On the Spectra of Commuting and Non Commuting Graph on Dihedral Group CAUCHY –JURNAL MATEMATIKA MURNI DAN APLIKASI Volume 4 (4) (2017), Pages 176-182 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: 17 May 2017 Reviewed: 20 May 2017 Accepted: 30 May 2017 DOI: http://dx.doi.org/10.18860/ca.v4i4.4211 On the Spectra of Commuting and Non Commuting Graph on Dihedral Group Abdussakir1, Rivatul Ridho Elvierayani2, Muflihatun Nafisah3 1,2,3) Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang Email: sakir@mat.uin-malang.ac.id ABSTRACT Study about spectra of graph has became interesting work as well as study about commuting and non commuting graph of a group or a ring. In this paper, we investigate adjacency spectrum, Laplacian spectrum, signless Laplacian spectrum, and detour spectrum of commuting and non commuting graph of dihedral group D2n. Keywords: graph, spectrum, commuting graph, non commuting graph, dihedral group. INTRODUCTION Let G be a graph of order n (n ≥ 1) and let its vertex set is V(G) = {v1, v2, …, vn}. The adjacency matrix A(G) = [aij] of G is nn matrix where aij = 1 if vivj  E(G) and aij = 0 if vivj  E(G)) [1]. So, the adjacency matrix of a graph is real and symmetric that contains 0 in all of main diagonal entries [2]. The degree matrix D(G) = [dij] of G is a diagonal matrix where dii = degG(vi) and dij = 0, for i ≠ j. The Laplacian matrix L(G) of G is defined by L(G) = D(G) – A(G) [3] and the signless Laplacian matrix Q(G) of G is defined by Q(G) = D(G) + A(G) [4]. The detour matrix DD(G) = [eij] of G is nn matrix where eij is equal to the length of the longest path between vi and vj [5]. Let 0, 1, …, 𝜆𝑘−1 where 0 > 1 > … > 𝑘−1 are distinct eigenvalues from a matrix of a graph, and let m(0), m(1), …, m(𝑘−1) are multiplicities of i, for i = 0, 1, 2, ..., k – 1. The multiplicity of i as roots of characteristic equation is equal to the dimension of the space of eigenvectors corresponding to i [6]. Matrix of order 2k that contains 0, 1, …, 𝑘−1 for the first row and m(0), m(1), …, m(𝑘−1) for the second row is called spectrum of graph G and denoted by Spec(G) [6]. According to Yin [7], we can write spectrum of graph G by Spec(G) = [ 𝜆0 𝜆1 ⋯ 𝜆𝑘−1 𝑚(𝜆0) 𝑚(𝜆1) ⋯ 𝑚(𝜆𝑘−1) ]. Spectrum that obtained from the adjacency matrix A(G), the Laplacian matrix L(G), the signless Laplacian matrix Q(G), and the detour matrix DD(G) are called the adjacency spectrum denoted by SpecA(G), the Laplace spectrum denoted by SpecL(G), the signless Laplacian spectrum denoted by SpecQ(G), and the detour spectrum denoted by SpecDD(G) respectively. Let G be a non Abelian group with center Z(G). The commuting graph C(G, X), where X is a subset of G, is the graph with vertex set X and distinct vertices being joined by an edge whenever xy = yx in G [8]. In the case X = G, we denote the commuting graph of G by C(G) http://dx.doi.org/10.18860/ca.v4i4.4211 mailto:sakir@mat.uin-malang.ac.id On the Spectra of Commuting and