d_pq= 2, if v_p, v_q∈ G₁, and

zero otherwise. By excluding e from the set of vertex, hence Γ_D

has

2n

2n−1 vertices. Using Deﬁnition 1, CL(Γ_D) of size (2n−1)×(2n−1)

2n

is as follows:





3n−2

1

4

1

2

1

2

. . .

.

−

. . .

.

−

4









.

1

−

4

3n−2

1

−

2

1

2

1

. . .

.

−

. . .

−

4

2

CL(Γ_D) =

1

−

2

1

2n

n − 1 . . .

−

2

.

1

−

2

1

−

2

1

2

. . .

−

. . . n − 1

−₂J_(n−1)×n

ꢊ

ꢋ

1

3n−1

1

−₄J_n−1

+

I

n−1

4

=

.

− J_n+ ²ⁿ⁻¹I_n

1

2

−₂J_(n−1)×n

2

Using Theorem 2, with s = ³ⁿ⁻², t = ¹₂, u = n − 1, v = ¹₄, then we

4

obtain the formula of P_CL(Γ

₎(λ),

n

D

2

ꢃ

ꢄ

ꢃ

ꢄ

n−2

n

1

2

P_CL(Γ

₎(λ) = λ λ − (3n − 1)

λ − n +

.

D

2n

4

M.U. Romdhini, A. Abdurahim, A.E.S.H. Maharani, A. Nawawi, F.

Al-Sharqi, I.H. Dani

153

n

2

2. Now for the even n case, by excluding e, a

from the vertex set of

Γ_{D n}. Therefore, Γ_{D n}consists of 2n − 2 vertices. Based on Theorem

2

1 and Deﬁnition 1, then CL(Γ_D) of size (2n − 2) × (2n − 2) is as

2n

follows:

CL(Γ_D

)

2n





1

4

1

−

2

1

2

1

2

1

2

³₄(n − 1) . . .

−

. . .

.

−

. . .

.

−





.

1

−

2

1

−

. . . ³₄(n − 1)

. . .

.

−

. . .

.

−

4

2

1

2

3

n −

4

1

−

. . .

.

−





2

4

2





.

=

.

1

2

1

2

1

−

2

3

4

1

2

1

4

1

−

. . .

.

−

. . . n −

−

. . .

.

−

1

−

4

1

2

3

4

−

. . .

.

−

n −

−

2

.

1

2

1

2

1

−

2

1

4

1

2

3

4

−

. . .

−

. . .

−

. . . n −

The form of CL(Γ_D) can be written as follows:

2n

CL(Γ_D

)

2n





ꢈ

ꢉ

³₄n −

I_n−2− J_n−2

− J_(n−2)×

−₂J_(n−2)×

1

2

1

4

1

n

2

n

2

n −²I − J

₄I − J

ꢈ

ꢉ









1

4

1

n

2

n

²1

−₂J

=

.

₂×(n−2)

2

n

2

²1

2

1

4

ꢈ

ꢉ

1

n

2

n

2

−₂J

₄I − J

n −

I − J

2

By Theorem 3 with r = ³₄(n − 1), s = n − , t = ¹₂, and u = ¹₄, we

3

4

then obtain

P_CL(Γ

₎(λ)

D

2n

n

2

ꢃ

ꢄ

ꢃ

ꢄ ꢃ

ꢄ

n−3

n

3

λ − n +

4

1

2

1

2

1

₂−1

=

(λ − n)

λ − n +

λ²− nλ − n .

2

We have the following theorem and theorem 6 to determine the CL−energy

of Γ_D

.

2n

Theorem 5. The CL−spectral radius for Γ_D

is

2n

1

1. for odd n: ρ_CL(Γ_D) = n − ,

2n

2

2. for even n: ρ_CL(Γ_D) = n.

2n

Energy of non-commuting graph for dihedral groups

154

Proof.

1. The result follows from Theorem 4 (1), where n is odd, implies

1

there are 3 eigenvalues. First, we have λ₁= (3n − 1) of multiplicity

4

1

2

(n − 2), λ₂= n −

of multiplicity n, and a single λ₃= 0. Hence, the

spectrum of Γ_D

as the following:

2n

(

)

ꢃ

ꢄ

ꢃ

ꢄ

n

n−2

1

2

1

4

Spec_CL(Γ_D) =

2n

n −

,

(3n − 1)

, (0)¹

.

