EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6319 ISSN 1307-5543 – ejpam.com Published by New York Business Global Spectral Properties of Coprime Graphs for Dihedral Groups Mamika Ujianita Romdhini1,∗, Abdurahim1, Andika Ellena Saufika Hakim Maharani1 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram 83125, Indonesia Abstract. For a finite group G, the coprime graph ΓG of G is defined as the graph with vertex set G, the group itself, and two distinct vertices u, v in ΓG are adjacent if and only if gcd(|u|, |v|) = 1, where |u| is the order of u. This study analyzes the characteristic polynomial of matrices for the dihedral group of order 2n, where n is a power of a prime number. In addition, this paper examines the characteristic polynomial of the matrices for a power of a prime number n. The energy of the graph is also obtained. 2020 Mathematics Subject Classifications: 05C25, 15A18 Key Words and Phrases: Coprime graph, Dihedral group, Spectral radius, Energy of a graph 1. Introduction In spectral graph theory, specific matrices provide information about graphs, such as adjacency, Laplacian, or signless Laplacian matrices. A graph can be characterized by the spectrum of one of these matrices. In general, the spectra of these various matrices can provide useful information about the graph. Apart from that, graphs can be involved in decision-making theory as seen in [1–3] and more terminologies in [4, 5]. Therefore, in this paper, we discuss a coprime graph of a finite group. Definition 1. [6] Let G be a finite group. Coprime graph of G is denoted by ΓG and G as the set of vertices and ∀ a, b ∈ G adjacent whenever gcd(|a|, |b|) = 1. The dihedral group is denoted by D2n = 〈 a, b : an = b2 = e, bab = a−1 〉 [7]. This re- search investigates the coprime graph for the non-abelian D2n, where n ≥ 3 and n ∈ N, denoted by ΓD2n . ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6319 Email addresses: mamika@unram.ac.id (M. U. Romdhini), abdurahim@staff.unram.ac.id (Abdurahim), a.ellena.saufika@staff.unram.ac.id (A. E. S. H. Maharani) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 2 of 13 The energy of ΓD2n is denoted by E(ΓD2n). Gutman [8] first defined the graph energy in 1978. The graphs on 2n vertices with E(ΓD2n) ≥ 2n is nonhypoenergetic, while if E(ΓD2n) < 2n− 1 is strongly hypoenergetic [9]. In recent years, numerous papers on spectral graph theory have been published. Romd- hini et al. obtained the spectral properties of the non-commuting graph for D2n corre- sponding with the Sombor matrix [10] and Wiener-Hosoya matrix [11]. Apart from that, the spectral and structural properties of the cubic power graph for D2n can be seen in [12, 13], the equal-square graph is presented by [14], the prime ideal graph [15], and the prime coprime graph [16]. Furthermore, in [17], it is shown that a precise formula can be derived for the calculation of the degree of the vertex in a coprime order graph of group D2n”. Additional terminology related to this discussion