Non Commuting Graph on Dihedral Group Abdussakir 177 instead of C(G, X). The non commuting graph Γ𝐺 of G, is the graph whose vertices are the non central elements G\Z(G) and whose edges join those vertices x, y  G\Z(G) for which xy  yx in G [9]. The non commuting graph Γ𝐺 of G is always connected [10]. The study of spectrum of graph was begun by work of Norman Bigg [6] in his famous book Algebraic Graph Theory. Some investigations about spectrum of graphs had been done before, for example Yin [7], Brouwer and Haemers [4], Ayyaswamy & Balachandran [5], and Jog & Kotambari [11]. Researches on commuting and non commuting graph of a group and ring had also been done before, for example Akbari et al. [12], Abdollahi et al. [10], Darafsheh [13], Abdollahi et al. [9], Vahidi & Talebi [14], Rahayuningtyas et al. [15], Chelvam et al. [16], Woodcock [17], and Nawawi & Rowley [8]. In this study, we determine the spectra of commuting and non commuting graph of dihedral group, including adjacency spectrum, Laplacian spectrum, signless Laplacian spectrum, and detour spectrum. RESULT AND DISCUSSION Here we present several results from our studies. For adjacency spectrum, we observe and get result just for non commuting graph of dihedral group. Lemma 1: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n > 3. The characteristic polynomial for adjacency matrix 𝐴(Γ𝐷2𝑛) of Γ𝐷2𝑛 is 𝑝(𝜆) = (𝜆)𝑛−2(𝜆 + 1)𝑛−1 (−𝜆2 + (𝑛 − 1)𝜆 + (𝑛(𝑛 − 1))). Proof: For dihedral group 𝐷2𝑛 = {1, r, r 2, …, rn-1, s, sr, sr2, …, srn-1} where n is odd positive integer, n > 3, we have Z(𝐷2𝑛) = {1}. So, the non commuting graph Γ𝐷2𝑛 of D2n has { r, r2, …, rn-1, s, sr, sr2, …, srn-1} as its vertex set. Because n is odd, we have 𝑠𝑟𝑖 ≠ 𝑟𝑖𝑠, 𝑖 = 1, 2, … , 𝑛 − 1, so s and 𝑟𝑖 are adjacent in Γ𝐷2𝑛. And also, 𝑠𝑟 𝑖𝑠𝑟𝑗 ≠ 𝑠𝑟𝑗𝑠𝑟𝑖, for i = 1, 2,..., n – 1 and 𝑗 = 1, 2, … , 𝑛 − 1, where 𝑖 ≠ 𝑗, so 𝑠𝑟𝑖 and 𝑠𝑟𝑗 are adjacent in Γ𝐷2𝑛 . Then, we have the adjacency matrix for Γ𝐷2𝑛 as the following 𝑟 𝑟2 ⋯ 𝑟𝑛−1 𝑠 𝑠𝑟 𝑠𝑟2 ⋯ 𝑠𝑟𝑛−1 𝐴(ΓD2𝑛) = 𝑟 𝑟2 ⋮ 𝑟𝑛−1 𝑠 𝑠𝑟 𝑠𝑟2 ⋮ 𝑠𝑟𝑛−1 [ 0 0 … 0 1 1 1 ⋯ 1 0 0 … 0 1 1 1 ⋯ 1 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 1 1 1 ⋯ 1 1 1 ⋯ 1 0 1 1 ⋯ 1 1 1 ⋯ 1 1 0 1 ⋯ 1 1 1 ⋯ 1 1 1 0 ⋯ 1 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 1 1 ⋯ 1 1 1 1 ⋯ 0] We know that the characteristic polynomial of 𝐴(ΓD2n) is de𝑡(𝐴(ΓD14) − 𝐼). Using the Gaussian elimination procedure we obtain this upper triangular matrix On the Spectra of Commuting and Non Commuting Graph on Dihedral Group Abdussakir 178 𝑟 𝑟2 ⋯ 𝑟𝑛−1 𝑠 𝑠𝑟 𝑠𝑟2 ⋯ 𝑠𝑟𝑛−1 𝑟 𝑟2 ⋮ 𝑟𝑛−1 𝑠 𝑠𝑟 𝑠𝑟2 ⋮ 𝑠𝑟𝑛−1 [ 1 1 … 1 −𝜆 1 1 ⋯ 1 0 𝜆 … 𝜆 1 − 𝜆2 1 + 𝜆 1 + 𝜆 ⋯ 1 + 𝜆 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 𝜆 (𝑛 − 2) − 𝜆2 (𝑛 − 2) + 𝜆 (𝑛 − 2) + 𝜆 ⋯ (𝑛 − 2) + 𝜆 0 0 ⋯ 0 1 + 𝜆 −𝜆 − 1 0 ⋯ 0 0 0 ⋯ 0 0 1 + 𝜆 −𝜆 − 1 ⋯ 0 0 0 ⋯ 0 0 0 1 + 𝜆 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 0 0 ⋯ −𝜆2 + (𝑛 − 1)𝜆 + (𝑛(𝑛 − 1))] Because the determinant for this upper triangular matrix is the multiplication of all entries in its main diagonal, so we obtain the characteristic polynomial of 𝐴(ΓD2n) as follows 𝑝(𝜆) = (𝜆)𝑛−2(𝜆 + 1)𝑛−1 (−𝜆2 + (𝑛 − 1)𝜆 + (𝑛(𝑛 − 1))).  