The desired result is the spectral radius of Γ_D, which we obtain by

2n

taking the greatest absolute eigenvalues.

2. Recall from Theorem 4 (2) for even n that Γ_D

consists of 5 eigen-

2n

1

values. It follows that λ₁= ³₄n −

of multiplicity n − 3, λ₂= n

n(n −²2) . Hence, the spectrum of Γ_D

2

n

1

2

n

2

of multiplicity

− 1 and λ₃= n −

of multiplicity

and λ_4,5

=

ꢅ

ꢆ

p

1

2

n ±

as the following:

2n

Spec_CL(Γ_D

)

2n

(

ꢃ

n

2

ꢃ

p

ꢄ

ꢃ

ꢄ

ꢃ

ꢄ

n−3

1

ꢅ

ꢆ

p

n

2

1

2

3

4

)

1

2

1

2

=

(n) ⁻¹, n −

,

n −

,

n +

n(n − 2)

,

ꢄ

1

ꢅ

ꢆ

1

2

n −

n(n − 2)

.

We ﬁnish the proof by determining the CL−spectral radius of Γ_D

as

2n

the maximum of |λ_i|, i = 1, 2, 3, 4.

Theorem 6. The CL−energy for Γ_D

is

2n

1

1. for odd n: E_CL(Γ_D) = (n − 1)(7n − 2)

2n

4

ꢈ

ꢉ

1

4

2. for even n: E_CL(Γ_D) =

2n

7n²− 12n + 6 .

Proof.

1. Let n be odd. By Theorem 5 (1), it follows that the CL−energy

of Γ_D

is

2n

E_CL(Γ_D

)

2n

ꢇ

1

2

1

4

1

= (n) n −

+ (n − 2)

(3n − 1) + (1) |0| = (n − 1)(7n − 2).

ꢇ

4

M.U. Romdhini, A. Abdurahim, A.E.S.H. Maharani, A. Nawawi, F.

Al-Sharqi, I.H. Dani

155

2. Let n be even. Then according to Theorem 5 (2), the CL−energy of

Γ_D

is

2n

ꢇ

ꢅ

ꢆ

n ±

ꢅ ꢆ

n(n − 2)

n

1

2

3

4

1

2

E_CL(Γ_D) =

− 1 |n| +

n −

+ (n − 3)

n +

2n

2

ꢇ

ꢅ

ꢆ

p

ꢇ

ꢈ

ꢉ

1

+

=

7n²− 12n + 6 .

2

4

3.3

Closeness signless Laplacian

We begin this subsection with the following theorem which describes the

characteristic formula of the closeness signless Laplacian matrix of Γ_D

,

2n

CSL(Γ_D).

2n

Theorem 7. The characteristic polynomial of CSL(Γ_D) is

2n

1. for n is odd:

ꢃ

ꢄ

ꢃ

ꢄ

n−2

n−1

3

2

P_CSL(Γ

₎(λ) = λ − (n − 1)

λ − n +

D

2n

4

ꢃ

ꢄ

5

1

λ²− (n − 1)λ + (5n − 6)(n − 1) ,

2

4

2. for n is even:

n

ꢃ

ꢄ

ꢃ

ꢄ

n−3

₂−1

n

3

2

P_CSL(Γ

₎(λ) = λ − n + 1

λ − n +

(λ − n + 1)

D

2n

4

ꢃ

ꢄ

ꢈ

ꢉ

5

1

λ²+ 3 − n λ +

5n²− 13n + 9

.

2

4

Proof. We divided the proof into two cases.

1. Suppose n is odd. Based on Theorem 1 , by the same argument of the

proof of Theorem 4 (1) and Deﬁnition 2 , we can construct CSL(Γ_D

)

2n

Energy of non-commuting graph for dihedral groups

156

of size (2n − 1) × (2n − 1) as follows:





3n−2

1

4

.

1

2

.

1

2

1

2

.

1

. . .

.

4









.

1

3n−2

. . .

.

. . .

4

1

2

.

1

2

CSL(Γ_D) =

1

2n

n − 1 . . .

