EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5776 ISSN 1307-5543 – ejpam.com Published by New York Business Global Eccentricity-Based Energies of Non-Commuting Graph for Dihedral Groups Mamika Ujianita Romdhini1,∗, Athirah Nawawi2, Faisal Al-Sharqi3,4, Abdurahim1, Andika Ellena Saufika Hakim Maharani1, Ifan Hasnan Dani1 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, 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. Spectral graph theory is a research topic that combines algebra and graph theory, with the intersection representing a graph as a matrix. The eigenvalues of the matrix give the value of graph energy. This research focuses on the non-commuting graph for dihedral groups corresponding to eccentricity-based matrices including eccentricity, sum eccentricity, and average degree eccentricity matrices. 2020 Mathematics Subject Classifications: 05C25, 15A18 Key Words and Phrases: Eccentricity-based matrices, Energy of a graph, Non-commuting graph, Dihedral groups 1. Introduction Spectral graph theory is a combined research topic between algebra and graph theory with the intersection being a matrix representation of a graph. Originally, the adjacency matrix was the first representation of a graph. The research has extended to the degree- based and distance-based matrices, and recently, eccentricity-based matrices have been developed. Wang, et al. [1] defined the eccentricity matrix and is inspired by the idea of Randic [2]. Later, Mahato [3] continued to discuss this type of matrix and presented the spectra perspectives in 2020. Meanwhile, the sum eccentricity was introduced by Sowaity ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5776 Email addresses: mamika@unram.ac.id (M. U. Romdhini), athirah@upm.edu.my (A. Nawawi), faisal.ghazi@uoanbar.edu.iq (F. Al-Sharqi), 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 et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 2 of 13 and Sharada [4] and the average degree eccentricity matrix was pioneered by Mathad et al. [5]. The matrix of a graph is a square matrix whose size depends on the order of the graph. Therefore, we can calculate the eigenvalues of a matrix, which are hereinafter referred to as the eigenvalues of the corresponding graph. The sum of absolute eigenvalues is the energy of a graph defined by Gutman [6] in 1978. Moreover, the graph energy value has been discussed in [7] and [8]. The graph energy can further be associated with the graph defined on the group including the non-commuting graph. It is shown in [9] who discussed the Wiener-Hosoya energy, and for Sombor energy can be found in [10]. The algebraic discussion also can be found in [11, 12]. Therefore, this research aims to analyze the non-commuting graph energy associated with the