Theorem 1: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n > 3. The adjacency spectrum of Γ𝐷2𝑛 is 𝑆𝑝𝑒𝑐𝐴(𝛤𝐷2𝑛) = [ 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 0 −1 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 1 𝑛 − 2 𝑛 − 1 1 ]. Proof: From Lemma 1, the characteristic polynomial of 𝐴(ΓD2n) is 𝑝(𝜆) = (𝜆)𝑛−2(𝜆 + 1)𝑛−1 (−𝜆2 + (𝑛 − 1)𝜆 + (𝑛(𝑛 − 1))). By solving the equation 𝑝(𝜆) = 0, we have 𝜆1 = 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , 𝜆2 = 0, 𝜆3 = −1, 𝜆4 = 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 as eigenvalues for 𝐴(ΓD2n). Now, we will determine the multiplicity for each eigenvalue. Because the multiplicity is equal to number of basis for space of eigenvectors corresponding to i, i = 1, 2, 3, and 4, we substitute i to 𝐴(ΓD2n) − 𝑖𝐼 and apply Gauss- Jordan elimination to this matrix to obtain row-reduced eselon matrix and then we can see the number of zero rows of it. For 𝜆1 = 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , by eliminating 𝐴(ΓD2n) − 1𝐼, we obtained row- reduced eselon matrix with one zero row. So, for 𝜆1 = 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , we have its algebraic multiplicity is 1. For 𝜆2 = 0, by eliminating 𝐴(ΓD2n) − 2𝐼, we obtain row-reduced eselon matrix with n – 2 zero rows. So, for 𝜆2 = 0, we have its algebraic multiplicity is n – 2. For 𝜆3 = −1, by eliminating 𝐴(ΓD2n) − 3𝐼, we obtain row-reduced eselon matrix with n – 1 zero rows. So, for 𝜆2 = −1, we have its algebraic multiplicity is n – 1. On the Spectra of Commuting and Non Commuting Graph on Dihedral Group Abdussakir 179 For 𝜆4 = 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , by eliminating 𝐴(ΓD2n) − 4𝐼, we obtain row- reduced eselon matrix with one zero rows. So, for 𝜆4 = 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 , we have its algebraic multiplicity is 1. We conclude that SpecA(𝛤𝐷2𝑛) = [ 𝑛−1 2 + √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 0 −1 𝑛−1 2 − √(𝑛 − 1)𝑛 + ( 𝑛−1 2 ) 2 1 𝑛 − 2 𝑛 − 1 1 ].  Theorem 2: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is even positive integer and n  6. The adjacency spectrum of Γ𝐷2𝑛 is SpecA(𝛤𝐷2𝑛 ) = [ 𝑛−2 2 + √ 5𝑛2−12𝑛+4 4 0 −2 𝑛−2 2 − √ 5𝑛2−12𝑛+4 4 1 𝑛−2 2 3𝑛−6 2 1 ]. The next results are Laplacian spectrum of commuting and non commuting graph of dihedral group. Lemma 2: Let 𝐶(𝐷2𝑛) be a commuting graph of dihedrral group 𝐷2𝑛 where n is positive integer and n  3. The characteristic polynomial for Laplacian matrix 𝐿(𝐶(𝐷2𝑛)) of 𝐶(𝐷2𝑛) is 𝑝(𝜆) = −𝜆(𝜆 − 𝑛)𝑛−2(𝜆 − 1)𝑛(𝜆 − 2𝑛), for odd n and 𝑝(𝜆) = −𝜆(𝜆 − 𝑛)𝑛−3((𝜆 − 4)(𝜆 − 2)) 𝑛 2(𝜆 − 2𝑛)2, for even n. Theorem 3: Let 𝐶(𝐷2𝑛) be a commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. The Laplacian spectrum of 𝐶(𝐷2𝑛) is 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷2𝑛)) = [ 2𝑛 𝑛 1 0 1 𝑛 − 2 𝑛 1 ]. Proof: From Lemma 2, for odd n, we have 𝜆1 = 2𝑛, 𝜆2 = 𝑛, 𝜆3 = 1, and 𝜆4 = 0 as eigenvalues for the Laplacian matrix of 𝐶(𝐷2𝑛). And the multiplicity for each eigenvalue is 𝑚(𝜆1) = 1, 𝑚(𝜆2) = 𝑛 − 2, 𝑚(𝜆3) = 𝑛, and 𝑚(𝜆4) = 1.  