EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 591-603 ISSN 1307-5543 – ejpam.com Published by New York Business Global Spectral Properties of Power Graph of Dihedral Groups Mamika Ujianita Romdhini1,∗, Athirah Nawawi2, Faisal Al-Sharqi3,4, Ashraf Al-Quran5 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 College of Engineering, National University of Science and Technology, Dhi Qar, Iraq 5 Basic Sciences Department, Preparatory Year Deanship, King Faisal University, Al-Ahsa, Saudi Arabia Abstract. This paper focuses on the power graph of dihedral groups of order 2n, D2n, where n ≥ 3. We show the characteristic polynomial of the power graph corresponding to the adjacency, Laplacian, signless Laplacian, and normalized form of these matrices. 2020 Mathematics Subject Classifications: 05C25, 05C50, 15A18, 20D99 Key Words and Phrases: Characteristic polynomial, Power graph, Dihedral Groups 1. Introduction Spectral graph theory describes graphs based on specific matrices, such as adjacency, Laplacian, or signless Laplacian matrices. The spectrum of these matrices can characterize a graph. These various matrices, in general, give insight into the graph based on their spectrum. This paper examines a power graph, one of the finite groups that can be represented by graphs. A power graph of the group G is denoted by ΓG and defined as a graph whose vertex set is all the elements of G and two distinct vertices vp and vq are adjacent if and only if vxp = vq or v y q = vp for positive integers x and y [5]. 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]. Let G1 = {e}, G2 = {ai : 1 ≤ i ≤ n − 1}, and G3 = {aib : 1 ≤ i ≤ n}. Note that ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5036 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 591 © 2024 EJPAM All rights reserved. M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 592 D2n = G1 ∪ G2 ∪ G3. Throughout this paper, the power graph for the dihedral group is denoted by ΓD2n . It is clear that ΓD2n is a connected graph [2]. Furthermore, the discussion on the degree formula of the power graph of some finite group can be found in [14]. Later, Takshak, et al. [15] showed the new finding that if ΓG is a power graph of a finite group G, then it is a divisor graph. Kumar et al. [7] have presented a complete and excellent survey of the power graph for some finite groups. Meanwhile, the degree of ΓD2n has been presented by [2] as in the following theorem: Theorem 1. [2] If ΓD2n is the power graph of D2n, then (i) the degree of a in ΓD2n is de = 2n− 1, (ii) the degree of ai in ΓD2n is dai = n− 1, (iii) the degree