can be found in [18–21]. They worked on finite groups, and the review on graphs on groups can be seen in [22]. In addition, the discussion on the vertex degree of the coprime graph for dihedral groups has been discussed in [23]. Therefore, this paper examines the characteristic polynomial of a coprime graph associated with the adjacency, Laplacian, and signless Laplacian matrices. The definition of them refers to a book from [24]. The basic definitions and notational conventions relevant to this research are summa- rized in the following table. Definition 2. [24] An n×n adjacency (A) matrix of ΓD2n is denoted by A(ΓD2n) = [aij ], where aij = { 1, if vi ̸= vj and they are adjacent 0, otherwise. Definition 3. [24] An n × n diagonal degree matrix of ΓD2n is D(ΓD2n) = [dij ] whose (i, j)-th entry dij = { deg(vi), if vi = vj 0, otherwise, where deg(vi) represent the degree of vi. Definition 4. An n× n Laplacian (L) matrix of ΓD2n is L(ΓD2n) = D(ΓD2n)−A(ΓD2n). Definition 5. [24] An n × n signless Laplacian (SL) matrix of ΓD2n is SL(ΓD2n) = D(ΓD2n) +A(ΓD2n). The energy of ΓD2n [8] associated with A(ΓD2n) is defined as EA(ΓD2n) = Σn i=1 |λi| , and A−spectral radius of ΓD2n are defined as ρA(ΓD2n) = max{|λ| : λ ∈ SpecA(ΓD2n)}, where λ1, λ2, . . . , λ2n are eigenvalues of A(ΓD2n) as the roots of characteristic polynomial PA(ΓD2n )(λ) = |λI2n −A(ΓD2n)| = 0, and SpecA(ΓD2n) is the spectrum of A(ΓD2n). These notations also apply for L(ΓD2n) and SL(ΓD2n). Some previous results on the degree of a vertex in ΓD2n are presented as follows. M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 3 of 13 Table 1: Notation and its Definition Symbol Definition G group ΓG coprime graph of G |u| order of u in G D2n dihedral group of order 2n ΓD2n coprime graph for D2n deg(u) degree of vertex u A(ΓD2n) adjacency matrix of ΓD2n D(ΓD2n) diagonal degree matrix of ΓD2n L(ΓD2n) Laplacian matrix of ΓD2n SL(ΓD2n) signless Laplacian matrix of ΓD2n PA(ΓD2n )(λ) characteristic polynomial of A(ΓD2n) PL(ΓD2n )(λ) characteristic polynomial of L(ΓD2n) PSL(ΓD2n )(λ) characteristic polynomial of SL(ΓD2n) λi eigenvalues of the matrix EA(ΓD2n) adjacency energy of ΓD2n EL(ΓD2n) Laplacian energy of ΓD2n ESL(ΓD2n) signless Laplacian energy of ΓD2n SpecA(ΓD2n) spectrum of A(ΓD2n) ρA(ΓD2n) spectral radius of ΓD2n associated with A(ΓD2n) ρL(ΓD2n) spectral radius of ΓD2n associated with L(ΓD2n) ρSL(ΓD2n) spectral radius of ΓD2n associated with SL(ΓD2n) Ri the i-th row of the matrix Ci the i-th column of the matrix Theorem 1. [23] Let D2n be the dihedral group with n is a prime number or n = pk, p ̸= 2 for k ∈ N, then ΓD2n is a complete tripartite graph and deg(e) = 2n− 1, deg(ai) = n+ 1, deg(aib) = n for 1 ≤ i ≤ n− 1. Theorem 2. [23] Let D2n be the dihedral group with n = 2k, k ∈ N, then ΓD2n is a complete bipartite graph and deg(ai) = deg(aib) = 1 and deg(e) = 2n− 1. Moreover, the determinant properties of a square matrix are useful to ease the char- acteristic polynomial of ΓD2n . Now, let Jm×n be an m × n matrix