MATCH MATCH Commun. Math. Comput. Chem. 71 (2014) 341-354 Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253 An Upper Bound for Energy of Matrices Associated to an Infinite Class of Fullerenes M. Ghorbani1, M. Faghani2, A. R. Ashrafi2,, S. HeidariRad1, A. Graovac3,4 1Department of Mathematics, Faculty of Science, ShahidRajaee Teacher Training University, Tehran, 16785-136, I. R. Iran 2Department of Mathematics, Payame Noor University, PO Box 19395-3697, Tehran, Iran 3Faculty of Science, University of Split, Nikole Tesle 12, HR21000 Split, Croatia 4The R. Bošković Institute, HR10002 Zagreb, POB 180, Croatia (Received December 10, 2012) Abstract The energy of a molecular graph Gis a popular parameter that is defined as the sum of the absolute values of the eigenvalues of G. This is a graph invariant stemming from the Hückel molecular orbital approximation for the total -electron energy. In this paper we consider an infinite class of graphs named fullerenes and matrices associated to this class such as adjacency matrix, PI matrix etc are investigated. The centrosymmetricity of these matrices are proved and an upper bound for the energy of these matrices is given. Our method is general and can be extended to other class of cubic graphs. 1. Introduction All graphs considered in this paper are simple and connected. The vertex and edge sets of a graph G are denoted by V(G) and E(G), respectively. Let G = (V, E) be a simple graph and WV.Then the induced subgraph by W is the subgraph of G obtained by taking the vertices in Corresponding author (Email: [email protected]). -342- W and joining those pairs of vertices in W which are joined in G. The notation G − {v1,...,vk} means a graph obtained by removing the vertices v1,…,vk from G and all edges incident to any of them. The adjacency matrix A(G)of graph G with vertex set V (G)= {v1, v2, ..., vn} is the n×n symmetric matrix [aij], such that aij= 1 if viand vjare adjacent and 0, otherwise. The characteristic polynomial (,)Gx of graph G was defined as (G , x ) det( A ( G ) xI ) .The roots of the characteristic polynomial are named the eigenvalues of graph G and form the spectrum of this graph. If α be an eigenvalue of matrix A, then there exist a vector such as V, in which A.V = αV. Let λ1, . ,λn be the eigenvalues of A(G), then the energy of G, denoted by E(G), is defined1,2,3as n EG( ) |i |. i 1 In theoretical chemistry, the energy is a graph parameter stemming from the Hückel molecular orbital approximation for the total -electron energy. So the graph energy has some specific chemical interests and has been extensively studied.4 The fullerene era was started in 1985 with the discovery of a stable C60 cluster and its interpretation as a cage structure with the familiar shape of a soccer ball, by Kroto and his co- 5 authors. The well-known fullerene, the C60 molecule, is a closed-cage carbon molecule with three-coordinate carbon atoms tiling the spherical or nearly spherical surface with a truncated icosahedral structure formed by 20 hexagonal and 12 pentagonal rings.6 Let p, h, n and m be the number of pentagons, hexagons, carbon atoms and bonds between them, in a given fullerene F. Since each atom lies in exactly 3 faces and each edge lies in 2 faces, the number of atoms is n = (5p+6h)/3, the number of edges is m = (5p+6h)/2 = 3/2n and the number of faces is f = p + h. By the Euler’s formula n − m + f = 2, one can deduce that (5p+6h)/3 – (5p+6h)/2 + p + h = 2, and therefore p = 12, v = 2h + 20 and e = 3h + 30. This implies that such molecules made up entirely of n carbon atoms and having 12 pentagonal and (n/2 10) hexagonal faces, where n 22 is a natural number equal or greater than 20. The goal of this paper is to compute some new results on fullerene graphs. We encourage the interested readers to consult7 for more information on this topic. -343- 2. Main Results A circulant matrix is a matrix where each row