T. Réti, I. László, D. Dimitrov, Structural Irregularity in Fullerenes, In Mathematical Chemistry Monographs, No. 16, Ante Graovac – Life and Works, Eds. I. Gutman, B. Pokrić, D. Vukičević, 2014 p. 291-302.
Structural Irregularity in Fullerenes Tamás Réti1, István László2, Darko Dimitrov3 1
Bánki Donát Faculty of Mechanical Engineering, Obuda University, Bécsi út 96/B, H-1034 Budapest, Hungary
[email protected]
2
Institute of Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary
[email protected] 3
Institut für Informatik, Freie Universität Berlin Takustrasse 9, D-14195 Berlin Germany
[email protected]
Abstract To characterize the topological structure of fullerene molecules, a novel approach is presented. The method proposed is based on the following concept. As a first step we consider the dual of the traditional fullerene graph, and as a second step, from the adjacency matrix of the corresponding dual graph, spectrum-based topological invariants designated for the characterization of fullerene structures are derived. Performing comparative tests on the sets of dual graphs of C40 and C66 fullerene isomers, it is verified that the Collatz-Sinogowitz irregularity index of dual graphs (CSdual) can be efficiently used for the structural characterization of fullerenes.
1. Introduction According to the classical definition, a fullerene is an all-carbon molecule where vertices represent the atoms of carbon, and edges between vertices realize the chemical bonds between carbon atom pairs. In structural chemistry, fullerene-like molecules are considered as polyhedra or planar polyhedral graphs having only pentagonal and hexagonal faces [1]. Several topological graph invariants have been proposed to evaluate and classify the topological structure of fullerene isomers and predict their stability: the pentagon adjacency index NP [1-4], the Wiener index WI [5], the resistance distance RT [5], the Kekulé structure count [6], the graph independence number [7], the number of spanning trees [8], the combinatorial curvature [9], the mean vibrational potential energy [10], and the occurrence number of different structural motifs in fullerenes [11-13]. It can be observed that for stability prediction the application of spectrum-based topological invariants has a growing importance (the smallest eigenvalue [14,15] and the separator of fullerene graphs [7, 16], the bipartivity measure [16], the Estrada index [17]).
In this study, based on the investigation of spectral properties of duals of fullerene graphs, a novel approach is suggested for the construction of global topological invariants. It is characterized by the following concept: As a first step we consider the dual of the traditional fullerene graph, and as a second step, from the adjacency matrix of the corresponding dual graph, spectrum-based topological invariants designated for the characterization of fullerene structures are derived. The discriminating performance of spectrum-based descriptors was tested on the set of C40 and C66 fullerene isomers.
2. Preliminaries Let G =G(V,E) be a simple graph with V(G ) vertices, E(G )
edges, and A(G) be its
adjacency matrix. The set of eigenvalues ρ1(G) ≥ ρ2(G) ≥......≥ ρn(G) of A(G) is the spectrum of the graph G and the largest eigenvalue ρ(G)=ρ1(G) of A(G) is called the spectral radius of G. An eigenvalue of a graph G is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. For a vertex u of G, the degree of
u, denoted by d(u) is the number of edges incident to u. We denote by Δ= Δ(G) and δ= δ(G) the maximum and the minimum degrees of vertices of G. A graph is called r-regular if all its vertices have the same degree r. A graph is called irregular if it contains at least two vertices with different degrees. Bidegreed graph is a non-regular graph whose vertices have exactly two degrees δ and Δ. By definition, the Estrada index [18] of a connected graph G is EE (G) exp i (G) i
where ρi(G) are the eigenvalues of the graph adjacency matrix A(G). The Estrada index has already found a remarkable variety of applications in structural chemistry [19-21], but Fowler and Graovac were the first to apply EE in structural characterization of fullerene graphs [17]. According to their findings, the Estrada index seems to be an efficient candidate among the spectrum based descriptors, for stability prediction. The Collatz–Sinogowitz irregularity index [22,23] of a connected graph G is a nonnegative number defined as CS(G ) (G )
2 E (G )
.