eccentricity-based matrices and dihedral groups as its vertex set. 2. Preliminaries In this section, we recall the fundamental definitions and theorems useful for our main results. We start with the definition of the non-commuting graph. Definition 1. [13] Let G be a finite group. The non-commuting graph of G is denoted by ΩG, in which the vertex set is G\Z(G), where Z(G) is the center of G, and two distinct vertices u and v are joined by an edge whenever uv ̸= vu. Throughout this paper, we denote the non-commuting graph for dihedral groups of order 2n, D2n, as ΩD2n , where n ≥ 3. The vertex set and edge set of ΩD2n are denoted by V (ΩD2n) and L(ΩD2n), respectively. Vertex x ∈ V (ΩD2n) is adjacent to y ∈ V (ΩD2n) if and only if edge xy ∈ L(ΩD2n). The distance between both vertices in ΩD2n is denoted by dxy and the degree of x is denoted by d(x). The eccentricity of x is given by e(x) = max{dxy|y ∈ V (ΩD2n)}. The construction of the graph matrices of ΩD2n is based on the definition of eccentricity, sum-eccentricity, and average degree-eccentricity matrices as presented below: Definition 2. [1] The eccentricity matrix of ΩD2n is E(ΩD2n) = [ϵij ] in which (i, j)-th entry is ϵij = { dxixj , if dxixj = min{e(xi), e(xj)} 0, if dxixj < min{e(xi), e(xj)}. Definition 3. [4] The sum eccentricity matrix of ΩD2n is SE(ΩD2n) = [sij ] in which (i, j)-th entry is sij = { e(xi) + e(xj), if xixj ∈ L(ΩD2n) 0, otherwise. Definition 4. [5] The average degree eccentricity matrix of ΩD2n is ADE(ΩD2n) = [aij ] whose (i, j)-th entry is aij = { 1 4(d(xi) + d(xj) + e(xi) + e(xj)), if xixj ∈ L(ΩD2n) 0, otherwise. M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 3 of 13 The characteristic polynomial of E(ΩD2n) is defined by PE(ΩD2n )(µ) = |µIn − E(ΩD2n)| , (1) where In is an n× n identity matrix. Furthermore, the eigenvalues of ΩD2n are the roots of PE(ΩD2n )(µ) = 0. The eccentricity energy definition is based on the eigenvalues of ΩD2n [6] as εE(ΩD2n) = n∑ i=1 |µi| . The eccentricity spectral radius of ΩD2n [14] is ρE(ΩD2n) = max{|µi| : i = 1, 2, . . . , n}, where µ1, µ2, . . . , µn are eigenvalues of E(ΩD2n). Similarly, one can apply the notation for SE and ADE-matrices in the same manner. The energy value of ΩD2n is classified as hyperenergetic if the energy of ΩD2n is greater than 4(n− 1) for odd n (or 2(2n− 3) for even n)[15]. Let D2n = 〈 a, b : an = b2 = e, bab = a−1 〉 . We denote Ω1 = {ap : 1 ≤ p ≤ n}\Z (D2n) and Ω2 = {apb : 1 ≤ p ≤ n}, where Z (D2n) is the center of D2n. We need the following results to determine the entries of the matrix of ΩD2n . Theorem 1. [16] In ΩD2n, the distance between xi and xj in V (ΩD2n) is (i) for odd n, dxixj = { 2, if xi, xj ∈ Ω1 1, otherwise, , and (ii) for the even n, dxixj =  2, if xi, xj ∈ Ω1 2, xi ∈ Ω2, xj ∈ { a n 