An Upper Bound for Energy of Matrices Associated to an Infinite Class of Fullerenes

MATCH MATCH Commun. Math. Comput. Chem. 71 (2014) 341-354 Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253

M. Ghorbani1, M. Faghani2, A. R. Ashrafi2,, S. HeidariRad1, 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, HR21000 Split, Croatia 4The R. Bošković Institute, HR10002 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 WV.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.

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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   .  ccnn21    cnn1 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 eigenvectorsUU1 ,, l  satisfying

k k k Bk U j  j U 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

. . . .

8 12

1 6 2 10-1 . 10n-4 . 5 3 . . 4

11 9

. . . .

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:

9 11 22 20 8 12

1 2 7 3 6 13 5 4

18 14 23 19 16 17 15

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). So, the energy of this class of fullerene can be obtained by the following: .

3. The Centrosymmetric Graphs In this section we begin with some basic notation which frequently used in the sequel.A 9 matrix A = [Ai,j] is called symmetric if aij  a ji1 , and it iscentrosymmetric when its entries satisfy aij  ani1,n j1 for 1 ≤ i,j ≤ n ,bisymmetric if A is symmetric centrosymmetric. If J denotes the n × n matrix with 1 on the counterdiagonal and 0 elsewhere (that is, Ji,n+1-i = 1; Ji,j = 0 if j ≠ n+1-i), then a matrix A is centrosymmetric if and only if AJ = JA. The matrix J is sometimes referred to as the exchange matrix. Our main goal is to investigate some properties of symmetric centrosymmetric matrices from the standpoint of computation using the partition of matrix. For this purpose we review the nether well-known result which can be found in [10, 11, 1217].

Lemma 2. (i) forn= 2m, a centrosymmetric matrix A can be written as the form

B J mCJm  A    ,where Band C are mm matrices and Jis the exchange matrix that is C J m BJ m  -349-

0 j  m  i 1 J ij   . 1 j  m  i 1 (ii) forn= 2m+1, a centrosymmetric matrix A can be partitioned into the

 B J m B J mCJ m  A  aT  aT J  ,  m   C b J m BJ m  whereB and C are mmreal matrices and a,b  R m1 and is a scalar.

Lemma 3. Let A be a centrosymmetric matrix, then A is orthogonal similar to block diagonal matrix. The block diagonal matrix can be described as follows:

2  I m I m  T B  J mC 0  (i) For n=2m,let P    , then P AP    . 2  J m J m   0 B  J mC (ii) For n=2m+1, we have:  I 0 I  B  J C 0 0  2 m m m P   0 2 0  and so PT AP   0  2aT  . 2      J m 0 J m   0 2J mb B  J mC

B J mCJm  Corollary 4. Let n=2m and A    Suppose that B  J mC has eigenvalues C J m BJ m 

1 ,2 ,...,m and B  J mC has eigenvalues 1 ,  2 ,..., m . Then the eigenvalues of Aare:

1 ,2 ,...,m , 1 , 2 ,...,m .

Suppose that G be a weighted graph with weighting function w:() E G R  and adjacency matrix A(G). We associate an n × n matrix A(w) to weighting function w, by replacing w(e) with 1 in A(G), i.e. w(e) e  ij  E(G) a(w)ij   .  0 ij  E(G) Let G be an nvertex graph and e=uv be an edge of G. The number of vertices closer to uthan to v is denoted by nu (e) and the number of vertices closer to vthan to u is denoted by

nv (e) . Now if w(e)  1, the associated matrix is the adjacency matrix A(G), if

w(e)  nu (e)  nv (e) the associated matrix is the vertex PI matrix of G, PI v (G) and if

w(e)  nu (e)  nv (e) the associated matrix is the Szeged matrix of G, Sz(G) and so on.

Suppose G=C10n be an infinite class of Fullerene graphs, see Figure 3. One can check that the matrices A(G), PIv (G) and and so on are symmetric centrosymmetric (bisymmetric) -350-

B J mCJm  and so by Lemma 3 these matrices have the form   . Note that the C J m BJ m  centrosymmetricity of above matrices depend on the labeling of vertices of Gbut energy of G is independent. Definition 5. A graph G is called centrosymmetric if there exist a labeling for G such that the adjacency matrix of G with respect to this labeling is a centrosymmetric matrix.

Theorem 6. The PI matrix of C10n,with respect to a special labeling, has the following form:

X Y 0 0 0 0 0  Y 0 P 0 0 0 0   1 

 0 P1 0 Q 0 0 0     0 0 Q 0 P1 0 0   0 0 P 0 Q 0 0   1   0 0 Q 0 P2 0 0   0 0 P 0 Q 0 0   2   0 0 Q 0 P2 0 0   0 0 P 0 Q 0 0   2   0 0 Q 0 P 0 0  PI v (C10n )     0 0 P 0 Q 0 0   0 0 Q 0 P 0 0   2   0 0 P2 0 Q 0 0   0 0 Q 0 P 0 0   2   0 0 P2 0 Q 0 0   0 0 Q 0 P 0 0   1   0 0 P1 0 Q 0 0    0 0 0 Q 0 P 0  1   0 0 0 0 P 0 Y   1   0 0 0 0 0 Y X  where, 0 24 0 0 24 10n  12 0 0 0 0  24 0 24 0 0 0 10n  12 0 0 0 X  0 24 0 24 0 , Y  0 0 10n  12 0 0 ,  0 0 24 0 24 0 0 0 10n  12 0 24 0 0 24 0 0 0 0 0 10n  12 -351-