V (G )
It follows that the irregularity index CS(G) is a spectrum-based invariant which is equal to 0 if and only if G is regular.
3. Characterization of structural irregularity of fullerenes A fullerene (fullerene graph) with k vertices, denoted by Ck exists for all even k≥ 20 except k=22, where the number of pentagon is 12, and the number of hexagons is k/2-10 [1]. As we have already mentioned, our method proposed for the structural characterization of fullerene isomers is based on “an indirect approach”. First, we consider the duals of the traditional fullerene graphs. Then, from the adjacency matrices of the dual graphs, we derive spectrum-based topological invariants associated with the fullerene structures. Because fullerene graphs Ck are polyhedral graphs, their dual graphs denoted by C dual exist k and are unambiguously defined.
Besides the well-known pentagon adjacency index Np [1] and the Estrada index
( EE EE(C k ) ) of original (trivalent) fullerene graphs, we computed 3 novel spectrumbased invariants: the Estrada index ( EEdual EE(C dual the spectral radius k ) ), dual ( Ro (C dual k ) ) and the Collatz-Sinogowitz irregularity index ( CSdual CS(C k ) ) of the
corresponding dual graphs, respectively. From the previous considerations it follows that
dual EEdual EE (C dual k ) exp i (C k )
i
CSdual CS(C
dual k
) (C
dual k
)
2 E(C dual k ) V(C dual k )
by definition. The duals of fullerene graphs are triangular polyhedral graphs, and they are all bidegreed except the dual of C20, which is a 5-regular graph. The dual graphs contain 12 vertices of degree 5, and any remaining vertices are of degree 6. This implies that for their spectral radii the inequality ( (Cdual k ) 6 ) holds. When testing the ability of a topological invariant for prediction purposes, it is necessary to take into consideration the following concept: it is required that a selected graph invariant (a topological descriptor of a fullerene) must be able not only to distinguish between isomers structures but also to rank the isomers in the order of decreasing stability. There are several theoretically well-founded physical models which are successfully applicable to rank and classify the isomers according to their relative stability [1-4]. For this purpose we used the density functional tight-binding (DFTB) method [24], by which the tight binding total energy TQ of a fullerene can be predicted. The most stable isomers are characterized by the lowest value of this energy. It should be noted that the majority of ab initio stability calculation methods is based on complicated physical models, in certain cases the computation time (CPU time) can increase dramatically. For testing purpose, a comparative study has been performed on the sets of C40 and C66 fullerenes.
3.1 Tests performed on the set of C40 isomers Several topological descriptors have been already calculated for C40 fullerene isomers [3-5, 8-10, 12,13]. Among the 40 isomers of C40, C40:38 is predicted to be the isomer of lowest energy by many methods, this is followed by C40:39 and C40:31.
Results of our calculations, the tight binding total energy TQ (eV), pentagon adjacency indices (NP), Estrada index of isomers (EE), the Estrada index and the spectral radius of dual graphs (EEdual and Ro) are summarized in Table 1. Each isomer is labeled according to Fowler and Manolopoulos [1].