of aib in ΓD2n is daib = 1, The above theorem gives the information that vertex e is adjacent to all other vertices in ΓD2n . Every vertex in G2 is adjacent to e and all other members in G2. Meanwhile, all vertices in G3 are only adjacent to e [2]. Several authors have discussed the graphs that are defined on dihedral groups. They worked on the spectral problem of the commuting and non-commuting graphs, which can be seen in [9–13], Accordingly, Romdhini et al. [8] investigated signless Laplacian spectral of interval-valued fuzzy graphs. Moreover, an analysis of the relationship between graphs and unitary commutative rings’ prime spectrum is presented by [1]. Motivated by this, this research aims to formulate the characteristic polynomial of the power graph of the dihedral group associated with the adjacency, Laplacian, signless Laplacian, and normalized form of these matrices. The definition of the various matrices can be seen in the following definitions. Definition 1. ([4]) The adjacency matrix of order n×n associated with ΓD2n is given by A(ΓD2n) = [aij ] whose (i, j)-th entry aij = { 1, if vi ̸= vj and they are adjacent 0, otherwise Definition 2. ([4]) The diagonal degree matrix of order n × n associated with ΓD2n is given by D(ΓD2n) = [dij ] whose (i, j)-th entry dij = { dvi , if vi = vj 0, otherwise where dvi is the degree of vertex vi, a number of vertices adjacent to vi in ΓD2n. Definition 3. ([4]) The Laplacian matrix of order n×n associated with ΓD2n is given by L(ΓD2n) = D(ΓD2n)−A(ΓD2n). Definition 4. ([4]) The signless Laplacian matrix of order n× n associated with ΓD2n is given by SL(ΓD2n) = D(ΓD2n) +A(ΓD2n). M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 593 Definition 5. ([4]) The normalized adjacency matrix of order n×n associated with ΓD2n is given by NA(ΓD2n) = √ D(ΓD2n) −1 A(ΓD2n) √ D(ΓD2n) −1 . Definition 6. ([4]) The normalized Laplacian (NL) matrix of order n×n associated with ΓD2n is given by NL(ΓD2n) = √ D(ΓD2n) −1 L(ΓD2n) √ D(ΓD2n) −1 = In −NA(ΓD2n). Definition 7. ([4]) The normalized signless Laplacian (NSL) matrix of order n × n associated with ΓD2n is given by NSL(ΓD2n) = √ D(ΓD2n) −1 SL(ΓD2n √ D(ΓD2n) −1 = In +NA(ΓD2n). The characteristic polynomial ofA(ΓD2n) is defined by PA(ΓD2n )(λ) = det (λI2n −A(ΓD2n)), where I2n is an 2n×2n identity matrix. Likewise, the notation for other matrices can also be applied in the same way. To formulate the characteristic polynomial of ΓD2n , row and column operations need to be performed. Assume that Ri and Ci are the i−th row and column of the matrix, respectively. In this case, R′ i and C ′ i will be the new i−th row and column of the matrix, respectively, as obtained from Ri and Ci. The following theorem is a result of [6] as our guideline to simplify the characteristic polynomial of ΓD2n . Theorem 2. [6] If a square matrix M can be partitioned into four blocks M = [ A B C D ] , where A is a nonsingular, then |M | = ∣∣∣∣ A B O D − CA−1B ∣∣∣∣ = |A| ∣∣D − CA−1B ∣∣ . 