Theorem 4: Let 𝐶(𝐷2𝑛) be a commuting graph of dihedrral group 𝐷2𝑛 where n is even positive integer and n  6. The Laplacian spectrum of 𝐶(𝐷2𝑛) is 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷2𝑛)) =[ 2𝑛 𝑛 4 2 0 2 𝑛 − 3 𝑛 2 𝑛 2 1]. Proof: From Lemma 2, for even n and n ≥ 6, we have 𝜆1 = 2𝑛, 𝜆2 = 𝑛, 𝜆3 = 4, 𝜆4 = 2, and 𝜆5 = 0 as eigenvalues for the Laplacian matrix of 𝐶(𝐷2𝑛). And the multiplicity for each eigenvalue is 𝑚(𝜆1) = 2, 𝑚(𝜆2) = 𝑛 − 3, 𝑚(𝜆3) = 𝑛 2 , 𝑚(𝜆4) = 𝑛 2 , and 𝑚(𝜆5) = 1. From our investigation, the Laplacian spectrum of 𝐶(𝐷8) is On the Spectra of Commuting and Non Commuting Graph on Dihedral Group Abdussakir 180 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷8)) = [ 8 4 2 0 2 3 2 1 ]. If we apply Theorem 4 for n = 4 then we have 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷8)) =[ 8 4 4 2 0 2 1 2 2 1 ] According to spectrum definition, we must write the Laplacian spectrum for 𝐶(𝐷8) as 𝑆𝑝𝑒𝑐𝐿 (𝐶(𝐷8)) =[ 8 4 2 0 2 3 2 1 ] So, by this reason, Theorem 4 also holds for n = 4. Lemma 3: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. The characteristic polynomial for the Laplacian matrix 𝐿(Γ𝐷2𝑛) of Γ𝐷2𝑛 is 𝑝(𝜆) = −𝜆(𝑛 − 𝜆)𝑛−2((−2𝑛 + 1) + 𝜆) 𝑛 . Theorem 5: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. The Laplacian spectrum of Γ𝐷2𝑛 is 𝑆𝑝𝑒𝑐𝐿 (𝛤𝐷2𝑛 ) = [ 2𝑛 − 1 𝑛 0 𝑛 𝑛 − 2 1 ]. Proof: From Lemma 3, for odd n and n ≥ 3, we have 𝜆1 = 2𝑛 − 1, 𝜆2 = 𝑛, and 𝜆3 = 0. as eigenvalues for the Laplacian matrix of Γ𝐷2𝑛 . And the multiplicity for each eigenvalue is 𝑚(𝜆1) = 𝑛, 𝑚(𝜆2) = 𝑛 − 2, and 𝑚(𝜆3) = 1. Theorem 6: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is even positive integer and n  6. The Laplacian spectrum of Γ𝐷2𝑛 is 𝑆𝑝𝑒𝑐𝐿 (𝛤𝐷2𝑛 ) = [ 2(𝑛 − 1) 2(𝑛 − 2) 𝑛 0 𝑛 2 𝑛 2 𝑛 − 3 1 ]. Proof: The characteristic polynomial for the Laplacian matrix 𝐿(Γ𝐷2𝑛) of Γ𝐷2𝑛 where n is even positive integer and n  6 is 𝑝(𝜆) = −𝜆(𝑛 − 𝜆)𝑛−3(2(𝑛 − 1) + 𝜆) 𝑛 2(2(𝑛 − 2) + 𝜆) 𝑛 2 So we have 𝜆1 = 2(𝑛 − 1), 𝜆2 = 2(𝑛 − 2), 𝜆3 = 𝑛, and 𝜆4 = 0 as eigenvalues for the Laplacian matrix of Γ𝐷2𝑛. And the multiplicity for each eigenvalue is 𝑚(𝜆1) = 𝑛 2 , 𝑚(𝜆2) = 𝑛 2 , 𝑚(𝜆3) = 𝑛 − 3, and 𝑚(𝜆4) = 1.  As for the adjacency spectrum, we just have the signless Laplacian spectrum for non commuting graph of dihedral group. We present the result without proof. Theorem 7: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is even positive integer and n  8. The signless Laplacian spectrum of Γ𝐷2𝑛 is 𝑠𝑝𝑒𝑐𝑄(𝛤𝐷2𝑛 ) = [ (2𝑛 − 3) + √2𝑛2 − 8𝑛 + 9 2(𝑛 − 2) 2(𝑛 − 3) 𝑛 (2𝑛 − 3) − √2𝑛2 − 8𝑛 + 9 1 𝑛 2 𝑛−2 2 𝑛 − 3 1 ] On the Spectra of Commuting and Non Commuting Graph on Dihedral Group Abdussakir 181 The last two theorems are results for detour spectrum of non commuting