Table 1 Computed topological invariants and energy values for forty C40 isomers. Isomer C40:38 C40:39 C40:31 C40:29 C40:26 C40:24 C40:37 C40:40 C40:14 C40:36 C40:30 C40:25 C40:22 C40:35 C40:21 C40:27 C40:15 C40:17 C40:34 C40:28 C40:16 C40:20 C40:9 C40:10 C40:12 C40:13 C40:19 C40:23 C40:6 C40:18 C40:5 C40:32 C40:8 C40:33 C40:4 C40:7 C40:11 C40:2 C40:3 C40:1
Ro 5.4890495 5.4897603 5.4913203 5.4917461 5.4918157 5.4922670 5.4910060 5.4927389 5.4946883 5.4910060 5.4939737 5.4945800 5.4947709 5.4917209 5.4956524 5.4937994 5.4944877 5.4960335 5.4936774 5.4944236 5.4964612 5.4957915 5.4977966 5.4977120 5.4970645 5.4976413 5.4962914 5.4972724 5.5011144 5.4989259 5.5020757 5.4994279 5.5046265 5.4999739 5.5039109 5.5029706 5.5019917 5.5086293 5.5160459 5.5260158
Topological parameter EEdual EE 384.0861429 132.0267238 384.1524340 132.0267262 384.3456462 132.0272051 384.3961455 132.0272065 384.3885577 132.0271643 384.4460111 132.0271656 384.2728637 132.0271218 384.5051255 132.0276837 384.7548057 132.0276523 384.2772077 132.0271218 384.6289867 132.0276467 384.6935372 132.0276048 384.7344424 132.0276060 384.3481603 132.0271220 384.8430166 132.0276076 384.6090975 132.0276044 384.6903195 132.0276088 384.8787435 132.0280898 384.5611023 132.0275623 384.6592604 132.0276058 384.9461311 132.0280952 384.8691424 132.0275654 385.1507334 132.0280982 385.0989738 132.0280519 384.9986241 132.0280504 385.0852416 132.0280517 384.8964782 132.0280911 384.9689797 132.0280055 385.5627727 132.0284988 385.1550042 132.0284895 385.7362473 132.0285480 385.2683608 132.0284853 386.0925238 132.0289921 385.3045578 132.0284858 385.9108375 132.0289449 385.7240240 132.0289380 385.5385376 132.0288945 386.6542055 132.0293939 387.8701024 132.0302887 389.9454188 132.0312134
Total energy TQ (eV) Np 10 10 11 11 11 11 11 12 12 11 12 12 12 11 12 12 12 13 12 12 13 12 13 13 13 13 13 13 14 14 14 14 15 14 15 15 15 16 18 20
–342,031 –341,631 –341,438 –341,345 –341,094 –341,022 –340,636 –340,580 –340,476 –340,431 –340,304 –340,277 –340,230 –340,196 –340,151 –340,126 –339,943 –339,884 –339,827 –339,777 –339,645 –339,627 –339,614 –339,558 –339,370 –339,347 –339,292 –338,690 –338,624 –338,341 –338,332 –338,270 –338,113 –337,922 –337,348 –337,330 –336,642 –336,489 –335,193 –333,806
As shown in Table 1, with the topological descriptors EE, EEdual and Ro, we get the same trends of relative stability: C40:38 > C40:39> C40:31. This corresponds to the theoretical results based on ab initio calculations [3-5]. It should be noted that isomers C40:38 and C40:39 have the smallest, but identical pentagon adjacency indices (NP=10). From our computed results it follows that the prediction performance of the descriptors EE,
EEdual and Ro are judged be equivalent for the set of C40 isomers. It is important to note that for trivalent fullerene graphs, the range of computed Estrada indices (EE) is very narrow. For example, considering the forty C40 isomers, one obtains EEmax= 132.03121 and EEmin= 132.02672, consequently EEmax - EEmin= 0.00449 [17]. For bidegreed dual graphs of fullernes, the spread of EEdual descriptors is relatively large: for C40 isomers one obtains 389.9454188 - 384.0861429= 5.8592759. (See Table 1). In Fig.1 the variation of the tight binding energy TQ with the four selected topological
-333
-333
-334
-334
-335
-335
-336
-336 TQ (eV)
TQ (eV)
descriptors (Np, EE, EEdual and CSdual) are illustrated.
-337 -338 -339
-337 -338 -339
-340
-340
-341
-341
-342
-342 10
12
14
16
18
20
132.027
Np
132.028
132.030
132.031
b)
-333
-333
-334
-334
-335
-335
-336
-336
-337
TQ (eV)
TQ (eV)
a)
132.029 EE
-338 -339
-337 -338 -339
-340
-340
-341
-341
-342
-342 384
385
386
387 EEdual
c)
388
389
390
0.03
0.04
0.05 CSdual
0.06
0.07
d)
Fig.1. Relations between the tight binding total energy and the 4 topological parameters
As can be seen, all computed topological descriptors have a good correlation with the physical stability represented by the tight binding energy parameter TQ. In Figure 2, the correlation of Estrada index (EE) and CSdual is depicted. As Figure 2 shows, the most stable isomer C40:38 is characterized by the minimum value of CSdual indices. The maximum values of Np, EE and CSdual indices belong to the least stable isomer C40:1. (See Table 1).
