Spectral clustering of combinatorial fullerene isomers based on their facet graph structure
SSpectral clustering of combinatorial fullerene isomers basedon their facet graph structure
Artur Bille
Victor Buchstaber
Evgeny Spodarev ... This spiritual experience, thisdiscovery of what Nature has in storefor us with carbon, is still ongoing.Richard E. Smalley, Discovering thefullerenes , Nobel lecture, Dec. 7, 1996.
Abstract
After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have beensubject of much research. One part of that research is the prediction of a fullerene’s stability usingtopological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facetson its surface, calculations mostly were performed on all isomers of C , C and C . This paper suggestsa novel method for the classification of combinatorial fullerene isomers using spectral graph theory. Theclassification presupposes an invariant scheme for the facets based on the Schlegel diagram. The mainidea is to find clusters of isomers by analyzing their graph structure of hexagonal facets only. We alsoshow that our classification scheme can serve as a formal stability criterion, which became evident from acomparison of our results with recent quantum chemical calculations [34]. We apply our method to classifyall isomers of C and give an example of two different cospectral isomers of C .Calculations are done with MATLAB. The only input for our algorithm is the vector of positions ofpentagons in the facet spiral. These vectors and Schlegel diagrams are generated with the software packageFullerene [33]. Keywords: convex polytope, fullerene, combinatorial isomer, facet spectrum, C , dual graph, eigenvalue,adjacency matrix MSC2010: Primary:
Secondary:
In this paper, we consider a fullerene C n as a convex polytope modeling a closed three-dimensional carbon-cage with n atoms, cf. [1]. Each vertex is connected to exactly three other vertices, such that the facetsare pentagons and hexagons only. Using the classical Euler relation and Eberhard’s theorem [19, § Artur Bille [email protected]
Victor Buchstaber [email protected]
Evgeny Spodarev [email protected] Ulm University Skoltech Steklov Mathematical Institute a r X i v : . [ m a t h . SP ] O c t qual to 12 and n is even. In this case, the number of hexagons is m = n −
10, cf. [17]. Theoretically, C n exists for n = 20 and all even n ≥
24, see [1]. In the sequel we call such a n feasible . However, up to nowFigure 1: IPR-Isomers of C (Buckminster fullerene) and C . Courtesy of Max von Deliusonly a few of them (some isomers of C n with n = 60 , , , , , ,
84) are separated (i.e., are chemicallystable and can be synthesized in considerable mass quantities), cf. [21, 27, 30, 38]. Another difficult problem isthe combinatorial constructive enumeration of all isomers of C n , see [8, 12, 35] and references in [16, Section3]. For instance, a set of four operations (including that of Endo-Kroto) enables the construction of allfullerenes for each even n starting with n = 24 (barrel), see [13, 16]. Other examples for such operations arethe generalized Stone-Wales operation (cf. [2], which we investigate in a forthcoming paper [5] in more detail)and the buckygen (introduced 2012 in [9]) which is up to now the fastest algorithm to generate all C n -isomers.Combinatorial isomer is a class of combinatorially equivalent polytopes. Two polytopes P and P (cid:48) arecalled combinatorially equivalent if there exists a one-to-one mapping between the lattice of all facets of P and P (cid:48) that is inclusion-preserving [19, p. 38]. Hence, this equivalence class is determined by the graphof vertices, or equivalently, by the dual graph of facets. Both are uniquely described by their adjacencymatrices. With increasing amount of atoms n , the number of isomers ISO( C n ) grows fast as O ( n ), cf. [35].For instance, there exist a unique C –isomer (we write: | C | = 1) which is dodecahedron , a Platonic solid,but C has already 1812 combinatorial isomers, see [7]. Among all isomers of C , only one (the so–called Buckminster fullerene, an Archimedean truncated icosahedron) has all pentagonal facets being not adjacent.Such isomers are called
IPR-fullerenes (from
Isolated Pentagon Rule ), cf. Figure 1. As illustrated in Figure2, the number ISO-IPR( C n ) of IPR-isomers also grows asymptotically as O ( n ) with increasing number ofatoms n [8, 32].The huge variety of possible fullerene isomers with large n (even in the IPR–class) makes the problem offinding molecules with remarkable chemical and physical attributes (including thermodynamic and kineticstability, permeability, electric conductivity, light diffraction, etc.) extremely difficult. Hence, a need for fastand computationally cheap classification methods of isomers arises. In the literature, there already exist anumber of functionals (called chemical descriptors ) allowing to find a certain order of isomers, cf. [4, 15, 20].These descriptors are of topological, geometric or physical nature. For the practical separation of fullerenes,their potential energetic level is of primary significance. The paper [34] computes the relative energies of all1812 isomers of C using the density functional theory (DFT) [31] and testing 26 chemical descriptors fortheir correlation with the energetic ordering of these isomers. The authors identify 7 rules among 26 whichthey call good stability criteria . These criteria are defined as those able to identify correctly the first twoenergetically most stable and the three energetically least stable isomers in the correct energetic order suchthat the correlation coefficient between the ordering according to these rules and the energetic order is atleast 0 .