2 +ib } , for i = 1, 2, . . . , n 1, otherwise. Theorem 2. [17] In ΩD2n, (i) the degree of ai on ΩD2n is dai = n, and (ii) the degree of aib on ΩD2n is daib = { 2(n− 1), if n is odd 2(n− 2), if n is even. The eccentricity of every vertex in ΩD2n can be found in [17] as follows. Theorem 3. [17] In ΩD2n, the eccentricity of x ∈ V (ΩD2n) is (i) for odd n, e(x) = { 2, if x ∈ Ω1 1, if x ∈ Ω2 and (ii) for even n, e(x) = 2. M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 4 of 13 The following theorems simplify the process of formulating the characteristic formula. Theorem 4. [18] If T =  a(J − I)n−2 cJ(n−2)×n 2 cJ(n−2)×n 2 cJn 2 ×(n−2) d(J − I)n 2 d(J − I)n 2 + bIn 2 cJn 2 ×(n−2) d(J − I)n 2 + bIn 2 d(J − I)n 2  , then for real numbers a, b, c, d, the characteristic polynomial of T is PT (µ) = (µ+ a)n−3 (µ− b+ 2d) n 2 −1 (µ+ b) n 2( µ2 − (b+ (n− 2)d+ a(n− 3))µ+ a(n− 3) (b+ (n− 2)d)− n(n− 2)c2 ) . Lemma 1. [19] Let a, b, c, and d be real numbers. Then the determinant of∣∣∣∣(µ+ a)In1 − aJn1 −cJn1×n2 −dJn2×n1 (µ+ b)In2 − bJn2 ∣∣∣∣ can be simplified as (µ+ a)n1−1(µ+ b)n2−1 ((µ− (n1 − 1)a)(µ− (n2 − 1)b)− n1n2cd) , where 1 ≤ n1, n2 ≤ n and n1 + n2 = n. 3. Main Results In this section, we find the eccentricity-based energies of ΩD2n . 3.1. Eccentricity Energy This part aims to determine the energy formula of ΩD2n associated with the eccentricity matrix. Theorem 5. In ΩD2n, the eccentricity energy of ΩD2n is εE(ΩD2n) = { 2(3n− 5), if n is odd 6(n− 2), if n is even . Proof. (i) Let n be odd. According to Theorem 1 (i) and Definition 2, we can construct the eccentricity matrix of ΩD2n . The matrix size is (2n − 1) × (2n − 1) excluding one center’s element of D2n. The entries of E(ΩD2n) = [ϵij ] are (a) for 1 ≤ i, j ≤ n− 1 and i ̸= j, ϵij = 2 since dxixj = min{e(xi), e(xj)} = 2; (b) for 1 ≤ i ≤ n − 1 and j = n, n + 1, . . . , 2n − 1 or vice versa, ϵij = 1 since dxixj = min{e(xi), e(xj)} = 1; M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 5 of 13 (c) for n ≤ i, j ≤ 2n− 1 , ϵij = 1 since dxixj = min{e(xi), e(xj)} = 1; (d) for i = j, ϵij = 0. Then E(ΩD2n) is as follows: E(ΩD2n) = a a2 . . . an−1 b ab . . . an−1b  a 0 2 . . . 2 1 1 . . . 1 a2 2 0 . . . 2 1 1 . . . 1 ... ... ... . . . ... ... ... . . . ... an−1 2 2 . . . 0 1 1 . . . 1 b 1 1 . . . 1 0 1 . . . 1 ab 1 1 . . . 1 1 0 . . . 1 ... ... ... . . . ... ... ... . . . ... an−1b 1 1 . . . 1 1 1 . . . 