The Laplacian and normalized Laplacian spectra of Mobius polyomino networks and their applications
TThe Laplacian and normalized Laplacian spectra of M¨obiuspolyomino networks and their applications
Zhi-Yu Shi , Jia-Bao Liu , , ∗ , Sakander Hayat School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China School of Mathematics, Southeast University, Nanjing 210096, China Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi 23460, Pakistan
Abstract.
Spectral theory has widely used in complex networks and solved some practical problems.In this paper, we investigated the Laplacian and normalized Laplacian spectra of M¨obius polyominonetworks by using spectral theory. Let M n denote M¨obius polyomino networks ( n ≥ M n are obtained. Moreover, it is surprising to find that the multiplicative degree-Kirchhoff index of M n is nine times as much as the Kirchhoff index. Keywords : Laplacian spectrum; Normalized Laplacian spectrum; M¨obius polyomino networks; Topo-logical indices Introduction
In 1964, Heilbronner [1] proposed M¨obius aromatic based on Huckel’s molecular orbital theory.Compared with Huckel’s system, the M¨obius system is stable because of its closed shell structure. Inrecent years, compounds with M¨obius aromatic have been synthesized. In particular, Ma et al. [2] studiedthe normalized Laplacian spectrum for the hexagonal M¨obius graphs. Then, Geng et al. [3] obtained theLaplacian spectrum of M¨obius phenylenes Chain. In 2019, Lei et al. [4] studied the normalized Laplacianspectrum of M¨obius phenylene chain. For other graphs, see [5–11]. Motivated by these, we investigatethe Laplacian spectrum and normalized Laplacian spectrum of M¨obius polyomino networks. The M¨obiuspolyomino networks are one kind molecular graphs embedded into the M¨obius strip, which each side isa polyomino (see Figure 1(1)). For example, the M¨obius polyomino network of length 6 is depicted inFigure 1(2).Since n = 1, the graph does not exist. When n = 2, the graph corresponds to the special graph inreference [12]. Therefore, we investigate the results of M¨obius polyomino networks which n ≥ G = ( E G , V G ) be a graph with edge set E G and vertex set V G , where its size are | E G | = m and | V G | = n . Let D ( G ) = diag { d , d , · · · , d n } represent a degree matrix, and A ( G ) be the adjacencymatrix, where d i is the degree of v i . The Laplacian matrix of G is given by L ( G ) = D ( G ) − A ( G ), which( i, j )-entry are equal to -1 when v i and v j are adjacent, d i when i = j , 0 otherwise. The normalizedLaplacian matrix of G is given by L ( G ) = D ( G ) − LD ( G ) − , which ( i, j )-entry are equal to − √ d i d j when v i and v j are adjacent, 1 when i = j , 0 otherwise. For more notation, one can be referred to [13].The traditional concept of distance is the length of the shortest path between vertices i and j , rep-resented by d ij . The Wiener index [14] was proposed as W ( G ) = (cid:80) i Given an n × n matrix B , submatrix of B is represented by B [ i , · · · , i k ], where B [ i , · · · , i k ] is formedby removing the i -th, · · · , i k -th rows and columns of B . Let P B ( x ) = det ( xI − B ) represent characteristicpolynomial of B .Label M¨obius polyomino networks as shown in the Figure 1(1). Evidently, | V ( M n ) | = 2 n, | E ( M n ) | =3 n and V = { , , · · · , n } , V = { (cid:48) , (cid:48) , · · · , n (cid:48) } is an automorphism of M n .Then L ( M n ) and L ( M n ) can be expressed by L ( M n ) = (cid:18) L V V L V V L V V L V V (cid:19) , L ( M n ) = (cid:18) L V V L V V L V V L V V (cid:19) , L V V = L V V , L V V = L V V , L V V = L V V , L V V = L V V . Let T = (cid:32) √ I n √ I n √ I n − √ I n (cid:33) , then T L ( M n ) T = (cid:18) L A L S (cid:19) , T L ( M n ) T = (cid:18) L A L S (cid:19) , where