6. However, the DFT calculations are based on the approximative numerical solution of electronicSchr¨odinger (linear partial differential) equations requiring from half an hour up to one day of calculationtime per isomer ( n = 60) on a usual personal computer. Moreover, the convergence of the DFT numericalmethod is not guaranteed.Following the famous question by Mark Kac (1966) Can one hear the shape of a drum? [22] we try to“hear” the shape of a fullerene from the spectrum of its adjacency matrix provided that C n –isomers canbe mapped bijectively onto their spectra. We also give an example of two different C –isomers with thesame spectrum of adjacency matrix of hexagonal facets. We propose a new method of clustering and of2lassification of C n –isomers for any n ≥ n (cid:54) = 44, based on combinatorial and graph theoretic structure ofdual graphs T n of their hexagonal facets. For n = 60 we show in this paper that it yields a good stabilitycriterion with correlation coefficient of 0 .
9. The spectral analysis of graphs based on adjacency matrices oftheir vertices has a long standing tradition [3, 10, 11]. However, we show that for the complete classificationand ordering of fullerenes it is sufficient to use • the dual graph T n of hexagons since the positions of 12 pentagons can be reconstructed out of cycleslarger than triangles and degrees of vertices in T n (cf. [5]). • Newton polynomials of the spectrum of the adjacency matrix A n of hexagons up to a certain degree k ∗ since it is well known that they are numerically more stable than the eigenvalues themselves. Thesepolynomials can be computed directly as a trace of (cid:0) A n (cid:1) k , k = 2 q , q = 1 , . . . , k ∗ /
2, whereas the matrixmultiplication is computationally less demanding than finding all eigenvalues of a matrix. Hereby, weuse a graph theoretical interpretation of Newton polynomials of adjacency matrices of graphs in termsof their cycle numbers.Our method is computationally very fast requiring O ( n log n ) operations with a total of 1.15s CPU timeon a Intel Core i5-8300H (2.3 GHz) ( n = 60). The high correlation of the obtained ordering with the DFTenergetic order allows to figure out few energetically stable isomers at a low computational cost. For theseisomer candidates, the detailed DFT analysis can be further performed.Figure 2: Logarithm of the numbers of C n -isomers ISO ( n ), IPR-isomers IP R − ISO ( n ) and their upperbound for all feasible n ∈ { , . . . , } , on a logarithmic scale.In order to construct the facet adjacency matrices, a certain enumeration algorithm of all facets is required.Since the spectra of these matrices are invariant with respect to the enumeration of facets, the choice of thisalgorithm does not matter from the mathematical point of view. For all fullerenes with 24 ≤ n < spiral rule first introduced in [26] (where it is called an orange peel scheme) to enumerate all pentagonsand hexagons and generate their adjacency matrices A n and A n , respectively. The first fullerene not obeyingthe spiral rule is a C –isomer, and the second counterexample is one of over 90 billion C -isomers. Allother isomers of C n with n ≤
450 stick to this rule, cf. [25]. For fullerenes without a facet spiral, a generalizedspiral [37] can be used for the one-to-one facet enumeration.3ur spectral approach is illustrated on all isomers of C which are the best studied fullerenes, especiallythe above mentioned famous Buckminster (soccer ball-like molecule). Such molecular structures are allallotropic forms of carbon [27]. C n For a feasible n a fullerene isomer P ∈ C n is a simple, compact and convex polytope in R with all m := n/ n/ −
10 hexagons: P := { x ∈ R | a i x + b i ≥ i = 1 , . . . , m } , a i (cid:54) = 0 , b i ∈ R , for all i. Its i th facet f i is given by f i := { x ∈ R | a i x + b i = 0 } ∩ P , i = 1 , . . . , m. P can be mapped on atwo–dimensional graph in a way that edge crossing is avoided and vertex connectivity information is retained.First, one has to choose a facet and rotate P so that this facet is located parallel to the ( x, y )–plane atsome distance below a fixed projection point q . Next, one draws a line starting in q to each vertex of thepolyhedron and extends this line until it crosses the ( x, y )–plane. The intersections are the vertices of thenew two-dimensional graph, also called Schlegel diagram . Although such a projection is not bijective, ityields a full combinatorial invariant of P . Each of f i can be chosen to be initially parallel to the x − y –plane.Depending on this choice, the resulting graphs can be very different, see [17]. In Figure 3(a) and 3(b), onecan see two possible Schlegel diagrams for the Buckminster fullerene ( n = 60). The graphs on these Schlegeldiagrams are equivalent in the sense that they have the same vertex connectivity. From the definition of afullerene it immediately follows that the corresponding planar graph is 3-regular. We denote a planar graphof a fullerene by F . (a) a pentagon was chosen initially (b) a hexagon was chosen initially Figure 3: Two different, but combinatorially equivalent, Schlegel diagrams of Buckminster fullereneAssume that C n has a facet spiral which is defined as an order of facets such that each facet sharesan edge with the previous and next one. This spiral can be presented by a facet spiral sequence . It is asequence of twelve integers, which determines the position of the twelve pentagons in the facet spiral. Anotherrepresentation is a sequence of fives and sixes such that the k –th number in the sequence indicates whetherthe k –th facet in the spiral is a pentagon or a hexagon [1]. We use the first approach in our software [6].For instance, the facet spiral sequence for Buckminster fullerene represented by its Schlegel diagram inFigure 3(a) is (1 , , , , , , , , , , , C n,i we mean the i th C n –isomer according to thelexicographical order of facet spiral sequences, see also [17, Chapter 2].4 .1 Dual facet graphs, adjacency matrices and their spectra Let G = ( V ( G ) , E ( G )) = ( V, E ) be a finite undirected graph with vertex set V and edge set E . Let | V | = m be the number of vertices in V . The adjacency matrix A G = A = ( a i,j ) of G is given by a i,j := (cid:40) , if ( i, j ) ∈ E, , otherwise , , ≤ i (cid:54) = j ≤ m, a i,i = 0 , ≤ i ≤ m. In matrix form, A is a symmetric m × m –matrix with zeros on the diagonal and (cid:80) mj =1 a i,j being equal to thevalency of the node i : A = a , . . . a ,m a , . . . a ,m ... ... . . . ... a m, a m, . . . . Define the spectrum σ ( A ) of A G as a set of its eigenvalues λ i ( A ) = λ i ( G ) = λ i , i = 1 , . . . , m . An inducedsubgraph H of G is a graph with vertices set V ( H ) ⊆ V ( G ) and all of the edges of G connecting pairs ofvertices in V ( H ).Due to the symmetry of A , it holds σ ( A ) ⊂ R . Let tr( A ) = (cid:80) mi =1 a i,i be the trace of A . It obviously holdstr( A ) = (cid:80) mi =1 λ i ( A ). Later the Newton polynomial N ( A, k ) := tr( A k ) = (cid:80) mi =1 λ ki ( A ) of degree k with k ∈ N and an adjacency matrix A , will be of interest to us. It is well–known that the spectrum of an m × m –matrix A can be uniquely restored from the values N ( A, k ) , k = 1 , . . . , m , cf. e.g. [18, p. 93]. Lemma 1.
Let k ≤ m be an integer and A be the adjacency matrix of a graph G with m vertices. Thena) the Newton polynomials can be calculated recursively as N ( A, k ) = − k (cid:88) | H | = k ( − e ( H )+ c ( H ) c ( H ) − k − (cid:88) j =2 N ( A, k − j ) (cid:88) H : | H | = j ( − e ( H )+ c ( H ) c ( H ) , (1) where the inner sum runs over all subgraphs H of G with j nodes and connected components being eitheredges or cycles, e ( H ) being the numbers of edges among these components and c ( H ) being the number ofcycles.b) the Newton polynomial of degree k can be interpreted as the number of all cycles of length k in G . Here, we call a cycle of length k any closed path (possibly with self–intersections) with k edges from avertex to itself. Proof. a) It is known that Newton polynomials can be represented as polynomials of elementary symmetricpolynomials of the eigenvalues λ i ( A ), i = 1 , . . . , m with integer coefficients, see [24, Chapter 11, § S k of λ ( A ) , . . . , λ m ( A ) is a sum of principal minors of A ofthe corresponding degree k ∈ N (cf. [29, p. 495]), and these minors have integer values due to a i,j ∈ { , } ,we get that the values of N ( A, k ) are integers. Moreover, S = N ( A,
1) = 0. For k ≥ N ( A, k ) = − kS j + k − (cid:88) j =2 ( − j − N ( A, k − j ) S j , (2)where we put S j = 0, j > m . By [3, Theorem 3.10], it holds S j = ( − j (cid:88) H : | H | = j ( − e ( H )+ c ( H ) c ( H ) . (3)5) This interpretation follows immediately from [10, Proposition 1.3.1], since the i th diagonal entry of the k th power of A is the number of walks of length k from vertex i to itself.For fullerenes, vertex adjacency matrices and their spectra are well-studied, see [1, Section 4.5]. Asmentioned above, we generate dual facet graphs T n out of the Schlegel diagrams of C n and consider thespectra of their adjacency matrices. In T n the original facets become vertices, and the original verticesbecome facets. The edges of the dual graph show adjacency relations between original facets: two nodesof the dual graph are connected by an edge if the corresponding facets of the fullerene are adjacent, i.e.share an edge. In Figure 4(a), one can see