whose entries are all 1. Lemma 1. [25] For real numbers a, b, c, and d, the determinant of∣∣∣∣(λ+ a)In1 − aJn1 −cJn1×n2 −dJn2×n1 (λ+ b)In2 − bJn2 ∣∣∣∣ of size n1 + n2 can be simplified as (λ+ a)n1−1(λ+ b)n2−1 ((λ− (n1 − 1)a)(λ− (n2 − 1)b)− n1n2cd) . M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 4 of 13 Additionally, we require some row and column operations in our proof. We shall introduce the i−th row of a matrix, Ri, and the i−th column, Ci. 2. Main results In this part, we discuss the spectral radius of the coprime graph for the dihedral group, D2n, with n as a prime number or n = 2k or n = pk, where p ̸= 2, p, k ∈ N corresponding with A(ΓD2n), L(ΓD2n), and SL(ΓD2n). 2.1. Adjacency energy We first prove the adjacency matrix of the coprime graph for D2n. Theorem 3. Let ΓD2n be the coprime graph for D2n with n is a prime number or n = pk, p ̸= 2 for a k ∈ N, then PA(ΓD2n )(λ) = λ2n−3 ( λ3 − ( n2 + n− 1 ) λ+ 2n(n− 1) ) . Proof. From Theorem 1, ΓD2n , for D2n with n is a prime number or n = pk, p ̸= 2 for a k ∈ N, is a complete tripartite graph, then we have a 2n× 2n adjacency matrix of ΓD2n , A(ΓD2n) = e a a2 . . . an−1 b ab . . . an−1b  e 0 1 1 . . . 1 1 1 . . . 1 a 1 0 0 . . . 0 1 1 . . . 1 a2 1 0 0 . . . 0 1 1 . . . 1 ... ... ... ... . . . ... ... ... . . . ... an−1 1 0 0 . . . 0 1 1 . . . 1 b 1 1 1 . . . 1 0 0 . . . 0 ab 1 1 1 . . . 1 0 0 . . . 0 ... ... ... ... . . . ... ... ... . . . ... an−1b 1 1 1 . . . 1 0 0 . . . 0 . (1) Moreover, A(ΓD2n) can be mentioned as A(ΓD2n) =  0 J1×(n−1) J1×n J(n−1)×1 0n−1 J(n−1)×n Jn×1 Jn×(n−1) 0n  . The characteristic formula of A(ΓD2n) is given in the following PA(ΓD2n )(λ) = ∣∣∣∣∣∣ λ −J1×(n−1) −J1×n −J(n−1)×1 λIn−1 −J(n−1)×n −Jn×1 −Jn×(n−1) λIn ∣∣∣∣∣∣ . The proof follows from the following sequential steps: M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 5 of 13 (i) For i = 1, 2, . . . , n − 1, we replace elements in the n + 1 + i-th row by subtracting the corresponding element of the n+ 1 + i-th row from the element in the n+ 1-th row, or in other words, Rn+1+i −→ Rn+1+i −Rn+1. (ii) We replace the elements of the n+ 1-th column with the sum of the corresponding elements in columns numbered n + 1, n + 2, ..., and 2n. It is written by Cn+1 −→ Cn+1 + Cn+2 + . . .+ C2n. (iii) We replace elements in the n+1-th row by subtraction between those elements and the elements in the first row, or equivalently, Rn+1 −→ Rn+1 −R1. (iv) We replace the first column by adding its own elements and λ+1 λ+n times the elements of the n+ 1-th column, or in other words, C1 −→ C1 + ( λ+1 λ+n ) Cn+1. (v) For i = 1, 2, . . . , n−2, the 2+i-th row is replaced by subtraction between the elements of the 2 + ith row and the second row, and is expressed as R2+i −→ R2+i −R2. (vi) The second column is replaced by the sum of the second, third, fourth,..., and n-th columns, and is stated as C2 −→ C2 + C3 + . . .