0 , and the characteristic formula of E(ΩD2n), PE(ΩD2n )(µ) = ∣∣∣∣ (µ+ 2)In−1 − 2Jn−1 −J(n−1)×n −Jn×(n−1) (µ+ 1)In − Jn ∣∣∣∣ . By Lemma 1, with a = 2, b = c = d = 1, n1 = n− 1 and n2 = n, we obtain PE(ΩD2n )(µ) = (µ+ 2)n−2(µ+ 1)n−1 ( µ2 − (3n− 5)µ+ (n− 1)(n− 4) ) . The eigenvalues of ΩD2n are µ1 = −2 of multiplicity n − 2, µ2 = −1 of multiplicity n− 1, and µ3,4 = 1 2 ( 3n− 5± √ 5n2 − 10n+ 9 ) . The eccentricity spectral radius of ΩD2n is ρE(ΩD2n) = 1 2 ( 3n− 5 + √ 5n2 − 10n+ 9 ) . The eccentricity energy of ΩD2n is εE(ΩD2n) =(n− 2)| − 2|+ (n− 1)| − 1|+ ∣∣∣∣12 (3n− 5± √ 5n2 − 10n+ 9 )∣∣∣∣ =2(3n− 5) (ii) Let n be even. Based on Theorem 1 (ii) and Definition 2, E(ΩD2n) = [ϵij ] is (2n − 2)× (2n− 2) excluding two center’s elements of D2n. The entries of E(ΩD2n) are (a) for i, j = 1, 2, . . . , n− 2 and i ̸= j, ϵij = 2 since dxixj = min{e(xi), e(xj)} = 2; (b) for i = 1, 2, . . . , n − 2 and j = n − 1, n, n + 1, . . . , 2n − 2 or vice versa, ϵij = 0 since dxixj = 1 < min{e(xi), e(xj)} = 2; M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 6 of 13 (c) for i = n − 2 + p and j = n − 2 + n 2 + p or vice versa where p = 1, 2, . . . , n2 , ϵij = 2 since dxixj = min{e(xi), e(xj)} = 2; (d) for i, j = n − 1, n, n + 1, . . . , 2n − 2 , ϵij = 1 except (i = n − 2 + p and j = n − 2 + n 2 + p for p = 1, 2, . . . , n2 ) or vice versa, and i ̸= j, ϵij = 0 since dxixj1 < min{e(xi), e(xj)} = 2; (e) for i = j, ϵij = 0. This implies that E(ΩD2n) is a a2 . . . an−1 b ab . . . a n 2 −1b a n 2 b a n 2 +1b . . . an−1b  a 0 2 . . . 2 0 0 . . . 0 0 0 . . . 0 a2 2 0 . . . 2 0 0 . . . 0 0 0 . . . 0 ... ... . . . ... ... ... . . . ... ... ... . . . ... an−1 2 2 . . . 0 0 0 . . . 0 0 0 . . . 0 b 0 0 . . . 0 0 0 . . . 0 2 0 . . . 0 ab 0 0 . . . 0 0 0 . . . 0 0 2 . . . 0 ... ... . . . ... ... ... . . . ... ... ... . . . ... a n 2 −1b 0 0 . . . 0 0 0 . . . 0 0 0 . . . 2 a n 2 b 0 0 . . . 0 2 0 . . . 0 0 0 . . . 0 a n 2 +1b 0 0 . . . 0 0 2 . . . 0 0 0 . . . 0 ... ... . . . ... ... ... . . . ... ... ... . . . ... an−1b 0 0 . . . 0 0 0 . . . 2 0 0 . . . 0 . Based on Theorem 4 with a = b = 2, c = d = 0, then we have PE(ΩD2n )(λ) = (λ+ 2) 3(n−2) 2 (λ− 2) n 2 −1 (λ2 − 2 (n− 2)λ+ 4(n− 3) ) . The roots of PE(ΩD2n )(λ) give the eigenvalues of ΩD2n . Therefore, the eccentricity energy of ΩD2n is εE(ΩD2n) = ( 3(n− 2) 2 ) |−2|+ (n 2 − 1 ) |2|+ |n− 2± (n− 4)| = 6(n− 2). 3.2. Sum Eccentricity Energy This part focuses on ΩD2n ’s sum eccentricity matrix for odd and even n. Theorem 6. In ΩD2n, the sum eccentricity spectral radius of ΩD2n is ρSE(ΩD2n) = { n− 1 + √ (n− 1)(10n− 1), if n is odd 2 ( n− 2 + √ (n− 2)(5n− 2) ) , if n is even . Proof. M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 7 of 13 (i) Let n be odd. According to Theorem 3 and Definition 3, we can construct the sum eccentricity matrix of ΩD2n . The entries of SE(ΩD2n) = [sij ] are (a) for 1 ≤ i, j ≤ n− 1 and i ̸= j, then sij = 0; (b) for 1 ≤ i ≤ n − 1 and j = n, n + 1, . . . , 2n − 1 or vice versa and i ̸= j, then sij = 2 + 1 = 3 since dxixj = min{e(xi), e(xj)} = 1; (c) for i, j = n, n+ 1, . . . , 2n− 1 and i ̸= j, then sij = 1 + 1 = 2; (d) for i = j, sij = 0 Then SE(ΩD2n) is as follows: SE(ΩD2n) = a a2 . . . an−1 b ab . . . an−1b  a 0 0 . . . 