the dual graph T of all facets of Buckminster fullerene with theSchlegel diagram in Figure 3(a). Notice that (for the sake of legibility) the facet f is displayed five timesin Figure 4(a), whereas it should occur just once. In this paper, we use red, green and white nodes in theimages of the dual graphs of C n for pentagons, hexagons and unspecified facets, respectively.Consider two important induced subgraphs T n and T n of T n . The graph T n illustrates the connectivitybetween pentagonal facets, i.e. it always contains 12 vertices. For instance, the graph T n of every IPR-isomerconsists of 12 disconnected vertices. This being said, it is evident that the number of isomers of C n with thevery same graph T n increases rapidly with increasing n , cf. Figure 2 for the IPR-case. Hence, consideringthe graph T n does not yield an invariant for all C n -isomers. In order to characterize all isomers we needthe graph T n showing the connectivity of all m hexagonal facets of a C n -isomer. As an example the graph T of the Buckminster fullerene is shown in Figure 4(b). It turns out that T n completely characterizes thegraph T n . We denote by A n , A n , A n the adjacency matrix of T n , T n and T n , respectively. Remark 1. a) For k > m a formula similar to (1) can be derived, by which it follows that traces of the k th power of A with k > m are linear combinations of traces of smaller powers. This can be explained by thefact that subgraphs have at most as much vertices as the whole graph.b) An alternative approach is to insert formula (2) into itself, which yields a representation of Newtonpolynomials as a polynom with several unknowns being Newton polynomials of lower degree.c) The condition that every vertex in a fullerene has valency three corresponds to the fact that the dual graph T n consists of triangles only. However, the subgraphs T n and T n may also have larger cycles.d) No 4-cycles exist neither in T n nor in T n and T n , see [13, Theorem 4.15 (1)].e) The problem of description of all simple cycles (i.e., closed loops without self–intersections) of pentagonalor hexagonal facets of length k is crucial to combinatorial classification of fullerenes. (a) Dual graph T of all facets (b) Hexagonal dual graph T Figure 4: Dual facet graphs of Buckminster fullerene6enote by N the set of natural numbers and zero. Obviously, it holds N ( A n , k ) ∈ N for all k ∈ N .For the dual graphs T n and T n we construct their adjacency matrices A n and A n . Since the number ofunit entries in each line does not exceed 5 for A n or 6 for A n it holds by Gershgorin’s theorem that σ ( A n ) ⊂ [ − , , σ ( A n ) ⊂ [ − , . In the sequel, let us concentrate on the properties of σ ( A n ). The classical Frobenius–Perron theory appliedto graph spectra [10, Proposition 3.1.1] yields a more accurate estimate for the largest eigenvalue of A n which is positive and of multiplicity one if T n is connected. Lemma 2.
Let G be a graph with m vertices and valencies κ , . . . , κ m , and H be an induced subgraph of G .Let λ max ( G ) and λ max ( H ) be the largest eigenvalues of the adjacency matrix A G and A H .a) If G is connected, then κ min ≤ ¯ κ ≤ λ max ( G ) ≤ κ max , with ¯ κ being its mean valency and κ max the maximumvalency of the vertices in G . In particular, λ max ( G ) = κ holds if G is κ -regular.b) It holds (cid:113) m (cid:80) mi =1 κ i ≤ λ max ( G ) ≤ κ max .c) It holds λ max ( H ) ≤ λ max ( G ) .Proof. a) See [10, Proposition 3.1.2].b) See [14, Theorem 1.2] and [10, Comment on Proposition 3.1.2].c) See [3, Lemma 3.16].Notice that for all connected irregular graphs T n , T n and T n we have ¯ κ > κ max ≤
6. For non–connected or irregular graphs we get only 0 ≤ ¯ κ ≤ κ max . For instance, the dual graph T of the Buckminsterfullerene (see Fig. 4(b)) is connected and regular with κ = 3, thus the largest eigenvalue of A , for it isequal to 3.In general, the value θ := κ max − ¯ κ ≥ asymmetry coefficient . Remark 2. If k → ∞ , we have N ( A n , k ) /λ kmax ∼ a max + ( − k ( − λ max ∈ σ ( A n )) , where ( B ) is theindicator function of B and a max is the multiplicity of the eigenvalue λ max . Indeed, by [3, Theorem 6.3], itholds − λ max ∈ σ ( A n ) iff our dual graph T n is bipartite, i.e., it has no cycles of all odd lengths, cf. [3, Corollary3.12]. In this case, − λ max has necessarily multiplicity one, see [3, Lemma 3.13]. Since for large n ≥ thedual graph T n of hexagons of any isomer of C n contains either a 3– or a 5–cycle, it holds − λ max (cid:54)∈ σ ( A n ) ,and the behavior of the the whole Newton polynomial tr (cid:16)(cid:0) A n (cid:1) k (cid:17) for large powers k is dominated by λ kmax . Let | A | be the cardinality of a finite set A . Lemma 3.