2. Main Results This section presents the main results on the characteristic polynomial of ΓD2n . We begin with the adjacency matrix as the matrix representation of ΓD2n . Theorem 3. Let ΓD2n be the power graph for D2n, then the characteristic polynomial of A(ΓD2n) is PA(ΓD2n )(λ) = λn−1(λ+ 1)n−2 ( λ3 − (n− 2)λ2 + (1− 2n)λ+ n(n− 2) ) . Proof. From Theorem 1 we know that vertex e is adjacent to all other vertices in ΓD2n and vertices in G3 are only adjacent to e. Meanwhile, every vertex in G2 is adjacent to e and all other vertices in G2. Following definition 1, we can construct A(ΓD2n) of the size M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 594 2n× 2n: A(ΓD2n) = e a a2 . . . an−1 b ab . . . an−1b  e 0 1 1 . . . 1 1 1 . . . 1 a 1 0 1 . . . 1 0 0 . . . 0 a2 1 1 0 . . . 1 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1 1 1 1 . . . 0 0 0 . . . 0 b 1 0 0 . . . 0 0 0 . . . 0 ab 1 0 0 . . . 0 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1b 1 0 0 . . . 0 0 0 . . . 0 . (1) Matrix A(ΓD2n) can be partitioned into nine block matrices as follows: A(ΓD2n) =  0 J1×(n−1) J1×n J(n−1)×1 (J − I)n−1 0(n−1)×n Jn×1 0n×(n−1) 0n  . The characteristic polynomial of A(ΓD2n) is PA(ΓD2n )(λ) = ∣∣∣∣∣∣ λ −J1×(n−1) −J1×n −J(n−1)×1 (λ+ 1)In−1 − Jn−1 0(n−1)×n −Jn×1 0n×(n−1) λIn ∣∣∣∣∣∣ . (2) We apply the following steps to simplify the determinant in Equation 2: (i) Rn+1+i −→ Rn+1+i −Rn+1, for i = 1, 2, . . . , n− 1. (ii) Cn+1 −→ Cn+1 + Cn+2 + . . .+ C2n. (iii) C1 −→ C1 + 1 λCn+1. (iv) R2+i −→ R2+i −R2, for i = 1, 2, . . . , n− 2. (v) C2 −→ C2 + C3 + . . .+ Cn. Then we get PA(ΓD2n )(λ) = ∣∣∣∣∣∣∣∣∣∣ λ2+n λ 1− n −J1×(n−2) −n −J1×n −1 λ− (n− 2) −J1×(n−2) 0 01×(n−1) 0(n−2)×1 0(n−2)×1 (λ+ 1)In−2 0(n−2)×1 0(n−2)×(n−1) 0 0 01×(n−2) λ 01×(n−1) 0(n−1)×1 0(n−1)×1 0n−1 0(n−1)×1 λIn−1 ∣∣∣∣∣∣∣∣∣∣ . M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 595 Consequently, by Theorem 2, we can obtain the characteristic polynomial of A(ΓD2n) as follows: PA(ΓD2n )(λ) = λn−1(λ+ 1)n−2 ( λ3 − (n− 2)λ2 + (1− 2n)λ+ n(n− 2) ) . The previous theorem is devoted to the adjacency matrix. Now we are moving to the Laplacian matrix as the representation of ΓD2n . Theorem 4. Let ΓD2n be the power graph for D2n, then the characteristic polynomial of L(ΓD2n) is PL(ΓD2n )(λ) = λ(λ− 2n)(λ− n)n−2(λ− 1)n. Proof. The Laplacian matrix of ΓD2n construction depends on the degree and adjacency matrices of ΓD2n . Now we need to construct a 2n× 2n degree matrix of ΓD2n as follows: D(ΓD2n) = e a a2 . . . an−1 b ab . . . an−1b  e 2n− 1 0 0 . . . 