The sum of the absolutes of these eigenvalues is the energy of graph

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ENERGY AND LAPLACIAN SPECTRUM OF C4C8(S) NANOTORI AND NANOTUBE

MAJID AREZOOMAND

Islamic Azad University, Majlesi Branch, Isfahan, Iran

The spectrum of a finite graph is by definition the spectrum of the adjacency matrix, that is, its set of eigenvalues together with their multiplicities. The sum of the absolutes of these eigenvalues is the energy of graph. The Laplace spectrum of a finite undirected graph without loops is the spectrum of the Laplace matrix. There are some topological indices related the Laplacian spectrum. In this paper, using a mathematical model for C4C8(S) that introduced in Ref.[26], we write a MATHEMATICA program to compute the energy and Laplacian spectrum of molecular graph of arbitrary C4C8(S) nanotori and nanotube.

(Received November 10, 2009; accepted November 20, 2009)

Keywords: Molecular graph, Laplacian Spectrum, Topological index, Energy.

1. Introduction

A topological index is a real number related to a structural graph of a molecule. It does not depend on the labeling or pictorial representation of a graph. In recent years there has been considerable interest in the general problem of determining topological indices of nanotubes, nanotori and fullerenes. It has been established, for example, that the Wiener and hyper-Wiener indices of polyhex nanotubes and tori are computable from the molecular graph of these structures.

Accordingly, some of the interest has been focused on computing topological indices of these nanostructures^1,2.

Let G be a undirected graph without directed and multiple edges, and without loops, the vertex and edge-sets of which are presented by V(G) and E(G), respectively. The adjacency matrix of G is the 0-1 matrix A=A(G)=[aij] indexed by the vertex set V(G) of G, where aij=1 when ij is an edge and i≠ j, and 0 otherwise. This matrix characterizes a graph up to isomorphism. It allows the reconstruction of a graph and is a symmetric matrix. The Laplacian matrix of G is the matrix L=L(G)=[l_ij] indexed by the vertex set of G, with zero row sums, where L_ij=-a_ij for i ≠ j. If D=D(G)=[dij] is the diagonal matrix, indexed by the vertex set of G such that dii is the degree of i, then L=D-A.

The Laplacian matrix is sometimes also called the Kirchhoff matrix³ of a graph because of its role in the matrix-tree theorem⁴ implicit⁵ in the work of Kirchhoff⁶. And sometimes L(G) is called the combinatorial Laplacian, to distinguish it from the normalized Laplacian earlier noted in connection with random walks.

The Laplacian matrix is a real symmetric matrix, so that diagonalization of the Laplacian matrix of a graph (molecule) G with N vertices (atoms) gives N real eigenvalues μ_i(G), i=1,2,…,N.

The smallest eigenvalue of the Laplacian spectrum is always 0, as a consequence of the special structure of the Laplacian matrix. The sum of these eigenvalues is the trace of L, which is twice the number of edges of G. The uses of the Laplacian matrix, its characteristic polynomial, its eigenspectrum, and related invariants have been explored in chemistry for at least the last decade^7-

20. There are many problems in physics and chemistry where the Laplacian matrices of graphs and their spectra play the central role. Some of the applications are mentioned in ²¹.

_____________________

*Corresponding author: [email protected]

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A graph G is called regular of degree k, when every vertex has precisely k neighbors. If G is regular of degree k, then for every eigenvalue θ we have | θ | ≤k and L(G)=kI-A(G). It follows that if G has ordinary eigenvalues k= θ₁≥θ₂≥…≥ θ_N and Laplace eigenvalues 0=μ₁≤ μ₂≤…≤ μ_N, then θ_i=k- μ_i for i=1,2,…, N. Moreover k is the largest eigenvalue of G, and its multiplicity equals the number of connected components of G.

The Quasi-Wiener index, W^*, defined as W^*=N

∑

= N

i 21/

λ

i , where λi, i=2,…,N denotes the positive eigenvalues of the Laplacian matrix^3,10.

The eigenvalues of the Laplacian matrix are used in calculating the number of spanning trees, t(G), in graph. We know that if G be an undirected graph with Laplacian matrix L(G) and eigenvalues 0=μ1≤ μ2≤…≤ μN, then the number of spanning trees of G, equals ³ t(G)=

N

N i

∏

i

=2

μ . Moreover Mohar defined two topological indices, TI1 and TI2 on the ground of Laplacian spectrum TI₁ =2N log(Q/N)

∑

= and TI₂=4/(N μ₂), where Q is the number of edges.