For all feasible n , consider a C n –isomer with graphs T n , T n and T n . Then it holds | E ( T n ) | = | E ( T n ) | + 3 n − . Proof.
Since the graph T n of a C n –isomer is 3–regular, it has n edges. An edge ( v, w ) ∈ E ( T n ) is eitheran edge between two pentagons, ( v, w ) ∈ E ( T n ), or between two hexagons, ( v, w ) ∈ E ( T n ), or between apentagon and a hexagon, ( v, w ) ∈ E ( T n \ (cid:0) T n ∪ T n (cid:1) ). Since the number of pentagons is always 12, there are60 − | E ( T n ) | edges between pentagons and thus | E ( T n ) | + | E ( T n ) | + 60 − | E ( T n ) | = 3 n , which finishes the proof. 7 Cospectral Isomers
Definition 1.
Let A G , A H be adjacency matrices of graphs G and H with m vertices each. G and H (or A G and A H ) are said to be cospectral if σ ( A G ) = σ ( A H ) . It is easy to prove that isomorphic graphs are cospectral. The inverse statement is in general not true(cf. e.g. a counterexample in [36]). The natural question arises:
Which graphs are determined by theirspectrum? [36]. Some specific graphs like paths, complete graphs, regular complete bipartite graphs, cyclesand their complements yield a positive answer to this question. In what follows, we provide new examples ofnon–isomorphic cospectral graphs coming from the world of fullerenes.We derive some theoretical results for sets of non-cospectral graphs and apply them to fullerene isomers.To begin with, recall the following
Lemma 4.
For graphs G and H with m vertices and adjacency matrices A G and A H the following statementsare equivalent:a) G and H are cospectral.b) A G and A H have the same characteristic polynomial.c) N ( A G , k ) = N ( A H , k ) for all k = 1 , . . . , m .Proof. Equivalence of a) and b) is obvious.Now prove the equvalence of c) and b). As stated in [24, Chapter 11, §
53] elementary symmetricpolynomials S k of degree k of eigenvalues of A can be expressed as S k = ( − k − N ( A, k ) k − k k − (cid:88) i =1 ( − i S k − i N ( A, i ) , ≤ k ≤ n. Since S = N ( A,
0) = 0 and S = − N ( A, /
2, every S k with 2 ≤ k ≤ n can be computed just knowing N ( A, , . . . , N ( A, k ). Hence, the symmetric polynomials are identical for both A G and A H . Finally, thecoefficients of the characteristic polynomials can be expressed as ( − k S k , so the characteristic polynomialsare equal as well.For all even 24 ≤ n ≤ C n -isomers within T n , T n and T n graphs. Using Lemma 4, we applied MATLAB functions eig() , charpoly() and trace() with doubleprecision, cf. [28]. In cases where two isomers seem to be cospectral with respect to T n or T n , we increasethe precision by using vpi format and MATLAB symbolic toolbox [28].For n <
32 no pair of cospectral isomers with respect to T n , T n and T n can be found. Our results for32 ≤ n ≤
60 are listed in Table 1. This table can be extended to n >
60 with all zeroes in its first and thirdrows, and positive integers in its second row.Considering the whole dual graph T n one finds only one pair of cospectral isomers with n = 44, compareFigures 5(a) and 5(b). To explain the difference within this pair, we need the following Definition 2.