L A = L V V + L V V , L S = L V V − L V V , L A = L V V + L V V , L S = L V V − L V V . In what follows, the theorems that we present will be used throughout the Section 3 and Section 4. Theorem 2.1. [24] If L A , L S , L A , L S are defined as above, the following formula can be obtained. Then P L ( M n ) ( x ) = P L A ( x ) P L S ( x ) , P L ( M n ) ( x ) = P L A ( x ) P L S ( x ) . Theorem 2.2. [25, 26] For a cycle with n vertices C n :(1) The Kirchhoff index of C n is Kf ( C n ) = n − n . (2) The Laplacian eigenvalues of C n is α i = 2 − πin , i ∈ [1 , n ] . Theorem 2.3. [27] For a connected graph G with vertices n . The number of spanning trees of G is τ ( G ) = 1 n n (cid:89) i =2 µ i , where µ i is the Laplacian eigenvalue of G . Theorem 2.4. [28] For a connected graph G with vertices n . Kemeny’s constant of G is Kc ( G ) = n (cid:88) i =2 λ i , where λ i is the normalized Laplacian eigenvalue of G . Evidently, from Theorem 2.4 and (1.4), the relation between the Kemeny’s constant and multiplicativedegree-Kirchhoff index is Kf ∗ ( G ) = 2 mKc ( G ). Theorem 2.5. [27] If graph G with | V G | = n and | E G | = m . The number of spanning trees of G is mτ ( G ) = n (cid:89) i =1 d i · n (cid:89) i =2 λ i , where λ i is the normalized Laplacian eigenvalue of G . For convenience, all p represents 2 + √ q represents 2 − √ . Laplacian spectrum of M n In this section, we mainly obtain the Laplacian spectrum of M n .According to Laplacian matrix of M n , we can get L V V and L V V : L V V = − − − − − 1. . . − − − n × n ,L V V = − − − − − − − n × n . Based on Theorem 2.1, Laplacian spectrum consists of the eigenvalues of L A and L S of M n can beobtained. L A = − − − − − − 1. . . − − − − n × n ,L S = − − − − − 1. . . − − − n × n . Assume that 0 = α < α ≤ α ≤ · · · ≤ α n are the roots of P L A ( x ) = 0, and 0 < β ≤ β ≤ β ≤· · · ≤ β n are the roots of P L S ( x ) = 0. Noticing that L A is the Laplacian matrix of cycle C n , the followinglemma can be obtained from (1.1) and Theorem 2.2(1). Lemma 3.1. For M¨obius polyomino networks M n ( n ≥ ), Kf ( M n ) = n − n n n (cid:88) i =1 β i , where β i is the eigenvalue of L S . Next, we first determine (cid:80) ni =1 1 β i . 4 emma 3.2. Let < β ≤ β ≤ β ≤ · · · ≤ β n be the eigenvalues of L S . Then n (cid:88) i =1 β i = n √ · p n − q n p n + q n + 2 . Proof. Let P L S ( x ) = det ( xI − L S ) = x n + a x n − + · · · + a n − x + a n . According to Vieta (cid:48) s theorem of P L S ( x ), one obtain n (cid:88) i =1 β i = ( − n − a n − det ( L S ) . (3.3)Since ( − n − a n − and det ( L S ), we focus on i -th order principal submatrix F i , which consists of thefirst i rows and columns of the following matrix L (cid:48) S , i ∈ [1 , n ]. Let f i = det ( F i ). L (cid:48) S = − − − − − 1. . . − − − n × n . It is easy to get that f = 4 , f = 15 , f = 56 and for 3 ≤ i ≤ n , f i = 4 f i − − f i − . For convenience,we let f = 1. Furthermore, one can verity that f i = p i +1 − q i +1 √ . Fact 1. ( − n − a n − = n √ ( p n − q n ) . Proof of Fact 1. Obviously, we obtain ( − n − a n − is the sum of all the principal minors of order n − L S . According to the property of L S , we know that det ( L S [ i ]) = (cid:40) f n − , i = 1 or n ; f i − f n − i − f i − f n − i − , i ∈ [2 , n − . Thus, we can obtain( − n − a n − = n (cid:88) i =1 det ( L S [ i ])= det ( L S [1]) + det ( L S [ n ]) + n − (cid:88) i =2 det ( L S [ i ])= n √ p n − q n ) . This result as desired. Fact 2. det ( L S ) = p n + q n + 2 . Proof of Fact 2. By expanding the last row of L S , we can arrive at det ( L S ) = f n − f n − + 2= p n + q n + 2 . Theorem 3.3. For M¨obius polyomino networks M n ( n ≥ ), Kf ( M n ) = n − n n √ · p n − q n p n + q n + 2 . Remark. We found that Theorem 3.3 can be obtained by different methods in [29] and [30].In view of the Laplacian spectrum of M n , the following theorem can be obtained. Theorem 3.4. For M¨obius polyomino networks M n ( n ≥ ), τ ( M n ) = n p n + q n + 2) . Proof. Based on Theorem 2.2(2), we know that the eigenvalues of L A are α i = 2 − πin (1 ≤ i ≤ n )and the product of α , α , · · · , α n are (cid:81) ni =2 α i = n . By Fact 2, we get n (cid:89) i =1 β i = det ( L S ) = p n + q n + 2 . Together with Fact 2 and Theorem 2.3, Theorem 3.4 follows immediately. Normalized Laplacian spectrum of M n In this section, we mainly obtain the normalized Laplacian spectrum of M n . Moreover, we find thatthe multiplicative degree-Kirchhoff index of M n is nine times of Kirchhoff index.According to normalized Laplacian matrix of M n , we can get L V V and L V V : L V V = − − − − − . . . − − − n × n , L V V = − − − − . . . − − − n × n . Based on Theorem 2.1, we can get normalized Laplacian spectrum which is obtained by the eigenvaluesof L A and L S of M n . 6 A = − − − 13 23 − − 13 23 − . . . − 13 23 − − − 13 23 n × n , L S = − 13 13 − 13 43 − − 13 43 − . . . − 13 43 − − 13 43 n × n . Assume that 0 = γ < γ ≤ γ ≤ · · · ≤ γ n are the roots of P L A ( x ) = 0, and 0 < δ ≤ δ ≤ δ ≤ · · · ≤ δ n are the roots of P L S ( x ) = 0. Next, we derive the formulas of (cid:80) ni =2 1 γ i and (cid:80) ni =1 1 δ i . Lemma 4.1. Let γ < γ ≤ γ ≤ · · · ≤ γ n be the eigenvalues of L A . Then n (cid:88) i =2 γ i = n − . Proof. Suppose that P L A ( x ) = det ( xI − L A ) = x n + b x n − + · · · + b n − x + b n − x. Applying Vieta (cid:48) s theorem, one can get n (cid:88) i =2 γ i = ( − n − b n − ( − n − b n − . (4.4)Before calculating ( − n − b n − and ( − n − b n − , we focus on i -th order principal submatrix G i ,which consists of the first i rows and columns of the following matrix L (cid:48) A , i ∈ [1 , n ]. Let g i = det ( G i ). L (cid:48) A = − − 13 23 − − 13 23 − . . . − 13 23 − − 13 23 n × n . It is easy to get that g = , g = , g = and for 3 ≤ i ≤ n , g i = g i − − g i − . For convenience,we let g = 1. Furthermore, we can get g i = i + 13 i . Fact 3. ( − n − b n − = n n − . roof of Fact 3. Evidently, we obtain ( − n − b n − is the sum of all the principal minors of order n − L A . According to the property of L A , we know that det ( L A [ i ]) = (cid:40) g n − , i = 1 or n ; g i − g n − i − g i − g n − i − , i ∈ [2 , n − . Therefore, one can get( − n − b n − = n (cid:88) i =1 det ( L A [ i ])= det ( L A [1]) + det ( L A [ n ]) + n − (cid:88) i =2 det ( L A [ i ])= n n − , which is the desired result. Fact 4. ( − n − b n − = n ( n − · n − . Proof of Fact 4. Obviously, we obtain ( − n − b n − is the sum of all the principal minors of order n − L A . In a similar way, we know that det ( L A [ i, j ]) = g n − , i = 1 and j = n ; g i − g n − i , j = 1 or n, and i ∈ [2 , n − g i − g j − i − g n − j − g i − g j − i − g n − j − , i < j and i, j ∈ [2 , n − . Thus, one can obtain( − n − b n − = (cid:88) ≤ i Lemma 4.2. Let < δ ≤ δ ≤ δ ≤ · · · ≤ δ n be the eigenvalues of L S . Then n (cid:88) i =1 δ i = √ n · p n − q n p n + q n + 2 . Proof. Let P L S ( x ) = det ( xI − L S ) = x n + k x n − + · · · + k n − x + k n . Applying Vieta (cid:48) s theorem, one can get n (cid:88) i =1 δ i = ( − n − k n − det ( L S ) . (4.5)In order to obtain ( − n − k n − and det ( L S ), we focus on i -th order principal submatrix H i , whichconsists of the first i rows and columns of the following matrix L (cid:48) S , i ∈ [1 , n ]. Let h i = det ( H i ).8 (cid:48) S = − − 13 43 − − 13 43 − . . . − 13 43 − − 13 43 n × n . It is easy to get that h = , h = , h = and for 3 ≤ i ≤ n , h i = h i − − h i − . For convenience,we let h = 1. Furthermore, one can verity that h i = p i +1 − q i +1 √ · i . Fact 5. ( − n − k n − = n √ · p n − q n n − . Proof of Fact 