0 0 0 . . . 0 a 0 n− 1 0 . . . 0 0 0 . . . 0 a2 0 0 n− 1 . . . 0 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1 0 0 0 . . . n− 1 0 0 . . . 0 b 0 0 0 . . . 0 1 0 . . . 0 ab 0 0 0 . . . 0 0 1 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1b 0 0 0 . . . 0 0 0 . . . 1 . (3) Based on Definition 3, the Laplacian matrix of ΓD2n is L(ΓD2n) =D(ΓD2n)−A(ΓD2n) = e a a2 . . . an−1 b ab . . . an−1b  e 2n− 1 −1 −1 . . . −1 −1 −1 . . . −1 a −1 n− 1 −1 . . . −1 0 0 . . . 0 a2 −1 −1 n− 1 . . . −1 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1 −1 −1 −1 . . . n− 1 0 0 . . . 0 b −1 0 0 . . . 0 1 0 . . . 0 ab −1 0 0 . . . 0 0 1 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1b −1 0 0 . . . 0 0 0 . . . 1 . M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 596 Moreover, L(ΓD2n) can be partitioned into six block matrices as given below: L(ΓD2n) =  2n− 1 −J1×(n−1) −J1×n −J(n−1)×1 nIn−1 − Jn−1 0(n−1)×n −Jn×1 0n×(n−1) In  . The characteristic polynomial of L(ΓD2n) can be obtained from the following determinant: PL(ΓD2n )(λ) = ∣∣∣∣∣∣ λ− (2n− 1) J1×(n−1) J1×n J(n−1)×1 (λ− (n− 2))In−1 − Jn−1 0(n−1)×n Jn×1 0n×(n−1) (λ− 1)In ∣∣∣∣∣∣ . (4) We apply the following steps to equation 4: (i) Rn+1+i −→ Rn+1+i −Rn+1, for i = 1, 2, . . . , n− 1. (ii) Cn+1 −→ Cn+1 + Cn+2 + . . .+ C2n. (iii) C1 −→ C1 − 1 λ−1Cn+1. (iv) R2+i −→ R2+i −R2, for i = 1, 2, . . . , n− 2. (v) C2 −→ C2 + C2+1 + . . .+ Cn, then we get PL(ΓD2n )(λ) = ∣∣∣∣∣∣∣∣∣∣ λ2−2nλ+n−1 λ−1 n− 1 J1×(n−2) n J1×(n−1) 1 λ− 1 J1×(n−2) 0 01×(n−1) 0(n−2)×1 0(n−2)×1 (λ− n)In−2 0(n−2)×1 0(n−2)×(n−1) 0 0 01×(n−2) λ− 1 01×(n−1) 0(n−1)×1 0(n−1)×1 0n−1 0(n−1)×1 (λ− 1)In−1 ∣∣∣∣∣∣∣∣∣∣ . Following Theorem 2, we then can obtain PL(ΓD2n )(λ) = λ(λ− 2n)(λ− n)n−2(λ− 1)n. The next theorem presents the characteristic polynomial of ΓD2n associated with the signless Laplacian matrix. Theorem 5. Let ΓD2n be the power graph for D2n, then the characteristic polynomial of SL(ΓD2n) is PSL(ΓD2n )(λ) = (λ−1)n−1(λ−n+2)n−2 ( λ3 + (3− 4n)λ2 + 2n(2n− 3)λ− 2(n− 1)(n− 2) ) . Proof. By Equations 1 and 4, and Definition 4, the signless Laplacian matrix of ΓD2n is SL(ΓD2n) =D(ΓD2n) +A(ΓD2n) M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 597 = e a a2 . . . an−1 b ab . . . an−1b  e 2n− 1 1 1 . . . 1 1 1 . . . 1 a 1 n− 1 1 . . . 1 0 0 . . . 0 a2 1 1 n− 1 . . . 