Let G and H be two graphs.a) A fragment F of G is a connected induced subgraph of G . In particular, for fullerenes we call fragmentsof T n and T n pentagon-fragments and hexagon-fragments of T n , respectively.b) Two non-isomorphic graphs G and H are called fragment flipped if two isomorphic fragments F G in G and F H in H exist such that the remaining graphs G \ F and H \ F are isomorphic. As one can see in Figure 5(a) and 5(b), the two cospectral C -isomers are fragment flipped. However,in general such a flip does not preserve the spectrum of a graph. To illustrate this, Figure 5(c) and 5(d)contains a non–cospectral fragment flipped pair of C –isomers.8or n ∈ { , , , } a pair of distinct isomers exists with the same spectrum σ (cid:0) T n (cid:1) , since their graphs T n are isomorphic. n
32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 T n T n T n T n , T n and T n of C n –isomers.For a fixed 40 ≤ n ≤
150 at least three and at most 298 non-unique spectra σ ( T n ) exist. Since the amountof different arrangements of 12 pentagons is limited and the number of isomers grows rapidly with increasing n , there must exist isomers with same subgraphs T n . We discuss the number and different configurations ofpentagon-fragments in [5] in more detail.The above empirical findings lead to the following Conjecture 1. a) For all feasible n (cid:54) = 44 two C n –isomers are isomorphic iff they are cospectral with respectto T n .b) For all feasible n ≥ two C n –isomers are isomorphic iff they are cospectral with respect to T n .c) For any feasible n at least one of the spectra σ ( T n ) , σ ( T n ) , σ ( T n ) is unique for all C n –isomers. For a graph G with m vertices we denote by | σ ( G ) | the set of absolute values of eigenvalues λ i ∈ σ ( G ) for i = 1 , . . . , m sorted in descending order, i.e. | σ ( G ) | := {| λ i | | λ i ∈ σ ( G ) , | λ i | ≥ | λ i +1 | ∀ i = 1 , . . . , m − } , andcall it the absolute spectrum of G . Denote by | σ [0 , ( G ) | and | σ [1 , ∞ ) ( G ) | the part of | σ ( G ) | with 0 ≤ | λ | < ≤ | λ | . It holds λ ( G ) = λ max ( G ) Theorem 1.
Let Γ be a set of graphs with m vertices and with distinct absolute spectra such that λ max ( G ) > for all G ∈ Γ . Then all graphs G ∈ Γ can be uniquely characterized by at most two Newton polynomials ofeven degrees k ∗ , k ∗ with m ≥ k ∗ ≥ k ∗ .Proof. Ordering all absolute spectra | σ ( G ) | lexicographically yields a unique order on the set Γ. Then allgraphs G from Γ can be distinguished either by | σ [1 , ∞ ) ( G ) | or by | σ [0 , ( G ) | .In the first case, let us consider all G ∈ Γ with distinct | σ [1 , ∞ ) ( G ) | . The sum (cid:80) λ ∈| σ [0 , ( G ) | | λ | k convergesto 0 for k → ∞ , so its influence on the Newton polynomials N ( A G , k ) for k large enough can be neglected.Since | σ [1 , ∞ ) ( G ) | is unique for all considered graphs G there must exist a degree k ∗ such that the values of N ( A G , k ) for all even k ≥ k ∗ are distinct.Now if some graphs G ∈ Γ have the identical part of absolute spectrum | σ [1 , ∞ ) ( G ) | , they must differ withinthe part | σ [0 , ( G ) | . Hence, there must always exist a degree k ∗ ≤ k ∗ such that the sum (cid:80) λ ∈| σ [0 , ( G ) | | λ | k ∗ distinguishes these graphs.Lemma 4 c) and Remark 1 a) yield the number of vertices in the graph G as an upper bound for k ∗ and k ∗ . Corollary 1.
Assuming Conjecture 1 to be true, all C n -isomers can be uniquely characterized by at mosttwo Newton polynomials N ( A G , k ∗ ) and N ( A G , k ∗ ) , k ∗ ≤ k ∗ even, with respect to at least one of the graphs G ∈ { T n , T n , T n } .Proof. This follows from Theorem 1 and Lemma 2, since λ max ( G ) > G ∈ { T n T n , T n } . Remark 3. (i) Note that the set of C n –isomers with distinct largest eigenvalues λ max ( G ) for some G ∈ { T n T n , T n } has distinct sets | σ [1 , ( G ) | . ii) It is important to stress that we consider even degrees k ≤ m only. In general, values of Newtonpolynomials N ( A G , k ) with odd degrees k do not distinguish between graphs G . To illustrate this point,consider the hexagon graphs of two specific isomers of C . The graph T , of the first isomer consistsof four isolated hexagons and the other graph T , of two pairs of two adjacent hexagons. One gets thefollowing spectra: σ ( T , ) = { , , , } , σ ( T , ) = {− , − , , } . It follows directly that Newton polynomials of any odd degree do not distinguish between C , and C , ,but N ( A , , k ) (cid:54) = N ( A , , k ) for every even k ≥ . (a) Dual graph T of C , (b) Dual graph T of C , (c) Dual graph T of C , (d) Dual graph T of C , Figure 5: Two examples of pentagon-fragment (marked with an X) flipped isomers. C , and C , arecospectral, but C , and C , are not. In this section, we decompose the family of all C n –isomers into subsets, which we call clusters , using theNewton polynomials N ( A G , k ), k = 2 , , , . . . , k ∗ with adjacency matrix A G . This