N i 21/

μ

i

The eigenvalues of adjacency matrix A(G) are called eigenvalues of graph G ⁴. Following Ivan Gutman ²², the energy, E(G), of a molecular graph G is defined to be the sum of the absolutes of the eigenvalues of G. We encourage the reader to consult papers ^22-24 and references therein for background material as well as basic computational techniques.

A C₄C₈ net is a trivalent decoration made by alternating squares C₄ and octagons C₈. It can cover either a cylinder or a torus. Such a covering can be derived from a square net by the leapfrog operation²⁵. Optimized C₄C₈ net covering a nanotube is illustrated in Figure 1. A nanotorus is a nanotube whose ends are connected.

Fig.1. TUC4C8(S).

a₁₁ a_{1 2} a_{1 3} a₁₄ a_1,2_q_-₁ a_1,2_q

a₂₁ a_{2 2} a₃₁ a_{3 2} a_{4 1} a_{4 2} a_{5 1}

a₄_p-1,1

a₄_p,1 a₄_p,2_q_-₁

a₄_p,2_q Fig.2. Two-dimensional lattice of nanotube G2=TUC4C8(S) [4,5].

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2. Results and discussions

Description of molecular graphs of nanotori and nanotubes

A carbon nanotorus may be described as a long rolled-up graphite sheet bent around to the form of torus. Let G1=TRC4C8(S)[p,q] and G2=TUC4C8(S)[p,q] be the molecular graphs of C4C8(S) nanotorus and nanotube, in which p and q are the number of octagons in vertical and horizontal directions, respectively, Figure 2. Note that the graph G1 has exactly 8pq vertices and is a 3-regular graph and so the Laplacian matrix of this graph is L(G1)=3I-A(G1) where I is the 8pq×8pq identity matrix. Arezoomand and Taeri²⁶proved that the set

is a mathematical model for vertices of C₄C₈(S) lattice and the mapping

}}

1 , 0 { },

1 , 0 {

| ) , , ,

{( ∈ ⁴ + + ∈ + + ∈

=

α β γ δ

Z

β γ δ α γ δ

l

N

f :l×l→ , f(u,v)=

∑

_i⁴=₁|u_i −v_i | is a distance function on the set of lattice vertices and give us the minimum distance between two vertices u and v. We assume that a_ij denotes (i,j)-entry of two-dimensional lattice of G₂ as shown in Figure 2. We put the origin point O at the a₄₁ and consider the vectors e₀, e₁, e₂ and e₃.

Consider the points a=a₁₁, b=a₂₁, c=a₃₁, d=a₄₁=O, e=a₁₂, f=a₂₂, g=a₃₂ and h=a₄₂. It is easy to see that every point of G₂ can be constructed by a translation of these points in two directions v=2e₀-e₂+e₃ and w=2e₁-e₂-e₃. This is the content of lemma below.

Lemma 1. Assume that a_ij, 1≤ i≤ 4p and 1≤ j≤ 2q, denotes the (i,j)-entry of the two- dimensional lattice of G₂, as shown in the Fig.2, in our model we have

odd j i

or even j i

even j i

or odd j i

odd j i

or even j i

even j i

or odd j i

a a a a a

ij ij ij ij

ij

), 4 (mod 0 ),

4 (mod 2

), 4 (mod 0 ),

4 (mod 2

), 4 (mod 3 ),

4 (mod 1

), 4 (mod 3 ),

4 (mod 1

4 3 2 1

≡

⎪⎪

⎩

⎪⎪

⎨

⎧

= (1)

where

1

aij=(j-1) e₀+(i-3)/2 e₁+(7-i-2j)/4 e₂+(3-i+2j)/4 e₃,

2

aij=(j-1) e₀+(i-3)/2 e₁+(9-i-2j)/4 e₂+(1-i+2j)/4 e₃,

3

aij=(j-1) e₀+(i-4)/2 e₁+(8-i-2j)/4 e₂+(-i+2j)/4 e₃ and

4

aij=(j-1) e₀+(i-4)/2 e₁+(6-i-2j)/4 e₂+(2-i+2j)/4 e₃. Proof. It is easy to see that the lattice points of G₂ lie in T₁U^T²U^…U^T⁸^{, where}