graph on dihedral group. We start by this lemma. Lemma 4: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is positive integer and n  3. The longest path between two distinct vertices x, y  V(Γ𝐷2𝑛 ) is 2n – 2 for odd n and 2n – 3 for even n. Proof: For dihedral group 𝐷2𝑛 = {1, r, r 2, …, rn-1, s, sr, sr2, …, srn-1} where n is odd positive integer, n  3, we have Z(𝐷2𝑛) = {1}. So, the non commuting graph Γ𝐷2𝑛 of D2n has {r, r2, …, rn-1, s, sr, sr2, …, srn-1} as its vertex set. Hence the order of Γ𝐷2𝑛 is 2n – 1. Because n is odd, we have 𝑠𝑟𝑖 ≠ 𝑟𝑖𝑠, 𝑖 = 1, 2, … , 𝑛 − 1, so s and 𝑟𝑖 are adjacent in Γ𝐷2𝑛 . And also, 𝑠𝑟𝑖𝑠𝑟𝑗 ≠ 𝑠𝑟𝑗𝑠𝑟𝑖, for i = 1, 2,..., n – 1 and 𝑗 = 1, 2, … , 𝑛 − 1, where 𝑖 ≠ 𝑗, so 𝑠𝑟𝑖 and 𝑠𝑟𝑗 are adjacent in Γ𝐷2𝑛. On the other hand r i and rj are not adjacent in Γ𝐷2𝑛 because r irj = rjri in D2n. We have that the sum of degree of any two non adjacent vertices in Γ𝐷2𝑛 is 2n > 2𝑛−1 2 and Γ𝐷2𝑛 is Hamiltonian graph (see [10]). So, the length of the longest path between two distinct vertices is 2n – 2 for odd n. For dihedral group 𝐷2𝑛 = {1, r, r 2, …, rn-1, s, sr, sr2, …, srn-1} where n is even positive integer, n > 3, we have 𝑍(𝐷2𝑛) = {1, 𝑟 𝑛 2}. So, the non commuting graph Γ𝐷2𝑛 of D2n has {r, r2, …, 𝑟 𝑛 2 −1 , 𝑟 𝑛 2 +1 , ..., rn-1, s, sr, sr2, …, srn-1} as its vertex set. Hence the order of Γ𝐷2𝑛 is 2n – 2. We also have that Γ𝐷2𝑛 is Hamiltonian graph. So, the length of the longest path between two distinct vertices is 2n – 3 for even n.  Theorem 8: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. The detour spectrum of Γ𝐷2𝑛 is SpecDD(Γ𝐷2𝑛) = [ (2𝑛 − 2)2 −(2𝑛 − 2) 1 2𝑛 − 2 ]. Proof: By Lemma 4, we have DD(Γ𝐷2𝑛) is (2n – 1)(2n – 1) matrix where all entries in its main diagonal is 0 and 2n – 2 elsewhere, for odd n. Then, we have the characteristic polinomial is p(𝜆) = (𝜆 + (2𝑛 − 2)) 2𝑛−2 (𝜆 − (2𝑛 − 2)2). From the characteristics polinomial, we have 𝜆1 = (2𝑛 − 2) 2 and 𝜆2 = 2𝑛 − 2 as eigenvalues of DD(Γ𝐷2𝑛) with their multiplicity are 𝑚(𝜆1) = 1 and 𝑚(𝜆2) = 2𝑛 − 2. So, we get the desired result.  Theorem 9: Let Γ𝐷2𝑛 be a non commuting graph of dihedral group 𝐷2𝑛 where n is odd positive integer and n  3. The detour spectrum of Γ𝐷2𝑛 is SpecDD(Γ𝐷2𝑛) = [ (2𝑛 − 3)2 −(2𝑛 − 3) 1 2𝑛 − 3 ]. Proof: By Lemma 4, we have DD(Γ𝐷2𝑛) is (2n – 2)(2n – 2) matrix where all entries in its main diagonal is 0 and 2n – 3 elsewhere, for even n. Then, we have the characteristic polinomial is p(𝜆) = (𝜆 + (2𝑛 − 2)) 2𝑛−2 (𝜆 − (2𝑛 − 2)2). From the characteristic polinomial, we have 𝜆1 = (2𝑛 − 3) 2 and 𝜆2 = 2𝑛 − 3 as eigenvalues of DD(Γ𝐷2𝑛) with their multiplicity are 𝑚(𝜆1) = 1 and 𝑚(𝜆2) = 2𝑛 − 3. This lead us to the detour spectrum.  On the Spectra of Commuting and Non Commuting Graph on Dihedral Group Abdussakir 182 CONCLUSION From our results, several spectrums for several cases of n are not determined yet. So, the remaining cases can be done in further research. 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