5. Similarly, we obtain ( − n − k n − is the sum of all the principal minors of order n − L S . Thus, we know that det ( L S [ i ]) = (cid:40) h n − , i = 1 or n ; h i − h n − i − h i − h n − i − , i ∈ [2 , n − . Therefore, one can obtain( − n − k n − = n (cid:88) i =1 det ( L S [ i ])= det ( L S [1]) + det ( L S [ n ]) + n − (cid:88) i =2 det ( L S [ i ])= n √ · p n − q n n − . This completes the proof. Fact 6. det ( L S ) = p n + q n +23 n . Proof of Fact 6. By expanding the last row of L S , we can arrive at det ( L S ) = h n − h n − + 23 n = p n + q n + 23 n . Thus Fact 6 holds.Substituting Facts 5 and 6 into (4.5) yields lemma 4.2.Based on Theorem 2.4, lemmas 4.1 and 4.2, the following theorem can be obtained. Theorem 4.3. For M¨obius polyomino networks M n ( n ≥ ), Kc ( M n ) = n − 14 + √ n · p n − q n p n + q n + 2 . In view of (1.2), lemmas 4.1 and 4.2, one get the multiplicative degree-Kirchhoff index. Theorem 4.4. For M¨obius polyomino networks M n ( n ≥ ), Kf ∗ ( M n ) = 32 ( n − n ) + 3 √ n ( p n − q n ) p n + q n + 2 . M n is nine times of Kirchhoff index. Since the degree of each point in the M¨obius polyomino networksis three, Kf ∗ ( M n ) = d i d j Kf ( M n ) = 9 Kf ( M n ) can be obtained from the definition of Kirchhoff indexand multiplicative degree-Kirchhoff index index. Therefore, it is not difficult to obtain and prove thecorrectness of results.In view of the normalized Laplacian spectrum of M n , the following theorem can be obtained. Theorem 4.5. For M¨obius polyomino networks M n ( n ≥ ), τ ( M n ) = n p n + q n + 2) . Proof. Based on Theorem 2.5, one can get (cid:81) ni =1 d i (cid:81) ni =2 1 γ i (cid:81) ni =1 1 δ i = 2 · n · τ ( M n ), where n (cid:89) i =1 d i = 3 n , n (cid:89) i =2 γ i = ( − n − b n − = n n − , n (cid:89) i =1 δ i = det ( L S ) = p n + q n + 23 n . Hence, τ ( M n ) = n p n + q n + 2) . The result as desired. Conclusion and discussion In this paper, based on the Laplacian matrix and normalized Laplacian matrix of M¨obius polyominonetworks, the Kirchhoff index, multiplicative degree-Kirchhoff index, Kemeny’s constant and spanningtrees of M¨obius polyomino networks are determined through the decomposition theorem and Vieta (cid:48) stheorem.Spectral theory has important applications in many fields. Wu [6] and Ding [31] obtained the meanfirst-passage time of Koch networks and 3-prism graph according to the Laplacian spectrum, respectively.Thus, we can explore the mean first-passage time of M¨obius polyomino networks. Funding This work was supported in part by National Natural Science Foundation of China Grant 11601006. References [1] E. Heilbronner, Huckel molecular orbitals of M¨obius-type conformations of annulenes, Tetrahedron Letters5(29) (1964) 1923-1928.[2] X. Ma, H. Bian, The normalized Laplacians, degree-Kirchhoff index and the spanning trees of hexagonalM¨obius graphs, Applied Mathematics and Computation 355 (2019) 33-46. 3] X. Geng, P. Wang, L. Lei, S. Wang, On the Kirchhoff indices and the number of spanning treesof M¨obius phenylenes chain and Cylinder phenylenes chain, Polycyclic Aromatic Compounds (2019)https://doi.org/10.1080/10406638.2019.1693405.[4] L. Lei, X. Geng, S.C. Li, Y.J. Peng, Y. Yu, On the normalized Laplacian of M¨obius phenylene chain and itsapplications, International Journal of Quantum Chemistry 119 (24) (2019) e26044.[5] S. Li, W. Yan, T. Tian, The spectrum and Laplacian spectrum of the dice lattice, Journal of StatisticalPhysics 164 (2016) 449-462.[6] B. Wu, Z.Z. Zhang, G.R. Chen, Properties and applications of Laplacian spectra for Koch networks, Journalof Physics A: Mathematical and Theoretical 45(2) (2012) 025102.[7] Y.J. Yang, D.J. Klein, Two-point resistances and random walks on stellated regular graphs, Journal ofPhysics A: Mathematical and Theoretical 52(7) (2019) 075201.