1 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1 1 1 1 . . . n− 1 0 0 . . . 0 b 1 0 0 . . . 0 1 0 . . . 0 ab 1 0 0 . . . 0 0 1 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1b 1 0 0 . . . 0 0 0 . . . 1 . Moreover, SL(ΓD2n) can be partitioned into nine block matrices as given below: SL(ΓD2n) =  2n− 1 J1×(n−1) J1×n J(n−1)×1 (n− 2)In−1 + Jn−1 0(n−1)×n Jn×1 0n×(n−1) In  . The characteristic polynomial of SL(ΓD2n) can be obtained from the following determi- nant: PSL(ΓD2n )(λ) = ∣∣∣∣∣∣ λ− (2n− 1) −J1×(n−1) −J1×n −J(n−1)×1 (λ− (n− 2))In−1 − Jn−1 0(n−1)×n −Jn×1 0n×(n−1) (λ− 1)In ∣∣∣∣∣∣ . (5) We apply the following steps into Equation 5: (i) Rn+1+i −→ Rn+1+i −Rn+1, for i = 1, 2, . . . , n− 1. (ii) Cn+1 −→ Cn+1 + Cn+2 + . . .+ C2n. (iii) C1 −→ C1 + ( 1 λ−1 ) Cn+1. (iv) R2+i −→ R2+i −R2, for i = 1, 2, . . . , n− 2. (v) C2 −→ C2 + C2+1 + . . .+ Cn, then we get PSL(ΓD2n )(λ) = ∣∣∣∣∣∣∣∣∣∣ λ2−2nλ+n−1 λ−1 1− n −J1×(n−2) −n −J1×(n−1) −1 λ− 2n+ 3 −J1×(n−2) 0 01×(n−1) 0(n−2)×1 0(n−2)×1 (λ− n+ 2)In−2 0(n−2)×1 0(n−2)×(n−1) 0 0 01×(n−2) λ− 1 01×(n−1) 0(n−1)×1 0(n−1)×1 0n−1 0(n−1)×1 (λ− 1)In−1 ∣∣∣∣∣∣∣∣∣∣ . M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 598 From Theorem 2, we derive the characteristic polynomial of SL(ΓD2n) as follows: PSL(ΓD2n )(λ) = (λ− 1)n−1(λ− n+ 2)n−2 ( λ3 + (3− 4n)λ2 + 2n(2n− 3)λ− 2(n− 1)(n− 2) ) . The normalized form of the adjacency, Laplacian, and signless Laplacian matrices of ΓD2n are presented in the following three theorems. Theorem 6. Let ΓD2n be the power graph for D2n, then the characteristic polynomial of NA(ΓD2n) is PNA(ΓD2n )(λ) = λn−1 ( λ+ 1 n− 1 )n−2( λ3 − (n− 2) n− 1 λ2 − n+ 1 2n− 1 λ+ n(n− 2) (n− 1)(2n− 1) ) . Proof. By Definition 5, we need to construct ( √ D)−1(ΓD2n). Using Equation 3, we can construct ( √ D)−1(ΓD2n) as follows: ( √ D)−1(ΓD2n) = e a a2 . . . an−1 b ab . . . an−1b  e 1√ 2n−1 0 0 . . . 0 0 0 . . . 0 a 0 1√ n−1 0 . . . 0 0 0 . . . 0 a2 0 0 1√ n−1 . . . 0 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1 0 0 0 . . . 1√ n−1 0 0 . . . 0 b 0 0 0 . . . 0 1 0 . . . 0 ab 0 0 0 . . . 0 0 1 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1b 0 0 0 . . . 0 0 0 . . . 1 . (6) Based on Definition 5, NA(ΓD2n) is a 2n× 2n matrix as given below: e a a2 . . . an−1 b ab . . . an−1b  e 0 1√ (2n−1)(n−1) 1√ (2n−1)(n−1) . . . 1√ (2n−1)(n−1) 1√ 2n−1 1√ 2n−1 . . . 1√ 2n−1 a 1√ (2n−1)(n−1) 0 1 n−1 . . . 1 n−1 0 0 . . . 0 a2 1√ (2n−1)(n−1) 1 n−1 0 . . . 1 n−1 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1 1√ (2n−1)(n−1) 1 n−1 1 n−1 . . . 0 0 0 . . . 0 b 1√ 2n−1 0 0 . . . 0 0 0 . . . 0 ab 1√ 2n−1 0 0 . . . 