0 3 3 . . . 3 a2 0 0 . . . 0 3 3 . . . 3 ... ... ... . . . ... ... ... . . . ... an−1 0 0 . . . 0 3 3 . . . 3 b 3 3 . . . 3 0 2 . . . 2 ab 3 3 . . . 3 2 0 . . . 2 ... ... ... . . . ... ... ... . . . ... an−1b 3 3 . . . 3 2 2 . . . 0 . SE-matrix of ΩD2n can be written as given below SE(ΩD2n) = ( 0n−1 3J(n−1)×n 3Jn×(n−1) 2(J − I)n ) , and the characteristic formula of SE(ΩD2n), PSE(ΩD2n )(µ) = ∣∣∣∣ µIn−1 −3J(n−1)×n −3Jn×(n−1) (µ+ 2)In − 2Jn ∣∣∣∣ . By Lemma 1, with a = 0, b = 2, c = d = 3, n1 = n− 1 and n2 = n, we obtain PSE(ΩD2n )(µ) = µn−2(µ+ 2)n−1 ( µ2 − 2(n− 1)µ− 9n(n− 1) ) . The eigenvalues of ΩD2n are µ1 = 0 of multiplicity n − 2, µ2 = −2 of multiplicity n− 1, and µ3,4 = n− 1± √ (n− 1)(10n− 1). Therefore, the SE-spectral radius of ΩD2n is ρSE(ΩD2n) = n− 1 + √ (n− 1)(10n− 1). (ii) Let n be even. The entries of SE(ΩD2n) = [sij ] are (a) for 1 ≤ i, j ≤ n− 2 and i ̸= i, sij = 0; M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 8 of 13 (b) for 1 ≤ i ≤ n− 2, n− 1 ≤ j ≤ 2n− 2 or vice versa, sij = 2 + 2 = 4; (c) for n− 1 ≤ i ≤ n+ n 2 − 2 and n+ n 2 − 1 ≤ j ≤ 2n− 2 where j ̸= n− 2 + n 2 + i or vice versa, sij = 2 + 2 = 4; (d) for n−1 ≤ i, j ≤ n+ n 2 −2, n+ n 2 −2 ≤ i, j ≤ 2n−2, and i ̸= j, sij = 2+2 = 4; (e) for i = j, j = n− 2 + n 2 + i, i = n− 2 + n 2 + j, sij = 0. Hence, the matrix construction is as follows. SE(ΩD2n) = a . . . an−1 b . . . a n 2 −1b a n 2 b . . . an−1b  a 0 . . . 0 4 . . . 4 4 . . . 4 ... ... . . . ... ... . . . ... ... . . . ... an−1 0 . . . 0 4 . . . 4 4 . . . 4 b 4 . . . 4 0 . . . 4 0 . . . 4 ... ... . . . ... ... . . . ... ... . . . ... a n 2 −1b 4 . . . 4 4 . . . 0 4 . . . 0 a n 2 b 4 . . . 4 0 . . . 4 0 . . . 4 ... ... . . . ... ... . . . ... ... . . . ... an−1b 4 . . . 4 4 . . . 0 4 . . . 0 . In other words, SE(ΩD2n) is as follows: SE(ΩD2n) =  0n−2 4J(n−2)×n 2 4J(n−2)×n 2 4Jn 2 ×(n−2) 4(J − I)n 2 4(J − I)n 2 4Jn 2 ×(n−2) 4(J − I)n 2 4(J − I)n 2  . Based on Theorem 4 with a = b = 0, c = d = 4, then PSE(ΩD2n )(µ) = µ 3(n−2) 2 (µ+ 8) n 2 −1 (µ2 − 4(n− 2)µ− 16n(n− 2) ) . The eigenvalues of ΩD2n are µ1 = 0 of multiplicity 3(n−2) 2 , µ2 = −8 of multiplicity n 2 − 1, and µ3,4 = 2 ( n− 2± √ (n− 2)(5n− 2) ) . Therefore, the SE-spectral radius of ΩD2n is ρSE(ΩD2n) = 2 ( n− 2 + √ (n− 2)(5n− 2) ) . Theorem 7. In ΩD2n, the sum eccentricity energy of ΩD2n is εSE(ΩD2n) =  2 ( n− 1 + √ (n− 1)(10n− 1) ) , if n is odd 4 ( n− 2 + √ (n− 2)(5n− 2) ) , if n is even . M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 9 of 13 Proof. (i) Let n be odd. According to Theorem 6, SE-energy of ΩD2n is εSE(ΩD2n) = (n− 2) |0|+ (n− 1) |−2|+ ∣∣∣n− 1± √ (n− 1)(10n− 1) ∣∣∣ = 2 ( n− 1 + √ (n− 1)(10n− 1) ) . (ii) Let n be even. According to Theorem 6, the SE-energy of ΩD2n is εSE(ΩD2n) = ( 3(n− 2) 2 ) |0|+ (n 2 − 1 ) |−8|+ ∣∣∣2(n− 2)± 2 √ (n− 2)(5n− 2) ∣∣∣ = 4 ( n− 2 + √ (n− 2)(5n− 2) ) . 