2

.

1

2

.

1

2

. . .

. . . n − 1

"

#

3(n−1)

¹₂J_(n−1)×n

¹₄J_n−1

+

I

¹₂J_(n−1)×n

n−1

4

=

.

¹J_n+ ²ⁿ⁻³I_n

2

Using Theorem 2, with s = ³ⁿ⁻², t = −₂, u = n − 1, v = −₄, then we

1

4

obtain the formula of P_CSL(Γ

₎(λ),

n

D

2

ꢃ

ꢄ

ꢃ

ꢄ

n−2

n−1

3

2

P_CSL(Γ

₎(λ) = λ − (n − 1)

λ − n +

D

2n

4

ꢃ

ꢄ

5

1

λ²− (n − 1)λ + (5n − 6)(n − 1) .

2

4

2. Consider the second case for even n, by Deﬁnition 2 we have CSL(Γ_D

)

2n

of size (2n − 2) × (2n − 2) as follows:

CSL(Γ_D

)

2n





1

4

.

1

2

.

1

2

1

2

.

1

2

.

1

2

.

1

³₄(n − 1) . . .

. . .

.

. . .

.





.

1

4

. . . ³₄(n − 1)

. . .

.

. . .

.

2

1

2

1

2

.

1

3

n −

4

1

. . .





2

.

4

2





.

=

.

1

3

4

1

2

1

. . .

. . . n −

. . .

.

2

4

1

3

4

1

. . .

n −

2

4

2

.

1

2

.

1

2

1

2

1

4

1

2

3

4

. . .

. . . n −

Hence, the block matrices of CSL(Γ_D) follows:

2n

CSL(Γ_D

)

2n

M.U. Romdhini, A. Abdurahim, A.E.S.H. Maharani, A. Nawawi, F.

Al-Sharqi, I.H. Dani

157





ꢈ

ꢉ

1

4

1

³n − 1 I_n−2+ J_n−2

₂J_(n−2)×

n

2

4

ꢈ

ꢉ









1

5

²1

1

n

2

n

2

1

2

₂J

n −

I + J

− I + J

=

.

₂×(n−2)

4

2

n

2

−₄I + J

n −⁴

I + J

2

ꢈ

ꢉ

1

2

5

4

n

2

n

₂J

₂×(n−2)

2

By Theorem 3 with r = ³₄(n − 1), s = n − , t = ¹₂, and u = ¹₄, we

3

4

then obtain

n

ꢃ

ꢄ

ꢃ

ꢄ

n−3

₂−1

n

3

2

P_CSL(Γ

₎(λ) = λ − n + 1

λ − n +

(λ − n + 1)

D

2n

4

ꢃ

ꢄ

ꢈ

ꢉ

5

1

λ²+ 3 − n λ +

5n²− 13n + 9

.

2

4

Theorem 8. The CSL−spectral radius for Γ_D

is

2n

ꢅ

ꢆ

p

1

4

1. for n is odd: ρ_CSL(Γ_D) =

2n

5(n − 1) +

(n − 1)(5n − 1) ,

ꢅ

ꢆ

p

1

4

2. for n is even: ρ_CSL(Γ_D) =

2n

5n − 6 +

n(5n − 8) .

Proof.

1. The results follows from Theorem 7 (1) for odd n. It can

be seen that the eigenvalues of CSL(Γ_D) are λ₁

=

³₄(n − 1) of

2n

3

2

multiplicity (n − 2), λ₂= n −

of multiplicity n − 1, and a single

ꢅ

ꢆ

p

1

4

λ_3,4

=

5(n − 1) ±

(n − 1)(5n − 1) . Thus, the spectrum of Γ

D_2n

is as follows:

Spec_CSL(Γ_D

)

2n

(

ꢃ

ꢄ

ꢃ

ꢄ

1

n−1

ꢅ

ꢆ

p

1

4

3

2

=

5(n − 1) +

(n − 1)(5n − 1)

, n −

,

)

ꢄ

ꢃ

ꢄ

n−2

1

ꢅ

ꢆ

3

4

1

(n − 1)

,

5(n − 1) −

(n − 1)(5n − 1)

.