approach can be appliedto any graph G ∈ { T n , T n , T n } with no cospectral isomers. Definition 3. a) For a feasible n and a given even integer k , we call a set of isomers with same value N ( A G , k ) of Newton polynomial of degree k a cluster of C n . For a fixed n , the family of all these clustersis called a clusterization of C n . A clusterization such that its every cluster has exactly one element iscalled a complete clusterization.b) We define k ∗ single as the minimal degree k of Newton polynomials which is needed for a complete clusteri-zation. As we have seen in Lemma 1 b), the Newton polynomial of degree two and three is equal to twicethe number of edges and six times the number of triangles in the considered graph. The interpretation of10 ( A n ,
4) is a bit more complex. We have to count all possible cycles of length four. In Figure 6, the idea ofcalculation of N ( A ,
4) is illustrated on Buckminster fullerene C , . There, all five possible cycles oflength four together with their frequencies are listed for one pentagon in C , . (a) A fragment fromBuckminster fullerene (b) Appears fivetimes in (a) (c) Appears 25times in (a) (d) Appears 20times in (a) (e) Appears ten timesin (a) (f) Appears ten timesin (a) Figure 6: Five ((b)-(f)) different kinds of cycles of length four beginning and starting in the central vertex(X) in (a)For the graph T n , one can show that the sum of these frequencies over all vertices in T n only depends on n , cf. [5]. So, N ( A n , k ) for k ≤ C n on the basis of T n , since theseNewton polynomials have the same value for all C n -isomers.Nevertheless, for T n we have to consider every Newton polynomial of even degree, since in T n the numberof vertices is fixed, but neither the number of edges between them nor the number of triangles in T n isdetermined by n .In the following section we use T in order to get a complete clusterization of C as an example. C using Newton polynomials Recall that by Euler’s formula each C -isomer has 90 edges and 32 facets with 12 pentagons and 20 hexagonsamong them.The Newton polynomial N ( A ,
2) can take on 18 distinct values 60 = t < . . . < t = 100. For values t = 60 , t = 64 , t = 92 , t = 96 and t = 100, there exists exactly one C -isomer with N ( A ,
2) = t i , i ∈ { , , , , } . These isomers are C , , C , , C , , C , , C , , respectively. Their Schlegeldiagrams and dual hexagonal graphs T are shown in Figure 7. Moreover, our numerical results showthat for any degree k ≥ T these isomers form a cluster with one single element.Ordering these five isomers according to N ( A , k ) does not change with increasing degree k ≥
2. Inaddition, Newton polynomials of all other C -isomers are bounded by N ( A , , k ) and N ( A , , k ), i.e. N ( A ,i , k ) ∈ (cid:0) N ( A , , k ) , N ( A , , k ) (cid:1) for all i / ∈ { , , , , } and all k ≥ a) C , (b) C , (c) C , (d) C , (e) C , (f) T , (g) T , (h) T , (i) T , (j) T , Figure 7: Schlegel diagrams and dual graphs of hexagons of five C -isomersWe checked that for any pair of two C -isomers with distinct Newton polynomials of degree k the Newtonpolynomials N ( A , ˜ k ) with k ≤ ˜ k ≤
100 are distinct as well. So, it holds k ∗ = k ∗ = k ∗ single = 12, where k ∗ and k ∗ are from Theorem 1. One can observe that the number of distinct Newton polynomials and so thenumber of clusters with one element is monotone growing with k . Numbers of clusters and clusters with oneelement for all even 2 ≤ k ≤
100 are listed in Table 2. k . . . . . . . . . N ( A , k ) for even 2 ≤ k ≤ C n , 28 ≤ n ≤
150 and plotted n against k ∗ single in Figure 8(a).Recall that a (pessimistic) upper bound for k ∗ single is the number of vertices in T n , i.e. k ∗ single ≤ m = n − k ∗ single is logarithmic with n .Using MATLAB curve fitting toolbox [28] we get k ∗ single ( n ) ≈ − .
13 + 7 .
801 log (0 . n − , with a coefficient of determination R = 0 . a) n ∼ k ∗ single (b) n ∼ k ∗ pair Figure 8: Minimal degree k ∗ single (a) and k ∗ pair (b) needed for the complete clusterization of C n -isomers with28 ≤ n ≤ (cid:0) N ( A n , k ) , N ( A n , k ) (cid:1) with k < k ≤ k ∗ single in order tocluster all C -isomers. By considering additionally a second Newton polynomial of lower degree, we hope todecrease the needed degree to get a complete clusterization. We define k ∗ pair as the minimal k such that acomplete clusterization is given. Indeed, this approach decreases the needed degree significantly (compareboth plots in Figure 8), and therefore, reduces computational costs. For C and T the following four tupleswith k < k ∗ single of degrees of Newton polynomials lead to a full classification: k = ( k , k ) ∈ (cid:8) (6 , , (4 , , (6 , , (8 , (cid:9) . We get k ∗ pair = 8. Next we plotted all values for k ∗ pair against n and assumed a logarithmic function asfor k ∗ single . Using MATLAB curve fitting toolbox we get the following approximation k ∗ pair ( n ) ≈ − .