T1 = {a+ (i-1)/4 w+ (j-1)/2 v | i≡1 (mod 4), j odd}

T2 = {e+ (i-1)/4 w+ (j/2-1) v | i≡1 (mod 4), j even}

T3 = {b+ (i-2)/4 w+ (j-1)/2 v | i≡2 (mod 4), j odd}

T₄ = {f+ (i-2)/4 w+ (j/2-1) v | i≡2 (mod 4), j even}

T₅ = {c+ (i-3)/4 w+ (j-1)/2 v | i≡3 (mod 4), j odd}

T₆ = {g+ (i-3)/4 w+ (j/2-1) v | i≡3 (mod 4), j even}

T₇ = {d+ (i/4-1) w+ (j-1)/2 v | i≡0 (mod 4), j odd}

T₈ = {h+ (i/4-1) w+ (j/2-1) v | i≡0 (mod 4), j even}

By considering the coordinates of the points a, b, c, d, e, f, g, h and the vectors v, w we can see that the relation (1) holds.

Note that in the C₄C₈(S) net the nearest neighbors of vertex v=(v₁,v₂,v₃,v₄) are v¹ = (v₁ + ε₁(v), v₂, v₃, v₄)

v² = (v₁, v₂ + ε₂(v), v₃, v₄) (1) v³ = (v₁, v₂, v₃ + ε₃₁(v), v₄ + ε₃₂(v))

where

4 3

)1

1 ( )

1(

v v

v = − v⁺ ⁻

ε

4 3

) 2

1 ( )

2(

v v

v = − v⁺ ⁺

ε

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⎩⎨

⎧ + + = + −

= otherwise

v v v v v v if v v

0 ) ) (

( ² ² ³ ⁴ ¹ ³ ⁴

31

ε ε

⎩⎨

⎧ + + ≠ + −

= otherwise

v v v v v v if v v

0 ) ) (

( ² ² ³ ⁴ ¹ ³ ⁴

32

ε ε

By above notations, it is obvious that if we fix p and q, then V(G1)=V(G2) and E(G1)=E(G2) U{a₁_ja₄_p_,_j|1≤ j ≤2q}.

By the geometry of nanotori we have

a_i¹₁ =a_i_,₂_q(and so a¹_i_,₂_q =a_i₁) when i=2 or 3 (mod 4) a₁²_j =a₄_p_,_j(and so a₁²_j =a₄_p_,_j) for all j.

Note that we can compute the other neighbors of these vertices and all of neighbors of other vertices by relations (1).

A MATHEMATICA program for computing the adjacency and Laplacian matrices of G₁ and G₂. Now we are ready to write a MATHEMATICA program for computing the adjacency and Laplacian matrices of C4C8(S) nanotori and nanotubes. Using these two important matrices we can compute some topological indices. In the output of our program V=V(G1)=V(G2), A1 and A2 are the adjacency matrices of G1 and G2, respectively. Also L1 and L2 are the Laplacian matrices of G1

and G2, respectively and Eig1, Eig2 are the set of their eigenvalues. Finally D2 is the degree matrix of G2.

p=4; q=5;(* for example*)

a = {0, -1, 1, 1}; b = {0, -1, 1, 0}; c = {0, 0, 1, 0}; d = {0, 0, 0, 0}; e = {1, -1, 1, 1}; f = {1, -1, 0, 1}; g = {1, 0, 0, 1}; h = {1, 0, 0, 0};v={2, 0, -1, 1}; w = {0, 2, -1, -1};

V = {};

For[i=1, i≤ 4p, For[j=1, j≤ 2q,

If[Mod[i,4]==1 && OddQ[j], AppendTo[V, a+ (i-1)/4w+ (j-1)/2v]];

If[Mod[i,4]==1 && EvenQ[j], AppendTo[V, e+ (i-1)/4w+ (j/2-1)v]];

If[Mod[i,4]==2 && OddQ[j], AppendTo[V, b+ (i-2)/4w+ (j-1)/2v]];

If[Mod[i,4]==2 && EvenQ[j], AppendTo[V, f+ (i-2)/4w+ (j/2-1)v]];

If[Mod[i,4]==3 && OddQ[j], AppendTo[V, c+ (i-3)/4w+ (j-1)/2v]];

If[Mod[i,4]==3 && EvenQ[j], AppendTo[V, g+ (i-3)/4w+ (j/2-1)v]];

If[Mod[i,4]==0 && OddQ[j], AppendTo[V, d+ (i/4-1)w+ (j-1)/2v]];

If[Mod[i,4]==0 && EvenQ[j], AppendTo[V, h+ (i/4-1)w+ (j/2-1)v]];

j++];

i++];