0 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1b 1√ 2n−1 0 0 . . . 0 0 0 . . . 0 . (7) M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 599 In other words, NA(ΓD2n) can be partitioned into nine block matrices as follows: NA(ΓD2n) =  0 1√ (2n−1)(n−1) J1×(n−1) 1√ 2n−1 J1×n 1√ (2n−1)(n−1) J(n−1)×1 1 n−1(J − I)n−1 0(n−1)×n 1√ 2n−1 Jn×1 0n×(n−1) 0n  . The characteristic polynomial of NA(ΓD2n) is PNA(ΓD2n )(λ) = ∣∣∣∣∣∣∣∣ λ − 1√ (2n−1)(n−1) J1×(n−1) − 1√ 2n−1 J1×n − 1√ (2n−1)(n−1) J(n−1)×1 ( λ+ 1 n−1 ) In−1 − 1 n−1Jn−1 0(n−1)×n − 1√ 2n−1 Jn×1 0n×(n−1) λIn ∣∣∣∣∣∣∣∣ . (8) We apply the following steps into Equation 8: (i) Rn+1+i −→ Rn+1+i −Rn+1, for i = 1, 2, . . . , n− 1. (ii) Cn+1 −→ Cn+1 + Cn+2 + . . .+ C2n. (iii) C1 −→ C1 + 1 λ √ 2n−1 Cn+1. (iv) R2+i −→ R2+i −R2, for i = 1, 2, . . . , n− 2. (v) C2 −→ C2 + C2+1 + . . .+ Cn, then we get PNA(ΓD2n )(λ) = ∣∣∣∣∣∣∣∣∣∣∣∣ (2n−1)λ2−n (2n−1)λ − n−1√ (2n−1)(n−1) − 1√ (2n−1)(n−1) J1×(n−2) − n√ 2n−1 − 1√ 2n−1 J1×n − 1√ (2n−1)(n−1) λ− (n−2) n−1 − 1 n−1J1×(n−2) 0 01×(n−1) 0(n−2)×1 0(n−2)×1 (λ+ 1 n−1 )In−2 0(n−2)×1 0(n−2)×(n−1) 0 0 01×(n−2) λ 01×(n−1) 0(n−1)×1 0(n−1)×1 0n−1 0(n−1)×1 λIn−1 ∣∣∣∣∣∣∣∣∣∣∣∣ . (9) By Theorem 2, we can obtain PNA(ΓD2n )(λ) as follows: PNA(ΓD2n )(λ) = λn−1 ( λ+ 1 n− 1 )n−2( λ3 − (n− 2) n− 1 λ2 − n+ 1 2n− 1 λ+ n(n− 2) (n− 1)(2n− 1) ) . Theorem 7. Let ΓD2n be the power graph for D2n, then the characteristic polynomial of NL(ΓD2n) is PNL(ΓD2n )(λ) = λ(λ− 1)n−1 ( λ− 1− 1 n− 1 )n−2( λ2 + (1− 2n) n− 1 λ+ n(n+ 1) (2n− 1)(n− 1) ) . M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 600 Proof. By Definition 6, and Equations 6 and 7, we can construct NL(ΓD2n) of the size 2n× 2n as follows: e a a2 . . . an−1 b ab . . . an−1b  e 1 − 1√ (2n−1)(n−1) − 1√ (2n−1)(n−1) . . . − 1√ (2n−1)(n−1) − 1√ 2n−1 − 1√ 2n−1 . . . − 1√ 2n−1 a − 1√ (2n−1)(n−1) 1 − 1 n−1 . . . − 1 n−1 0 0 . . . 0 a2 − 1√ (2n−1)(n−1) − 1 n−1 1 . . . − 1 n−1 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1 − 1√ (2n−1)(n−1) − 1 n−1 − 1 n−1 . . . 1 0 0 . . . 0 b − 1√ 2n−1 0 0 . . . 0 1 0 . . . 0 ab − 1√ 2n−1 0 0 . . . 0 0 1 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1b − 1√ 2n−1 0 0 . . . 0 0 0 . . . 1 . NL(ΓD2n) can be partitioned into nine block matrices as follows: NL(ΓD2n) =  1 − 1√ (2n−1)(n−1) J1×(n−1) − 1√ 2n−1 J1×n − 1√ (2n−1)(n−1) J(n−1)×1 ( 1 + 1 n−1 ) In−1 − 1 n−1Jn−1 0(n−1)×n − 1√ 2n−1 Jn×1 0n×(n−1) In  . The characteristic polynomial of NL(ΓD2n) is PNL(ΓD2n )(λ) = ∣∣∣∣∣∣∣∣ λ− 1 1√ (2n−1)(n−1) J1×(n−1) 1√ 2n−1 J1×n 1√ (2n−1)(n−1) J(n−1)×1 ( λ− 1− 1 n−1 ) In−1 + 1 n−1Jn−1 0(n−1)×n 1√ 2n−1 Jn×1 0n×(n−1) (λ− 1)In ∣∣∣∣∣∣∣∣ . (10) By applying row and column operations into Equation 10: (i) Rn+1+i −→ Rn+1+i −Rn+1, for i = 1, 2, . . . , n− 1. (ii) Cn+1 −→ Cn+1 + Cn+2 + . . .