4

Using the greatest absolute eigenvalues, we can obtain the desired

spectral radius of Γ_D

.

2n

2. It was shown in Theorem 7 (2) for even n that the roots of P_CSL(Γ

₎(λ) =

D

2n

0 deliver ﬁve eigenvalues. First, λ₁= ³₄n − 1 of multiplicity n − 3,

3

2

n

2

n

2

λ₂= n −

of multiplicity

− 1 and λ₃= n − 1 of multiplicity

and

ꢅ

ꢆ

p

1

4

λ_4,5

=

5n − 6 ±

n(5n − 8) . Hence, the spectrum of Γ_D

as the

2n

following:

Spec_CSL(Γ_D

)

2n

Energy of non-commuting graph for dihedral groups

158

(

ꢃ

n

ꢃ

ꢄ

ꢃ

ꢆ

ꢄ

ꢃ

ꢄ

₂−1

n−3

1

ꢅ

ꢆ

p

n

2

3

2

3

4

1

4

=

n −

, (n − 1)

,

n − 1

,

5n − 6 +

n(5n − 8)

,

)

ꢄ

1

ꢅ

p

1

4

5n − 6 −

n(5n − 8)

.

The CSL−spectral radius of Γ_D

is the maximum of |λ_i|, where i =

2n

1, 2, 3, 4. This completes the proof.

Theorem 9. The CSL−energy for Γ_D

is

2n

1

1. for n is odd: E_CSL(Γ_D) = (n − 1)(7n − 2),

2n

4

ꢈ

ꢉ

1

4

2. for n is even: E_CSL(Γ_D) =

2n

7n²− 12n + 6 .

Proof.

1. We shall consider the ﬁrst case where n is odd. By Theorem 8

(1), the CSL−energy of Γ_D

can be calculated as follows:

2n

E_CSL(Γ_D

)

2n

ꢇ

ꢅ

ꢆ

p

3

2

3

4

1

4

ꢇ

=(n − 1) n −

+ (n − 2)

(n − 1) +

5(n − 1) ±

(n − 1)(5n − 1)

ꢇ

1

= (n − 1)(7n − 2).

4

2. For even n, by Theorem 8 (2), then the CSL−energy of Γ_D

is

2n

ꢇ

ꢅ

ꢆ

ꢅ ꢆ

n

3

2

p

n

3

4

E_CSL(Γ_D) =

− 1 n −

+

|n − 1| + (n − 3)

ꢆ

n − 1 +

2n

ꢇ

2

ꢇ

ꢅ

1

ꢇ

5n − 6 ±

n(5n − 8)

ꢉ

ꢇ

4

ꢈ

1

4

=

7n²− 12n + 6 .

4 Discussions

Based on Theorem 6 and Theorem 9, in the following, we derive the catego-

rization of the closeness Laplacian and closeness signless Laplacian energies

of Γ_D

.

2n

M.U. Romdhini, A. Abdurahim, A.E.S.H. Maharani, A. Nawawi, F.

Al-Sharqi, I.H. Dani

159

Corollary 1. Γ_D

is hyperenergetic corresponding with closeness Laplacian

2n

and closeness signless Laplacian matrices.

Furthermore, we can draw the following conclusion:

Corollary 2. Closeness Laplacian and closeness signless Laplacian energies

of Γ_Dare never an odd integer.

2n

Corollary 2’s statements align with the well-acknowledged facts found

in [4 ] and [9]. Moreover, the following assertion can be used to compare the

energy in Theorem 6 and Theorem 9. We conclude this section with the

following corollary.

Corollary 3. E_CL(Γ_D) = E_CSL(Γ_D).

2n

It can be seen that the closeness Laplacian energy is always similar to

closeness signless Laplacian energy of Γ_D

.

2n

5 Conclusions

This present research is concentrated on the energy of non-commuting graphs

based on the closeness Laplacian and closeness signless Laplacian matrices.

Instead of these, many important properties of graphs can be explored.

Therefore, further research is recommended to examine the same groups in

diﬀerent types of degree-based matrices.

Acknowledgments. We would like to express our gratitude to Uni-

versity of Mataram, Indonesia, for funding assistance through the Overseas

Collaborative Research Scheme PNBP No.2490/UN18.L1/PP/2025.

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