3.3. Average Degree-Eccentricity Matrix Next, we show the energy of ΩD2n concerning the average degree-eccentricity matrix for odd and even n. Theorem 8. In ΩD2n, the average degree-eccentricity spectral radius of ΩD2n is ρADE(ΩD2n ) =  1 2 ( (n− 1) ( n− 1 2 ) + √ (n− 1)2 ( n− 1 2 )2 + 1 4n(n− 1)(3n+ 1)2 ) , if n is odd 1 2 ( (n− 2)(n− 1) + √ (n− 2)2(n− 1)2 + 9 4n 3(n− 2) ) , if n is even. Proof. (i) Let n be odd. Based on Theorems 2 and 3, and Definition 4, we have the average degree-eccentricity matrix with entries of [aij ] are (a) for i, j = 1, 2, . . . , n− 1 and i ̸= j, then aij = 0; (b) for i = 1, 2, . . . , n− 1 and j = n, n+ 1, . . . , 2n− 1 or vice versa and i ̸= j, then aij = 1 4(n+ 2(n− 1) + 2 + 1) = 3n+1 4 ; (c) for i, j = n, n+1, . . . , 2n−1 and i ̸= j, then aij = 1 4(2(n−1)+2(n−1)+1+1) = n− 1 2 ; (d) for i = j, aij = 0 M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 10 of 13 Then ADE(ΩD2n) is as follows: ADE(ΩD2n) = a a2 . . . an−1 b ab . . . an−1b  a 0 0 . . . 0 3n+1 4 3n+1 4 . . . 3n+1 4 a2 0 0 . . . 0 3n+1 4 3n+1 4 . . . 3n+1 4 ... ... ... . . . ... ... ... . . . ... an−1 0 0 . . . 0 3n+1 4 3n+1 4 . . . 3n+1 4 b 3n+1 4 3n+1 4 . . . 3n+1 4 0 n− 1 2 . . . n− 1 2 ab 3n+1 4 3n+1 4 . . . 3n+1 4 n− 1 2 0 . . . n− 1 2 ... ... ... . . . ... ... ... . . . ... an−1b 3n+1 4 3n+1 4 . . . 3n+1 4 n− 1 2 n− 1 2 . . . 0 . ADE-matrix of ΩD2n can be written as given below ADE(ΩD2n) = ( 0n−1 ( 3n+1 4 ) J(n−1)×n( 3n+1 4 ) Jn×(n−1) ( n− 1 2 ) (J − I)n ) , and the characteristic formula of ADE(ΩD2n), PADE(ΩD2n )(µ) = ∣∣∣∣ µIn−1 − ( 3n+1 4 ) J(n−1)×n − ( 3n+1 4 ) Jn×(n−1) ( µ+ n− 1 2 ) In − ( n− 1 2 ) Jn ∣∣∣∣ . By Lemma 1, with a = 0, b = n − 1 2 , c = d = 3n+1 4 , n1 = n − 1 and n2 = n, we obtain PADE(ΩD2n )(µ) = µn−2 ( µ+ n− 1 2 )n−1( µ2 − (n− 1) ( n− 1 2 ) µ− 1 16 n(n− 1)(3n+ 1)2 ) . The eigenvalues of ΩD2n are µ1 = 0 of multiplicity n− 2, µ2 = 1 2 − n of multiplicity n − 1, and µ3,4 = 1 2 ( (n− 1) ( n− 1 2 ) ± √ (n− 1)2 ( n− 1 2 )2 + 1 4n(n− 1)(3n+ 1)2 ) . Therefore, the ADE-spectral radius of ΩD2n is ρADE(ΩD2n) = 1 2 (n− 1) ( n− 1 2 ) + √ (n− 1)2 ( n− 1 2 )2 + 1 4 n(n− 1)(3n+ 1)2  . (ii) Let n be even. The entries of ADE(ΩD2n) = [aij ] are (a) for 1 ≤ i, j ≤ n− 2 and i ̸= i, aij = 0; (b) for 1 ≤ i ≤ n−2, n−1 ≤ j ≤ 2n−2 or vice versa, aij = 1 4 (n+ 2(n− 2) + 2 + 2) = 3n 4 ; (c) for n− 1 ≤ i ≤ n+ n 2 − 2 and n+ n 2 − 1 ≤ j ≤ 2n− 2 where j ̸= n− 2 + n 2 + i or vice versa, aij = 1 4 (2(n− 2) + 2(n− 2) + 2 + 2) = n− 1; M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 11 of 13 (d) for n − 1 ≤ i, j ≤ n + n 2 − 2, n + n 2 − 2 ≤ i, j ≤ 2n − 2, and i ̸= j, aij = 1 4 (2(n− 2) + 2(n− 2) + 2 + 2) = n− 1; (e) for i = j, j = n− 2 + n 2 + i, i = n− 2 + n 2 + j, aij = 0. Thus, ADE(ΩD2n ) = a . . . an−1 b . . . a n 2 −1b a n 2 b . . . an−1b  a 0 . . . 