+ Cn. Hence, PA(ΓD2n )(λ) = λ2n−3 ( λ3 − ( n2 + n− 1 ) λ+ 2n(n− 1) ) . Theorem 4. Let ΓD2n be the coprime graph for D2n with n = 2k, k ∈ N, then PA(ΓD2n )(λ) = λ2n−2 ( λ− √ 2n− 1 ) ( λ+ √ 2n− 1 ) . Proof. According to Theorem 2, ΓD2n , for D2n with n = 2k, k ∈ N, is a complete bipartite graph, then we provide a 2n× 2n adjacency matrix of ΓD2n . A(ΓD2n) =  0 1 1 . . . 1 1 1 . . . 1 1 0 0 . . . 0 0 0 . . . 0 1 0 0 . . . 0 0 0 . . . 0 ... ... ... . . . ... ... ... . . . ... 1 0 0 . . . 0 0 0 . . . 0 1 0 0 . . . 0 0 0 . . . 0 1 0 0 . . . 0 0 0 . . . 0 ... ... ... . . . ... ... ... . . . ... 1 0 0 . . . 0 0 0 . . . 0  . Observe that A(ΓD2n) is A(ΓD2n) = ( 0 J1×(2n−1) J(2n−1)×1 02n−1 ) . M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 6 of 13 It can be seen that PA(ΓD2n )(λ) = ∣∣∣∣ λ −J1×(2n−1) −J(2n−1)×1 λI2n−1 ∣∣∣∣ . Based on Lemma 1 with a = b = 0, c = d = 1, n1 = 1, and n2 = 2n− 1, then we obtain PA(ΓD2n )(λ) = λ2n−2 ( λ− √ 2n− 1 ) ( λ+ √ 2n− 1 ) . 2.2. Laplacian energy In the next two theorems, we focus on the Laplacian matrix of ΓD2n . Theorem 5. Let ΓD2n be the coprime graph for D2n with n is a prime number or n = pk, p ̸= 2 for a k ∈ N, then PL(ΓD2n )(λ) = λ(λ− n)n−1(λ− 2n)(λ− (n+ 1))n−2 (λ− 2n) . Proof. Since ΓD2n , for D2n with n is a prime number or n = pk, p ̸= 2 for a k ∈ N, has deg(e) = 2n− 1, deg(ai) = n+ 1, and deg(aib) = n corforming from Theorem 1, then we have D(ΓD2n) =  2n− 1 0 0 . . . 0 0 0 . . . 0 0 n+ 1 0 . . . 0 0 0 . . . 0 0 0 n+ 1 . . . 0 0 0 . . . 0 ... ... ... . . . ... ... ... . . . ... 0 0 0 . . . n+ 1 0 0 . . . 0 0 0 0 . . . 0 n 0 . . . 0 0 0 0 . . . 0 0 n . . . 0 ... ... ... . . . ... ... ... . . . ... 0 0 0 . . . 0 0 0 . . . n  . (2) According to Equation 1 and following Definition 4, then we obtain L(ΓD2n) =D(ΓD2n)−A(ΓD2n) =  2n− 1 −1 −1 . . . −1 −1 −1 . . . −1 −1 n+ 1 0 . . . 0 −1 −1 . . . −1 −1 0 n+ 1 . . . 0 −1 −1 . . . −1 ... ... ... . . . ... ... ... . . . ... −1 0 0 . . . n+ 1 −1 −1 . . . −1 −1 −1 −1 . . . −1 n 0 . . . 0 −1 −1 −1 . . . −1 0 n . . . 0 ... ... ... . . . ... ... ... . . . ... −1 −1 −1 . . . −1 0 0 . . . n  . M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 7 of 13 Hence, it is L(ΓD2n) =  2n− 1 −J1×(n−1) −J1×n −J(n−1)×1 (n+ 1)In−1 −J(n−1)×n −Jn×1 −Jn×(n−1) nIn  . The characteristic equation of L(ΓD2n) is given below PL(ΓD2n )(λ) = ∣∣∣∣∣∣ λ− (2n− 1) J1×(n−1) J1×n J(n−1)×1 (λ− (n+ 1))In−1 J(n−1)×n Jn×1 Jn×(n−1) (λ− n)In ∣∣∣∣∣∣ . We follow the following operational steps in the same manner as the proof of Theorem 3: (i) Rn+1+i −→ Rn+1+i −Rn+1, for i = 1, 2, . . . , n− 1. (ii) Cn+1 −→ Cn+1 + Cn+2 + . . .+ C2n. (iii) Rn+1 −→ Rn+1 −R1. (iv) C1 −→ C1 + Cn+1. (v) For i = 1, 2, . . . , n− 2, R2+i −→ R2+i −R2. (vi) C2 −→ C2 + C2+1 + . . .+ Cn. Therefore, PL(ΓD2n )(λ) = λ(λ− n)n−1(λ− 2n)(λ− (n+ 1))n−2 (λ− 2n) . Theorem 6. Let ΓD2n be the coprime graph for D2n with n = 2k, k ∈ N, then PL(ΓD2n )(λ) = (λ− 1)2n−2 (λ+ 1) (λ− (2n− 1)) . Proof. Based on Theorem 2, ΓD2n , for D2n with n = 2k, k ∈ N, has deg(ai) = deg(aib) = 1 and deg(e) = 2n − 1, then we can provide a 2n × 2n degree matrix of ΓD2n . D(ΓD2n) =  2n− 1 0 0 . . . 0 0 0 . . . 0 0 1 0 . . . 0 0 0 . . . 