A1 = Table[x[i, j], {i, 1, 8p*q}, {j, 1, 8p*q}];

ff[u_, v_] :=Sum[Abs[u[[i]] - v[[i]]], {i, 1, 4}];

For[i = 1, i ≤ 8p*q, For[j = 1, j ≤ 8p*q,

If[ff[V[[i]], V[[j]]] = = 1, x[i, j] = 1, x[i, j] = 0];

j++];

i++];

For[i = 1, i ≤ 2q, x[i, (8p-2)q + i] = 1;

x[(8p-2)q+ i, i] = 1;

i++];

For[i = 0, i ≤p-1 ,

x[(8i+2)q+1,(8i+4)q] = 1;

x[(8i+4)q, (8i+2)q+1] = 1;

x[(8i+4)q+1,(8i+6)q]=1;

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x[(8i+6)q, (8i+4)q+1]=1;

i++];

A2 = Table[z[i, j], {i, 1, 8p*q}, {j, 1, 8p*q}];

For[i= 1, i≤ 8p*q, For[j = 1, j ≤ 8p*q,

If[ff[V[[i]], V[[j]]] = = 1, z[i, j] = 1, z[i, j] = 0];

j++];

i++];

For[i = 0, i ≤p-1 ,

z[(8i+2)*q+1,(8i+4)*q] = 1;

z[(8i+4)*q, (8i+2)*q+1] = 1;

z[(8i+4)*q+1,(8i+6)*q]=1;

z[(8i+6)*q, (8i+4)*q+1]=1;

i++];

D2 = Table[dd[i, j], {i, 1, 8p*q}, {j, 1, 8p*q}];

B=A2.A2;

For[i = 1, i≤ 8p*q, For[j = 1, j≤ 8p*q,

If[i= =j, dd[i,j]=B[[i]][[j]], dd[i,j]=0]

j++]

i++]

L1=3*IdentityMatrix[8p*q] - A1;

L2=D2-A2;

Eig1=Eigenvalues[L1]//N Eig2=Eigenvalues[L2]//N

Topological indices and energy of nanotori and nanotube

It is easy to see that |E(G1)|= 12pq and so |E(G2)|=12pq-2q. Note that MATHEMATICA arranges the Laplacian eigenvalues decreasing. So if we append these below lines to the program, then we have numerical values of the topological indices W^*, TI1, TI2, the number of spanning trees t(G), μ2

(as defined in the section 1) and energy of nanotori and nanotube : (*for nanotori*)

Eig1[[8p*q-1]]

QuasiWiener1 = 8p*q*Sum[1/Eig1[[i]], {i, 1, 8p*q - 1}]

tG1=Product[Eig1[[i]], {i, 1, 8p*q - 1}]/(8p*q)

TI1G1 =16p*q*Log[4/3]*Sum[1/Eig1[[i]], {i, 1, 8p*q - 1}]

TI2G1 =4/(8p*q*Eig1[[8p*q - 1]]) E1=Eigenvalues[A1];

EnG1=Sum[Abs[E1[[i]]],{i,1,8p*q}]//N (*for nanotube*)

Eig2[[8p*q-1]]

QuasiWiener2=8p*q*Sum[1/Eig2[[i]], {i, 1, 8p*q - 1}]

tG2=Product[Eig2[[i]], {i, 1, 8p*q - 1}]/(8p*q)

TI1G2=16p*q*Log[4/3-1/(4p)]*Sum[1/Eig2[[i]], {i, 1, 8p*q - 1}]

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Table1. Numerical data for nanotorus G1=TRC4C8(S)[p.q].

(p,q) μ2 W^* t(G) TI1 TI2 Energy

(2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (3,2) (3,3) (3,4) (3,5) (3,6) (5,2) (5,3) (5,4) (5,5)

0.585786 0.267949 0.152241 0.097887 0.0681483 0.0501442 0.0384294

0.267949 0.267949 0.152241 0.097887 0.0681483 0.097887 0.097887 0.097887 0.097887

550.476 1431.07 2887.15 5046.16 8036.07 11984.9 17020.6 1431.07 3472.83 6681.15 11240.5 17340.5 5046.16 11240.5 20573.1 33369.5