[8] Y.J. Yang, H.P. Zhang, Kirchhoff index of linear hexagonal chains, International Journal of Quantum Chem-istry 108(3) (2008) 503-512.[9] J. Huang, S.C. Li, X. Li, The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linearpolyomino chains, Applied Mathematics and Computation 289 (2016) 324-334.[10] Z.X. Zhu, J.B. Liu, The normalized Laplacian, degree-Kirchhoff index and the spanning tree numbers ofgeneralized phenylenes, Discrete Applied Mathematics 254 (2019) 256-267.[11] Y. Pan, J. Li, Kirchhoff index, multiplicative degree-Kirchhoff index and spanning trees of the linear crossedhexagonal chains, International Journal of Quantum Chemistry 118 (24) (2018) e25787.[12] Y. Pan, C. Liu, J. Li, Kirchhoff indices and numbers of spanning trees of moleculargraphs derived from linear crossed polyomino chain, Polycyclic Aromatic Compounds (2020)https://doi.org/10.1080/10406638.2020.1725898.[13] J.A. Bondy, U.S.R. Murty, Graph theory, Springer, New York 2008.[14] H. Wiener, Structural determination of paraffin boiling points, Journal of the American Chemical Society69 (1947) 17-20.[15] A.A. Dobrymin, R. Entriger, I. Gutman, Wiener index of trees: theory and applications, Acta ApplicandaeMathematicae 66 (2001) 211-249.[16] A.A. Dobrymin, I. Gutman, S. Klavˇzar, P. ˇZigert, Wiener index of hexagonal systems, Acta ApplicandaeMathematicae 72 (2002) 247-294.[17] M. Knor, R. Skrekovski, A. Tepeh, Orientations of Graphs with maximum Wiener index, Discrete AppliedMathematics 211 (2016) 121-129.[18] S.C. Li, Y.B. Song, On the sum of all distances in bipartite graphs, Discrete Applied Mathematics 169 (2014)176-185.[19] I. Gutman, Selected properties of the schultz molecular topological index, Journal of Chemical Informationand Computer Sciences 34 (1994) 1087-1089.[20] D.J. Klein, M. Randi´c, Resistance distances, Journal of Mathematical Chemistry 12 (1993) 81-95.[21] H.Y. Chen, F.J. Zhang, Resistance distance and the normalized Laplacian spectrum, Discrete Applied Math-ematics 155 (2007) 654-661.[22] I. Gutman, B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, Journal of Chemical Informationand Computer Sciences 36 (1996) 982-985.[23] H.Y. Zhu, D.J. Klein, I. Lukovits, Extensions of the Wiener number, Journal of Chemical Information andComputer Sciences 36 (1996) 420-428.[24] Y.L. Yang, T.Y. Yu, Graph theory of viscoelasticities for polymers with starshaped, multiple-ring and cyclicmultiple-ring molecules, Macromolecular Chemistry and Physics 186 (1985) 609-631.[25] Y.J. Yang, X.Y. Jiang, Unicyclic graphs with extremal Kirchhoff index, MATCH Communications in Math-ematical and in Computer Chemistry 60 (2008) 107-120.[26] R.B. Bapat, Graphs and matrices, Springer, New York 2010.[27] F.R.K. Chung, Spectral graph theory, American Mathematical Society Providence, RI, 1997.[28] S. Butler, Algebraic aspects of the normalized Laplacian, in: A. Beveridge, J. Griggs, L. Hogben, G. Musiker,P. Tetali (eds.), Recent Trends in Combinatorics, The IMA Volumes in Mathematics and its Applications,IMA, 2016.[29] Z. Cinkir, Effective resistances and Kirchhoff index of ladder graphs, Journal of Mathematical Chemistry 54(2016) 955-966. 30] G.A. Baigonakova, A.D. Mednykh, Elementary formulas for Kirchhoff index of M¨obius ladder and prismgraphs, Siberian Electronic Mathematical Reports 16 (2019) 1654-1661.[31] Q. Ding, W. Sun, F. Chen, Applications of Laplacian spectra on a 3-prism graph, Modern Physics LettersB 28 (2) (2014) 1450009.30] G.A. Baigonakova, A.D. Mednykh, Elementary formulas for Kirchhoff index of M¨obius ladder and prismgraphs, Siberian Electronic Mathematical Reports 16 (2019) 1654-1661.[31] Q. Ding, W. Sun, F. Chen, Applications of Laplacian spectra on a 3-prism graph, Modern Physics LettersB 28 (2) (2014) 1450009.