0 0 0 1 . . . 0 0 0 . . . 0 ... ... ... . . . ... ... ... . . . ... 0 0 0 . . . 1 0 0 . . . 0 0 0 0 . . . 0 1 0 . . . 0 0 0 0 . . . 0 0 1 . . . 0 ... ... ... . . . ... ... ... . . . ... 0 0 0 . . . 0 0 0 . . . 1  . (3) M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 8 of 13 Based on Equation 1 and by Definition 4, then L(ΓD2n) =D(ΓD2n)−A(ΓD2n) =  2n− 1 −1 −1 . . . −1 −1 −1 . . . −1 −1 1 0 . . . 0 0 0 . . . 0 −1 0 1 . . . 0 0 0 . . . 0 ... ... ... . . . ... ... ... . . . ... −1 0 0 . . . 1 0 0 . . . 0 −1 0 0 . . . 0 1 0 . . . 0 −1 0 0 . . . 0 0 1 . . . 0 ... ... ... . . . ... ... ... . . . ... −1 0 0 . . . 0 0 0 . . . 1  . Hence, we have L(ΓD2n) = ( 2n− 1 −J1×(2n−1) −J(2n−1)×1 I2n−1 ) . Furthermore, as PL(ΓD2n )(λ) = |L(ΓD2n)− λI2n|, then PL(ΓD2n )(λ) = ∣∣∣∣ λ− (2n− 1) J1×(2n−1) J(2n−1)×1 (λ− 1)I2n−1 ∣∣∣∣ . We follow the following operational steps (i) R2+i −→ R2+i −R2, for i = 1, 2, . . . , 2n− 2. (ii) C2 −→ C2 + C3 + . . .+ C2n. Therefore, PL(ΓD2n )(λ) = λ(λ− 1)2n−2 (λ− 2n) . 2.3. Signless Laplacian energy The next theorems are the results of the signless Laplacian matrix of ΓD2n . Theorem 7. Let ΓD2n be the coprime graph for D2n with n is a prime number or n = pk, p ̸= 2 for a k ∈ N, then PSL(ΓD2n )(λ) =(λ− n)n−1(λ− (n+ 1))n−2 ( λ3 − 4nλ2 + 4n2λ− 4n(n− 1) ) . M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 9 of 13 Proof. As the same argument with Theorem 5 and according to Equation 2 and Definition 5, then SL-matrix of ΓD2n is SL(ΓD2n) =D(ΓD2n) +A(ΓD2n) =  2n− 1 1 1 . . . 1 1 1 . . . 1 1 n+ 1 0 . . . 0 1 1 . . . 1 1 0 n+ 1 . . . 0 1 1 . . . 1 ... ... ... . . . ... ... ... . . . ... 1 0 0 . . . n+ 1 1 1 . . . 1 1 1 1 . . . 1 n 0 . . . 0 1 1 1 . . . 1 0 n . . . 0 ... ... ... . . . ... ... ... . . . ... 1 1 1 . . . 1 0 0 . . . n  . Thus, SL(ΓD2n) is a partitioned matrix, SL(ΓD2n) =  2n− 1 J1×(n−1) J1×n J(n−1)×1 (n+ 1)In−1 J(n−1)×n Jn×1 Jn×(n−1) nIn  . Since PSL(ΓD2n )(λ) = |SL(ΓD2n)− λI2n|, then PSL(ΓD2n )(λ) = ∣∣∣∣∣∣ λ− (2n− 1) −J1×(n−1) −J1×n −J(n−1)×1 (λ− (n+ 1))In−1 −J(n−1)×n −Jn×1 −Jn×(n−1) (λ− n)In ∣∣∣∣∣∣ . By the same argument of the proof of Theorem 3, we apply the following steps: (i) Rn+1+i −→ Rn+1+i −Rn+1, for i = 1, 2, . . . , n− 1. (ii) Cn+1 −→ Cn+1 + Cn+2 + . . .+ C2n. (iii) Rn+1 −→ Rn+1 −R1. (iv) C1 −→ C1 + ( λ−2(n−1) λ ) Cn+1. (v) For i = 1, 2, . . . , n− 2, R2+i −→ R2+i −R2. (vi) C2 −→ C2 + C2+1 + . . .+ Cn. Hence, we obtain PSL(ΓD2n )(λ) =(λ− n)n−1(λ− (n+ 1))n−2 ( λ3 − 4nλ2 + 4n2λ− 4n(n− 1) ) . M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 10 of 13 Theorem 8. Let ΓD2n be the coprime graph for D2n with n = 2k, k ∈ N, then PSL(ΓD2n )(λ) = (λ− 1)2n−2 (λ+ 1) (λ− (2n− 1)) . Proof. Based on the adjacency matrix in Equation 1, the degree matrix in Equation 3, and Definition 5, then we can produce a 2n× 2n signless Laplacian matrix of ΓD2n , SL(ΓD2n) =D(ΓD2n) +A(ΓD2n) =  2n− 1 1 1 . . . 1 1 1 . . . 1 1 1 0 . . . 0 0 0 . . . 0 1 0 1 . . . 0 0 0 . . . 0 ... ... ... . . . ... ... ... . . . ... 1 0 0 . . . 1 0 0 . . . 0 1 0 0 . . . 0 1 0 . . . 