vector is rotated one element to the right relative to the preceding row vector. In other words, a circulant matrix1is specified by one vector c which appears as the first column of C. The remaining columns of C are each cyclic permutations of the vector c with offset equal to the column index. The last row of C is the vector c in reverse order, and the remaining rows are each cyclic permutations of the last row. In generally, an n×ncirculant matrix C takes the following form: c0 cn 1 c 2 c 1 c c c c 1 0n 1 2 C c10 c . ccnn21 cnn1 c 2 c 1 c 0 The eigenvectors of a circulant matrix are given by 21nT vj(1, j , j , , j ) , j 0,1, , n 1, 2k i n 2 where, j e are the n-throots of unity and i = 1. The corresponding eigenvalues are then given by 21n jc0 c n 1 j c n 2 j c 1 j , j 0 n 1. Let A and B be matrices of dimensions nm and n' m' , respectively. Then their tensor product is a nn' mm' matrix with block forms A B aij B . Theorem 1[8]. Let Aij, 1 ≤ i,j ≤ l be square matrices of order n that have the complete set of k k eigenvectors {V1, …, Vn} with Aij V k ij . Let also, Bk [ ij ], 1 ≤ k ≤ n be square kk matrices of order l, each with a complete set of eigenvectorsUU1 ,, l satisfying k k k Bk U j jU j for 1 ≤ j ≤ l. Then a complete set of eigenvectors {W1, W2, …,Wnl} for the square matrix AAA11 12 1l AAA A 21 22 2l AAAl12 l ll -344- k is given by W(k 1) l j U j V k for k = 1, 2, …, n and j = 1, 2, …, l. The corresponding k eigenvalues are (k-1)l+j j .We will apply this theorem to the case where all blocks in the adjacency matrix are ciculant matrices. An l – level circulant is one whose adjacency matrix has an l × l block form A, all Aij being circulant. For example, a 2 – level circulant, 1 2 12 G Cn({ n i },{ n i },{ m i }), would consist of two vertex sets S1 = {v1, …, vn} and S2 = {w1, …, wn} such that 1 2 (a) G induces circulantsCnni({ }) and Cnni({ }) on S1 and S2, respectively. (b) Edges between the two circulants are of the form viwk, where 12 k j mi (mod n ) , for some i. In this section by using Theorem 1, we compute the energy of some fullerenes.Consider an infinite class of fullerene with 10n vertices, as depicted in Figure 1. 10n 10n-5 . 7 8 12 1 6 2 10-1 . 10n-4 . 5 3 . 4 11 9 10 . 10n-3 10n-2 Figure 1. The schlegel diagram of C24n. -345- The first member of this class of fullerenes is C24, see Figure 2. In [8], Lee and his co – authors computed its eigenvalues. Here we compute the eigenvalues of this class of fullerenes in general form of its adjacency matrix by means of block matrix. According to the main result of the mentioned article, one can see that the block form of adjacency matrix of C24 is as follows: 21 10 9 11 22 20 8 12 1 2 7 3 6 13 5 4 18 14 23 19 16 17 15 24 Figure 2. The schlegel diagram of C24. ACI(6 ) 0 0 IM00 AC()24 0 0ACI (6 ) t 00MI In generally, one can see that the adjacency matrix of this class of fullerenes is as follows: ACI(6 ) 0 0 0 0 t IIM0 IIM00t MII00 IIM00t MII00 IIM0 0 0 t MII0 0 0 0 A ACI(6 ) 0 . t IIM0 0IIM 0 0 t 00MII 00III MMI0 0 0 -346- The eigenvalues of A(C6) are: -2, -1 (two times), 1 (two times) and 2, the eigenvalues of I are: 1 (six times). On the other hand, M is a circulant matrix with eigenvalues 1, 0.5 0.866i , , 0.5 0.866i and . By this notations, the 4n × 4nblocksB1, B2, …, B6 are as follows: 1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 B 1 1 0 , 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 B 1 1 0 , 2 1 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 -347- 2 1 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 0 1 0 B 2 1 0 , 3 1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 B 1 1 0 , 4 1 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 -348- 1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 B 1 1 0 , 5 1 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 Since B1, B4 and B2, B3 are conjugated matrix, thenSpec(B1) =Spec(B4) and Spec(B2) = Spec(B3).