+ C2n. (iii) C1 −→ C1 − 1 (λ−1) √ 2n−1 Cn+1. (iv) R2+i −→ R2+i −R2, for i = 1, 2, . . . , n− 2. (v) C2 −→ C2 + C2+1 + . . .+ Cn, then we get PNL(ΓD2n )(λ) = ∣∣∣∣∣∣∣∣∣∣∣∣ −n (λ−1)(2n−1) + λ− 1 n−1√ (2n−1)(n−1) 1√ (2n−1)(n−1) J1×(n−2) n√ 2n−1 1√ 2n−1 J1×n 1√ (2n−1)(n−1) λ− 1 + (n−2) n−1 1 n−1J1×(n−2) 0 01×(n−1) 0(n−2)×1 0(n−2)×1 (λ− 1− 1 n−1 )In−2 0(n−2)×1 0(n−2)×(n−1) 0 0 01×(n−2) λ− 1 01×(n−1) 0(n−1)×1 0(n−1)×1 0n−1 0(n−1)×1 (λ− 1)In−1 ∣∣∣∣∣∣∣∣∣∣∣∣ . (11) M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 591-603 601 Consequently, based on Theorem 2, we can obtain PNL(ΓD2n )(λ) as follows: PNL(ΓD2n )(λ) = λ(λ− 1)n−1 ( λ− 1− 1 n− 1 )n−2( λ2 + (1− 2n) n− 1 λ+ n(n+ 1) (2n− 1)(n− 1) ) . Theorem 8. Let ΓD2n be the power graph for D2n, then the characteristic polynomial of NSL(ΓD2n) is PNSL(ΓD2n )(λ) = (λ−1)n−1 ( λ− 1 + 1 n− 1 )n−2 ( λ3 + (5− 4n) n− 1 λ2 + (9n2 − 19n+ 8) (2n− 1)(n− 1) λ− 2(n− 2) 2n− 1 ) . Proof. By Definition 7, and Equations 6 and 7, we can construct NSL(ΓD2n) of the size 2n× 2n as given below: e a a2 . . . an−1 b ab . . . an−1b  e 1 1√ (2n−1)(n−1) 1√ (2n−1)(n−1) . . . 1√ (2n−1)(n−1) 1√ 2n−1 1√ 2n−1 . . . 1√ 2n−1 a 1√ (2n−1)(n−1) 1 1 n−1 . . . 1 n−1 0 0 . . . 0 a2 1√ (2n−1)(n−1) 1 n−1 1 . . . 1 n−1 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1 1√ (2n−1)(n−1) 1 n−1 1 n−1 . . . 1 0 0 . . . 0 b 1√ 2n−1 0 0 . . . 0 1 0 . . . 0 ab 1√ 2n−1 0 0 . . . 0 0 1 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1b 1√ 2n−1 0 0 . . . 0 0 0 . . . 1 . In other words, NSL(ΓD2n) can be partitioned into nine block matrices as follows: NSL(ΓD2n) =  1 1√ (2n−1)(n−1) J1×(n−1) 1√ 2n−1 J1×n 1√ (2n−1)(n−1) J(n−1)×1 ( 1− 1 n−1 ) In−1 + 1 n−1Jn−1 0(n−1)×n 1√ 2n−1 Jn×1 0n×(n−1) In  . The characteristic polynomial of NSL(ΓD2n) is PNSL(ΓD2n )(λ) = ∣∣∣∣∣∣∣∣ λ− 1 − 1√ (2n−1)(n−1) J1×(n−1) − 1√ 2n−1 J1×n − 1√ (2n−1)(n−1) J(n−1)×1 ( λ− 1 + 1 n−1 ) In−1 − 1 n−1Jn−1 0(n−1)×n − 1√ 2n−1 Jn×1 0n×(n−1) (λ− 1)In ∣∣∣∣∣∣∣∣ . (12) We apply the row and column operations to Equation 12: (i) Rn+1+i −→ Rn+1+i −Rn+1, for i = 1, 2, . . . , n− 1. REFERENCES 602 (ii) Cn+1 −→ Cn+1 + Cn+2 + . . .+ C2n. (iii) C1 −→ C1 + 1 (λ−1) √ 2n−1 Cn+1. (iv) R2+i −→ R2+i −R2, for i = 1, 2, . . . , n− 2. (v) C2 −→ C2 + C2+1 + . . .