0 3n 4 . . . 3n 4 3n 4 . . . 3n 4 ... ... . . . ... ... . . . ... ... . . . ... an−1 0 . . . 0 3n 4 . . . 3n 4 3n 4 . . . 3n 4 b 3n 4 . . . 3n 4 0 . . . n− 1 0 . . . n− 1 ... ... . . . ... ... . . . ... ... . . . ... a n 2 −1b 3n 4 . . . 3n 4 n− 1 . . . 0 n− 1 . . . 0 a n 2 b 3n 4 . . . 3n 4 0 . . . n− 1 0 . . . n− 1 ... ... . . . ... ... . . . ... ... . . . ... an−1b 3n 4 . . . 3n 4 n− 1 . . . 0 n− 1 . . . 0 . In other words, ADE(ΩD2n) is as follows: ADE(ΩD2n) =  0n−2 3n 4 J(n−2)×n 2 3n 4 J(n−2)×n 2 3n 4 Jn 2 ×(n−2) (n− 1)(J − I)n 2 (n− 1)(J − I)n 2 3n 4 Jn 2 ×(n−2) (n− 1)(J − I)n 2 (n− 1)(J − I)n 2  . Based on Theorem 4 with a = b = 0, c = 3n 4 , d = n− 1, then PADE(ΩD2n )(µ) = µ 3(n−2) 2 (µ+ 2(n− 1)) n 2 −1 ( µ2 − (n− 1)(n− 2)µ− 9n2 16 n(n− 2) ) . The eigenvalues of ΩD2n are µ1 = 0 of multiplicity 3(n−2) 2 , µ2 = −2(n − 1) of multiplicity n 2 − 1, and µ3,4 = 1 2 ( (n− 2)(n− 1)± √ (n− 2)2(n− 1)2 + 9 4n 3(n− 2) ) . Therefore, the ADE- spectral radius of ΩD2n is ρADE(ΩD2n) = 1 2 ( (n− 2)(n− 1) + √ (n− 2)2(n− 1)2 + 9 4 n3(n− 2) ) . Theorem 9. In ΩD2n, the average degree-eccentricity energy of ΩD2n is εADE(ΩD2n ) =  (n− 1) ( n− 1 2 ) + √ (n− 1)2 ( n− 1 2 )2 + 1 4n(n− 1)(3n+ 1)2, if n is odd (n− 2)(n− 1) + √ (n− 2)2(n− 1)2 + 9 4n 3(n− 2), if n is even. Proof. M. U. Romdhini et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5776 12 of 13 (i) Let n be odd. Based on Theorems 8, the ADE-energy of ΩD2n is εADE(ΩD2n ) = (n− 2) |0|+ (n− 1) ∣∣∣∣12 − n ∣∣∣∣+∣∣∣∣∣∣12 (n− 1) ( n− 1 2 ) ± √ (n− 1)2 ( n− 1 2 )2 + 1 4 n(n− 1)(3n+ 1)2 ∣∣∣∣∣∣ =(n− 1) ( n− 1 2 ) + √ (n− 1)2 ( n− 1 2 )2 + 1 4 n(n− 1)(3n+ 1)2. (ii) Let n be even. Based on Theorem 8, the ADE-energy of ΩD2n is εADE(ΩD2n) = ( 3(n− 2) 2 ) |0|+ (n 2 − 1 ) |−2(n− 1)|+∣∣∣∣∣12 ( (n− 2)(n− 1)± √ (n− 2)2(n− 1)2 + 9 4 n3(n− 2) )∣∣∣∣∣ =(n− 2)(n− 1) + √ (n− 2)2(n− 1)2 + 9 4 n3(n− 2). 4. Discussion We can conclude several interesting statements from the results of the previous section. Corollary 1. The eccentricity energy of ΩD2n is always an even integer. Corollary 2. The energy of ΩD2n is never an odd integer associated with the sum eccen- tricity and average degree eccentricity matrices. Corollary 3. ΩD2n is hyperenergetic associated with the eccentricity-based matrices. Acknowledgements The authors thank the referees for their helpful comments or recommendations on this article. We also thank the University of Mataram, Indonesia, for funding assistance through the Overseas Collaborative Research Scheme No.2490/UN18.L1/PP/2025. References [1] J Wang, M Lu, F Belardo, and M Randic. The anti-adjacency matrix of a graph: Eccentricity matrix. 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