10nn 2 10 2 0 0 0 10nn 10 0 0 0  0 10nn 2 10 2 0 0 0 10nn 10 0 0 P1  0 0 10nn 2 10 2 0 , P  0 0 10nn 10 0  0 0 0 10nn 2 10 2 0 0 0 10nn 10 10nn 2 0 0 0 10 2 10nn 0 0 0 10 10nn 1 10 1 0 0 0 10n 0 0 0 0  0 10nn 1 10 1 0 0 0 10n 0 0 0 P2  0 0 10nn 1 10 1 0 and Q  0 0 10n 0 0 .  0 0 0 10nn 1 10 1 0 0 0 10n 0 10nn 1 0 0 0 10 1 0 0 0 0 10n

Proof. Consider a special labeling of vertices of C10n in which the central pentagon has labels 1 to 5. For the labeling of the other vertices, see Figure 3.

22 16 17 6

27 12 11 21 30 1 23 10 7 5

4 20 18 13 3 15

8 9 14 24 25 19 28 29

Figure 3. The schlegel diagram of C30.

38 37

27 32

33 22

16 17 6

28 12 11 21 26 1 23 10 7 5

4 20 18 13 3 15 34 31 36 39 8 9 14 24 25 19 29 30

40 Figure 4. The schlegel diagram of C40. -352-

We can draw all members of this class of fullerenes HyperChemSoftware.18 Now apply MATLAB to compute the adjacency and distance matrices of these graphs.

Theorem 7. Suppose that G=C10n be a weighted graph with weighting function w:() E G R  and adjacency matrix A(G). Then the n × n matrix A(w) associate to weighting function w, is symmetric centrosymmetrc matrix.

Proof: According to definition 5, it is clear that the adjacency matrix of any graph is symmetric, for centrosymmetricity, the adjacency matrix of G=C10n, A(G),with respect to labeling A has the following form:

XQ0 0 0 0 0  QP0 0 0 0 0 0PQ 0 0 0 0  0 0QP 0 0 0 0 0PQ 0 0 0  0 0QP 0 0 0 0 0PQ 0 0 0  0 0QP 0 0 0  0 0PQ 0 0 0 0 0QP 0 0 0 AC()10n   0 0PQ 0 0 0 0 0QP 0 0 0   0 0PQ 0 0 0   0 0QP 0 0 0     0 0PQ 0 0 0     0 0QP 0 0 0   0 0PQ 0 0 0     0 0 0QP 0 0   0 0 0 0PQ 0     0 0 0 0 0 QX where

0 1 0 0 1 1 1 0 0 0 1 0 0 0 0  1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 X  0 1 0 1 0 , P  0 0 1 1 0 and Q  0 0 1 0 0 .  0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1

So, by using Theorem 6 the graph C10n is centrosymmetric. -353-

B J mCJm  Theorem 8. Let A be a 2m × 2msemmetriccentrosymmetric matrix same as A    C J m BJ m 

,P=B+JmChas eigenvalues 1 ,2 ,...,m andQ=B-JmC has eigenvalues 1 ,  2 ,..., m . Then m m i  i E()()() A E P  E Q  m   in which   i1 and   i1 . PQ P m Q m m (i  i ) Proof: Let  i1 , then we have: A 2m

m EA() i   A   i   A  i1 m m   m m           mmi  i    i  i    i1 i  1      i  1 i  1  ii22mm ii11

mm1 1 1 1    i2 P 2 Q i 2 P 2 Q ii11 mm11       i P22 P Q i Q Q P  ii11

E()(). P  E Q  m PQ  

Corollary 9. Theenergy of matrices associated to Fullerene graph G=C10n have an upper bound as E()() P E Q  m   PQ.

Apply some MATLAB programs to compute the energy of some graphs in C10n class. Our data is recorded in Table 1, for matrices P and Q. These codes are accessible from authors upon request.

Table 1. The Energy of Fullerene C10n, for n = 2, …,17. n Energy of Energy(P)+Energy(Q)+m 2 39. 4107 36. 0295 3 45.7036 47.7036 4 61.6085 65.4085 5 77.4113 79.4113 6 93.1815 97.0481 7 108.9406 110.9406 8 124.6957 128.5957 9 140.4494 142.4494 10 156.2026 160.1226 11 171.9556 173.9556 12 187.7085 191.6419 13 203.4614 205.4614 -354-

14 219.2143 223.1572 15 234.9672 236.9672 16 250.7201 254.6701 17 266.4730 268.4730

Acknowledgment: The authors are indebted to professorMircea V. Diudea for some critical discussion on the problem and for editing our draft. The research of the second and third authors is partially supported by the University of Kashan under grant no 159020/34.

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