+ Cn, consequently we have PNSL(ΓD2n )(λ) = ∣∣∣∣∣∣∣∣∣∣∣∣ −n (λ−1)(2n−1) + λ− 1 − n−1√ (2n−1)(n−1) − 1√ (2n−1)(n−1) J1×(n−2) − n√ 2n−1 − 1√ 2n−1 J1×n − 1√ (2n−1)(n−1) λ− 1− (n−2) n−1 − 1 n−1 J1×(n−2) 0 01×(n−1) 0(n−2)×1 0(n−2)×1 (λ− 1 + 1 n−1 )In−2 0(n−2)×1 0(n−2)×(n−1) 0 0 01×(n−2) λ− 1 01×(n−1) 0(n−1)×1 0(n−1)×1 0n−1 0(n−1)×1 (λ− 1)In−1 ∣∣∣∣∣∣∣∣∣∣∣∣ . According to Theorem 2, we can obtain PNSL(ΓD2n )(λ) as follows: PNSL(ΓD2n )(λ) = (λ−1)n−1 ( λ− 1 + 1 n− 1 )n−2 ( λ3 + (5− 4n) n− 1 λ2 + (9n2 − 19n+ 8) (2n− 1)(n− 1) λ− 2(n− 2) 2n− 1 ) . Acknowledgements We wish to express our gratitude to Universitas Mataram, Indonesia, for providing partial funding assistance. References [1] B Alharbi. Graphs and the Prime Spectrum of Unitary Commutative Rings. European Journal of Pure and Applied Mathematics, 16(1):314–318, 2023. [2] F Ali, S Fatima, and W Wang. On the Power Graphs of Certain Finite Groups. Linear and Multilinear Algebra, pages 1–15, 2020. [3] M Aschbacher. Finite Group Theory. Cambridge University Press, Cambridge, 2000. [4] A E Brouwer and W H Haemers. Spectra of graphs. Springer, New York, 2011. [5] T T Chelvam and M Sattanathan. Power Graph of Finite Abelian Groups. Algebra and Discrete Mathematics, 16(1):33–41, 2013. [6] F R Gantmacher. The Theory of Matrices. Chelsea Publishing Company, New York, 1959. [7] A Kumar, L Selvaganesh, P J Cameron, and T Chelvam. Recent developments on the power graph of finite groups – a survey. AKCE International Journal of Graphs and Combinatorics, 18(2):65–94, 2021. REFERENCES 603 [8] M U Romdhini, F Al-Sharqi, A Nawawi, A Al-Quran, and H Rashmanlou. Sign- less Laplacian energy of interval-valued fuzzy graph and its applications. Sains Malaysiana, 52(7):2127–2137, 2023. [9] M U Romdhini and A Nawawi. Degree Sum Energy of Non-Commuting Graph for Dihedral Groups. Malaysian Journal of Science, 41(sp1):34–39, 2022. [10] M U Romdhini and A Nawawi. Maximum and minimum degree energy of commuting graph for dihedral groups. Sains Malaysiana, 51(12):4145–4151, 2022. [11] M U Romdhini and A Nawawi. Degree subtraction energy of commuting and non- commuting graphs for dihedral groups. Journal of Mathematical and Computational Science, 18(3):497–508, 2023. [12] M U Romdhini, A Nawawi, and C Y Chen. Degree Exponent Sum Energy of Com- muting Graph for Dihedral Groups. Malaysian Journal of Science, 41(sp1):40–46, 2022. [13] M U Romdhini, A Nawawi, and C Y Chen. Neighbors degree sum energy of com- muting and non-commuting graphs for dihedral groups. Malaysian Journal of Math- ematical Sciences, 17(1):53–65, 2023. [14] A Sehgal and S N Singh. The Degree of a Vertex in the Power Graph of a Finite Abelian Group. Southeast Asian Bulletin of Mathematics, 47:89–296, 2023. [15] N Takshak and A Sehgal A Malik. Power Graph of a Finite Group is Always Divisor Graph. Asian-European Journal of Mathematics, 16(1):2250236, 2023.