0.07
CSdual
0.06
0.05
0.04
0.03 132.027
132.028
132.029 EE
132.030
132.031
Fig. 2 Relationship between topological indices EE and CSdual for C40 isomers It is becoming clear that CSdual can be considered as a measure of irregularity (structural heterogeneity) of fullerene isomers. 0.07
CSdual
0.06
0.05
0.04
0.03 384
385
386
387 EEdual
388
389
390
Fig. 3 Relationship between topological indices EEdual and CSdual for C40 isomers In Figure 3, the correspondence between indices EEdual and CSdual is shown. It seems surprising that EEdual correlates strongly with CSdual.
3.2 Tests performed on the set of C66 isomers The number of topologically different C66 isomers is 4478. Computed topological indices and energy values for some C66 isomers are given in Table 2.
Table 2 Computed topological invariants and energy values for some selected C66 isomers. Isomer Ro 5.68919 5.68868 5.68898 5.69001 5.72646 5.72564 5.72282 5.72715 5.72553 5.73909
C66:4169 C66:4348 C66:4466 C66:4007 C66:15 C66:63 C66:2 C66:13 C66:12 C66:1
Total energy TQ (eV)
Topological parameter EEdual EE 650.02146 217.0792620 649.99237 217.0792606 649.99727 217.0792607 650.13592 217.0796576 654.54847 217.0842631 654.22684 217.0842133 654.23735 217.0843123 654.87027 217.0843130 654.55787 217.0843129 657.56383 217.0851123
Np 2 2 2 3 14 14 14 14 14 16
-583.0067 -582.8916 -582.7047 -582.3229 -569.4201 -569.4119 -569.3728 -569.3614 -569.0544 -567.0107
We can conclude that there exist only 3 isomers with lowest pentagon adjacency index Np=2, and 5 isomers with Np=14. There is no isomer with Np=15, and there exists only one isomer (C66:1) with maximum pentagon adjacency index Np=16. (See Table 2.) 658
0.08
657
0.07
655
CSdual
EEdual
656 654 653 652
0.06 0.05 0.04
651 650
0.03
0
2
4
6
8 Np
10
12
14
0
16
2
4
0.08
0.08
0.07
0.07 CSdual
CSdual
8 Np
10
12
14
16
b)
a)
0.06 0.05
0.06 0.05
0.04
0.04
0.03
0.03
217.079 217.080 217.081 217.082 217.083 217.084 217.085
EE
c)
6
650
651
652
653 654 655 EEdual
656
657
658
d)
Fig. 4 Relationships between topological indices Np, EE, EEdual and CSdual for C66 isomers
According to energetic calculations, the most stable isomer with minimum total energy is C66:4169. The next three isomers with increasing energies are C66:4348, C66:4466 and C66:4007. This finding is confirmed by the computational results published in [25,26]. The “least stable” isomer is C66:1 with Np=16. Relations between the computed topological
indices are demonstrated in Fig.4. A common feature of diagrams depicted in Fig.4 a), b) and c) is that the point set representing 4478 isomers can be divided into 14 well-separated clusters. These 14 clusters correspond basically to the isomers having identical pentagon adjacency indices Np. Consequently, the first cluster represents the most stable 3 isomers, namely C66:4169, C66:4348 and C66:4466 with Np=2. The second cluster includes the isomers with Np=3, and so on. There is no cluster corresponding to Np=15, because there are no isomers with Np=15. The last cluster belongs to the lonely isomer C66:1 with highest, Np=16 pentagon adjacency index. Considering the correspondence between topological invariants EEdual and CSdual, a parabolic relationship has been found between EEdual and CSdual (See Fig.4.d)). This is characterized by a very strong correlation (R2 =0.9856). As can be observed, in all cases, the maximum values of topological invariants Np, EE, EEdual and CSdual belong to the lonely C66:1 isomer with Np=16.