0 1 0 0 . . . 0 0 1 . . . 0 ... ... ... . . . ... ... ... . . . ... 1 0 0 . . . 0 0 0 . . . 1  . In this case, SL(ΓD2n) is SL(ΓD2n) = ( 2n− 1 J1×(2n−1) J(2n−1)×1 I2n−1 ) . Hence, using |SL(ΓD2n)− λI2n|, we have PSL(ΓD2n )(λ) = ∣∣∣∣ λ− (2n− 1) −J1×(2n−1) −J(2n−1)×1 (λ− 1)I2n−1 ∣∣∣∣ . We follow the following operational steps (i) R2+i −→ R2+i −R2, for i = 1, 2, . . . , 2n− 2. (ii) C2 −→ C2 + C3 + . . .+ C2n. Then we obtain PSL(ΓD2n )(λ) = λ(λ− 1)2n−2 (λ− 2n) . 3. Further discussions In comparing the result from Theorems 6 and 8, we derive the following fact: Corollary 1. Let ΓD2n be the coprime graph for D2n with n = 2k, k ∈ N, then PL(ΓD2n )(λ) = PSL(ΓD2n )(λ). M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 11 of 13 According to Theorems 4, 6, and 8, we can determine the energy of ΓD2n as presented in Theorems 9 and 10. Theorem 9. Let ΓD2n be the coprime graph for D2n with n = 2k, k ∈ N, then EA(ΓD2n) = 2 √ 2n− 1 = 2ρA(ΓD2n). Proof. Theorem 4 gives the formula of PA(ΓD2n )(λ). The roots of this polynomial are λ1 = 0 with multiplicity 2n − 2, and λ2,3 = ± √ 2n− 1. Thus, the spectral radius and energy of ΓD2n corresponds with adjacency matrix is ρA(ΓD2n) = √ 2n− 1, EA(ΓD2n) = (2n− 2)|0|+ ∣∣±√ 2n− 1 ∣∣ = 2 √ 2n− 1. Theorem 10. Let ΓD2n be the coprime graph for D2n with n = 2k, k ∈ N, then EL(ΓD2n) = ESL(ΓD2n) = 2(2n− 1) = 2ρL(ΓD2n) = 2ρSL(ΓD2n). Proof. Theorem 1 gives PL(ΓD2n )(λ) = PSL(ΓD2n )(λ). The roots of this polynomial are λ1 = 1 with multiplicity 2n − 2, and a single λ2 = 0, λ3 = 2n. Therefore, the spectral radius and the energy of ΓD2n corresponds with Laplacian and signless Laplacian matrices is ρL(ΓD2n) = ρSL(ΓD2n) = 2n, EL(ΓD2n) = ESL(ΓD2n) = (2n− 2)|1|+ (1)|0|+ (1)|2n| = 2(2n− 1). According to the two previous theorems, we can conclude that ΓD2n with n = 2k, k ∈ N is always twice their spectral radius. Moreover, it is shown that the energy of ΓD2n is never an odd integer and strongly hypoenergetic, meanwhile the L and SL-energies are always even integers and are nonhypoenergetic. These results are consistent with the previous results of Bapat and Pati [26]. 4. Conclusion This paper presented the spectral properties of the coprime graph for dihedral groups. The spectral radius and the energy of ΓD2n have been studied. We also found a correlation between them and the graph classification based on the obtained energies. Acknowledgements The authors thank the referees for their helpful comments and recommendations on this article. We also thank the University of Mataram, Indonesia, for partial funding assistance. M. U. Romdhini, Abdurahim, A. E. S. H. Maharani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6319 12 of 13 References [1] S Kosari, X Qiang, J Kacprzyk, Q T Ain, and H Rashmanlou. A Study on Topological Indices in Fuzzy Graphs with Application in Decision Making Problems. Journal of Multiple-Valued Logic & Soft Computing, 42(5-6):567–589, 2024. [2] Y Rao, S Kosari, S Hameed, and Z Yousaf. 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