6.04231×10¹⁰ 1.58582×10¹⁶ 3.68402×10²¹ 8.02232×10²⁶ 1.67705×10³² 3.40847×10³⁷ 6.78606×10⁴² 1.58582×10¹⁶ 2.78463×10²⁴ 4.17439×10³² 5.84076×10⁴⁰ 7.84154×10⁴⁸ 8.02232×10²⁶ 5.84076×10⁴⁰ 3.12427×10⁵⁴ 1.49478×10⁶⁸

316.724 823.386 1661.16 2903.38 4623.67 6895.67 9793.03 823.386 1998.14 3844.09 6467.38

9977.1 2903.38 6467.38 11837 19199.6

0.213388 0.311004 0.410533 0.510793 0.611411 0.712232 0.813179 0.311004 0.207336 0.273689 0.340529 0.407608 0.510793 0.340529 0.255397 0.204317

47.5717 70.4683 94.9648 118.667 141.951 166.048 189.752 70.4683 105.091 141.202 176.487 211.343 118.667 176.487 235.557 294.493

Table2. Numerical data for nanotube G2=TUC4C8(S)[p.q].

(p,q) μ2 W^* t(G) TI1 TI2 Energy

(2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (3,2) (3,3) (3,4) (3,5) (3,6) (5,2) (5,3) (5,4) (5,5) (5,6)

0.152241 0.152241 0.152241 0.097887 0.0681483 0.0501442 0.0384294 0.0681483 0.0681483 0.0681483 0.0681483 0.0681483 0.0246233 0.0246233 0.0246233 0.0246233 0.0246233

768.834 1873.15 3656.07 6242.33 9757.05 14326.9 20079.4 2079.59 4639.42 8587.9 14131.3 21459.5 7833.78 15752.5 27238.3 42748.7 62673.6

6.25766×10⁸ 1.8522×10¹³ 4.61365×10¹⁷ 1.06635×10²² 2.36124×10²⁶ 5.08126×10³⁰ 1.07106×10³⁵ 1.09013×10¹⁴ 2.07274×10²¹ 3.05379×10²³ 4.05566×10³⁵ 5.10528×10⁴² 3.30835×10²⁴ 2.59415×10³⁷ 1.32995×10⁵⁰ 5.74189×10⁶² 2.27372×10⁷⁵

290.991 708.975 1383.76 2362.62 3692.89 5422.5 7599.75 928.094 2070.51 3832.67 6306.62 9577.08 3908.44 7859.26 13589.8 21328.3 31269.2

0.821067 0.547378 0.410533 0.510793 0.611411 0.712232 0.813179 1.22282 0.815215 0.611411 0.489129 0.407608 2.0306 1.35373 1.0153 0.812238 0.676865

45.5573 68.0914 90.3743 113.593 136.419 159.025 181.503 69.5174 103.625 138.068 172.728 206.981 116.802 174.045 231.991 290.519 348.268 TI2G2=4/(8p*q*Eig2[[8p*q - 1]])

E2=Eigenvalues[A2];

EnG2=Sum[Abs[E2[[i]]],{i,1,8p*q}]//N

We give a numerical data for these indices, number of spanning trees and energies of the graphs G1 and G2 in Tables 1, 2. After running the above programs, we can guess some conjectures as follows:

Conjecture1. After running our program in many cases for p and q, we guess that the characteristic polynomial of Laplacian matrix of G1 is of the form

where 0=μ₁<μ₂<…<μ_k=6 are distinct eigenvalues of Laplacian matrix with multiplicity α_i, 1≤ i≤ k, respectively,

∑

^=|V(G¹^{)|, α}¹^=α^k=1 and for 1≤ j ≤ [k/2], α_k-j+1=α_j.

k

x k

x x

x

f( )=( −

μ

₁)^α¹( −

μ

₂)^α²...( −

μ

)^α

= k i1

α

i

Conjecture2. If we fix p and q, then we have E(G1)>E(G2), where E(G) is the energy of graph G.

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3. Conclusions

In this work, we give a MATHEMATICA program for calculating two important matrices adjacency and Laplacian of arbitrary C4C8(S) nanotorus and nanotubes. Many structural properties of these nanostructures are depending to these matrices. The molecular graph of these structures is given as an algebraic definition. Similarly one can write simple MATHEMATICA program and compute many topological indices related with these matrices, such as Randic index, Zagreb group indices, etc. Moreover according Huckel theory we can equate the eigenvectors of the adjacency matrix for atomic orbitals in the construction of molecular orbitals. Also we can use the eigenvectors for analyze some physico-chemical properties and interesting intra-and inter- molecular ordering (for more details see Refs. [27, 28]) .

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