4. Dual fullerene graphs as 2-walk linear graphs In the sequel we investigate some fullerene isomers having special spectral properties. It will be shown that there exist fullerene isomers the dual graphs of which belong to the family of 2-walk linear graphs. For that purpose, first we need to introduce the following definitions [27]: Let d2(v) denote the sum of the degrees of all vertices adjacent to a vertex v in a connected graph G. Then, G is called 2-walk (a,b) linear graph if there exist unique rational numbers a, b such that
d 2 (v) a d(v) b holds for every vertex of G.. From the above definition it follows that a 2-walk linear graph is irregular, and is characterized by the following properties [27]: A connected graphs G has two main eigenvalues if and only if G is 2-walk linear. Moreover, if G is 2-walk (a,b) linear graph, then parameters a and b must be integers, and the spectral radius of G is
(G)
1 a a 2 4b . 2
Studying the topological structure of fullerenes we found 5 isomers whose dual graphs are 2-walk linear, namely: C24, C28(Td), C60(Ih), C80(Ih) and C140(I). It is known that C24 is the smallest fullerene containing 2 isolated hexagons. (See Fig.5).
Fig.5 The graph of C24 fullerene and its dual graph Fullerene C28(Td) which includes 4 isolated hexagons is the smallest tetrahedral fullerene. Isomer C60(Ih) (known as Buckminsterfullerene), moreover isomers C80(Ih) and C140(I) have only isolated pentagonal faces, they belong to the particular class of fullerenes with icosahedral symmetry. After determining the graph parameters a and b, the computed results related to the 5 fullerene isomers investigated can be summarized as follows: The dual of C24 fullerene: a 2-walk (4,6) graph, (C dual 24 ) 2 10 5.162278 . The dual of C28(Td) isomer: a 2-walk (3,12) graph, (C dual 28 (Td )) 5.274917 . The dual of C60(Ih) isomer: a 2-walk (3,15) graph, (C dual 60 (I h )) 5.653312 . dual (I h )) 2 14 5.741657 . The dual of C80(Ih) isomer: 2-walk (4,10) graph, (C80 dual The dual of C140(I) isomer: 2-walk (5,5) graph, (C140 (I)) 5.854102 .
Knowing the spectral radii of the above mentioned dual graphs the corresponding Collatz– Sinogowitz irregularity indices can be simply calculated.
5. Summary, final remarks Topological index CSdual can be considered as an indirect quantitative measure of heterogeneity of original fullerene structures. In other words, CSdual characterizes the irregularity of the local arrangement of pentagonal and hexagonal faces in fullerene isomers. Because the Collatz-Sinogowitz irregularity index is equal to 0 if and only if the graph is regular, this implies that CSdual is a positive number, and the most stable fullerenes are characterized by the smallest CSdual indices. An advantage of using the topological index CSdual is that its application can be directly extended to the structural characterization of non-traditional fullerenes including square and heptagonal faces.
References [1]
P. W. Fowler, D. E. Manolopoulos, An Atlas of Fullerenes, Calendron Press, Oxford, 1995.
[2]
A. T. Balaban, X. Liu, D.J. Klein, D. Babics, T. G. Schmalz, W.A. Seitz, M. Randic, Graph invariants for fullerenes, J. Chem. Inf. Comput. Sci. 35 (1995) 396-404.
[3]
E.E.B. Campbell, P.W. Fowler, D. Mitchell, F. Zerbetto, Increasing cost of pentagon adjacency for larger fullerenes, Chem. Phys. Lett. 250 (1996) 544-548.
[4]
E. Albertazzi, C. Domene, P.W. Fowler, T. Heine, G. Seifert, C. Van Alsenoy, F. Zerbetto, Pentagon adjacency as a determinant of fullerene stability, Phys. Chem. Chem. Phys. 1, (1999) 2913-2918.
[5]
P. W. Fowler, Resistance distances in fullerene graphs, Croat. Chem. Acta 75 (2002) 401-408.
[6]
F. Torrens, Computing the permanent of the adjacency matrix for fullerenes, Internet Electron. J. Mol. Des. 1 (2002) 351-359.
[7]
S. Fajtlowitz, C.E. Larson, Graph-theoretical independence as a predictor of fullerene stability, Chem. Phys. Lett. 377 (2003) 485-490.
[8]
P. W. Fowler, Complexity, spanning trees and relative energies, in fullerene isomers, MATCH Commun. Math. Comput. Chem. 48 (2003) 87-96.
[9]
T. Réti, E. Bitay, Prediction of fullerene stability using topological descriptors, Mater. Sci. Forum 537-538 (2007) 439-448.
[10] E. Estrada, N. Hatano, A.R. Matamala, A graph theoretic approach to atomic displacements in fullerenes, in: F. Cataldo, A. Graovac, O. Ori, (Eds.) The Mathematics and Topology of Fullerenes, Spinger Dordrecht, 2011, pp.171-185. [11] M. Alcami, G. Sachez, S. Diaz-Tendero, Y. Wang, F. Martin, Structural patterns in fullerenes showing adjacent pentagons: C20 to C72, J. Nanosci. Nanotechnol. 7 (2007) 1329-1338. [12] T. Réti, I. László, A. Graovac, Local combinatorial characterization of fullerenes, in: F. Cataldo, A. Graovac, O. Ori, (Eds.) The Mathematics and Topology of Fullerenes, Spinger Dordrecht, 2011, pp. 61-83. [13] T. Réti, I. László, E. Bitay, T. Došlić, Classification of fullerene isomers using local topological descriptors, Mater. Sci. Forum 659 (2010) 447-451. [14] P.W. Fowler, P. Hansen, D. Stevanović, A note on the smallest eigenvalue of
fullerenes, MATCH Commun. Math. Chem. 48 (2003) 37-48. [15] T. Došlić, The smallest eigenvalue of fullerene graphs – Closing the gap, MATCH Commun. Math. Chem. 70 (2013) 73-78. [16] T. Došlić, Bipartivity of fullerene graphs and fullerene stability, Chem. Phys. Lett. 412 (2005) 336-340. [17] P.W. Fowler, A. Graovac, The Estrada index and fullerene isomerism, in: F. Cataldo, A. Graovac, O. Ori, (Eds.) The Mathematics and Topology of Fullerenes, Spinger Dordrecht, 2011, pp. 265-280. [18] E. Estrada, Characterization of 3D molecular structure, Chem. Phys. Lett. 319 (2000) 713-718. [19] J.A. de la Penas, I. Gutman, J. Rada, Estimating the Estrada Index, Lin. Algebra Appl. 427 (2007) 70-76. [20] I. Gutman, E. Estrada, J.A. Rodrígez-Velásquez, On a graph-spectrum-based structure descriptor, Croat. Chem. Acta 80 (2007) 151-154. [21] I. Gutman, H. Deng, S. Radenković, The Estrada index: An updated survey, in: D. Cvetković, I. Gutman (Eds.) Applications of Graph Spectra, Math. Inst. Belgrade 2009 pp. 123-140. [22]
L. Collatz, U. Sinogowitz, Spektren endlicher Graphen, Abh. Math. Sem. Univ. Hamburg, 21 (1957) 63-77.
[23] H. Abdo, D. Dimitrov, The total irregularity of a graph, arxiv.org/abs/1207.4804 2012. [24] D. Porezag, Th. Frauenheim, Th. Köhler, G. Seifert, R. Kaschner, Construction of tight-binding-like potentials on the basis of density-functional theory: Application to carbon, Phys. Rev. B51, (1995) 12947-12957. [25] T. Réti, I. László, On the combinatorial characterization of fullerne graphs, Acta Polytech. Hung. 6 (2009) 85-93. [26] QB. Yan, QR. Zheng, G. Su, Theoretical study on the structures, properties and spectroscopies of fullerene derivatives C66X4 (X =H, F, Cl), Carbon 45 (2007) 1821-1827. [27] Z. Tang, Y. Hou, The integral graphs with index 3 and exactly two main eigenvalues, Lin. Algebra Appl. 433 (2010) 984-993.