83 + 10 .
29 log (0 . n + 9 . R = 0 . k ≤ k ∗ single . Analogously to the first two approaches, we define k ∗ hierarchical asthe minimal k which yields a complete clusterization. This approach decreases e.g. the degree for n = 78from k ∗ pair = 12 to k ∗ hierarchical = 10. So, this approach does not change the needed degree significantly.Nevertheless, we performed the same interpolation using MATLAB curve fitting Toolbox and got13 ∗ hierarchical ( n ) ≈ − .
05 + 19 .
83 log (1 . n + 125 . . with R = 0 . k ∗ single (sharper than m ) using the generalized Stone-Walesoperation introduced in [2]. In the paper, we give a combinatorial interpretation of k ∗ and derive an equationsystem with Newton polynomials which determines whether a fullerene with given n can be constructed. A fullerene isomer, which can be chemically separated with a significant mass quantity and uniquelycharacterized, is called stable . In order to decide which C -isomer can be stable the relative energy ofall of them was calculated with high-accuracy quantum chemistry methods and discussed in [34]. Here, relative means compared with the Buckminster fullerene C , which has the lowest DFT-energy at thePW6B95-D3 ATM / def2-QZVP level (cf. [34]), i.e. Buckminster fullerene has a relative energy of 0. In thesequel, we say that an isomer C n,i is energetically more stable than C n,j , i (cid:54) = j , if C n,i has a smaller energythan C n,j .According to [34] the most stable isomer is C , and the second stable one is C , . At the otherend of the ranking the three least stable ones are C , , C , and C , . It has been assumed for a long timethat an isomer is the more stable the less adjacent pentagon it has. Indeed, calculations of [34] allow theconclusion that each pair of two adjacent pentagons leads to a increase in the relative energy of an isomer ofabout 20 to 25 kcal mol − . The amount of such pentagon pairs can be described with Fowler-Manolopoulospentagon indices p i := { pentagons which are adjacent to i other pentagons } , such that the sum of p up to p is equal to 12 for every fullerene, cf. [17]. Based on these values, the pentagon signature P = 1 / (cid:80) i =1 ip i can be calculated, which quantifies the amount of connected pentagons. Clustering all C -isomers accordingto the pentagon signature, five isomers stand out, namely C , ( P = 0), C , ( P = 2), C , ( P = 16), C , ( P = 18) and C , ( P = 20). The signature can be easily read from Figures 7(a)-7(e). Pentagonssignatures of the remaining isomers lie between 2 and 16. For each of these values, at least two isomers existwith the same pentagon signature. By Lemma 1 b) and 3 we get the following Proposition 1.
For any C n -isomer it holds P = N ( A n , − n . In Table 3, C -isomers are listed in the same order given by their relative energy, by their pentagonsignature and their Newton polynomial of degree 2.Isomer C , C , C , C , C , N ( A ,
2) 60 64 92 96 100 P C -isomers with unique Newton polynomial N ( A ,
2) and Pentagon signature P sorted bytheir relative energy in ascending order.Indeed, one gets more information about a fullerene structure looking on hexagons than on pentagons.This becomes clear looking at fullerenes with large n . For example, C has 7 IPR-isomers, so their pentagonstructure and pentagon signature are the same. Nevertheless, only two of them have been produced in pureform, although DFT calculations have been done for all of them, see [23]. As a result of [23], only twoIPR–isomers can be claimed stable. Hence, all descriptors based on the pentagon structure do not properlypredict stability. 14n [34] a good stability criterion is defined as the one which can identify C , and C , as the moststable and C , , C , and C , as the least stable isomers in the correct energetic order. Additionally, thePearson coefficient ρ of linear correlation between the relative energies of all C –isomers and their criterionvalues should be larger than 0 .
6. Finally, the slope and the Pearson correlation coefficient in the linearregression of relative energy vs. the criterion for C –isomers with P ∈ { , . . . , } should have the samesign.As we have seen in Table 3, Newton polynomials yield the correct order of the most and least stable C -isomers. Next, we perform a linear regression (using MATLAB curve fitting Toolbox) of N ( A , k )vs. relative energies of all C isomers for all even 4 ≤ k ≤
12. For the case k = 2 Newton polynomial isequivalent to the 1st moment hexagon Signature H , which is listed in [34, Table 3] as a good stabilitycriterion. Hence, N ( A ,
2) is a good stability criterion as well and can be neglected in further considerations.Table 4 shows that Pearson correlation coefficient ρ is much higher than 0 . k . For degrees k = 8 , ,
12, Newton polynomials N ( A , k ) get very large, and therefore we took a logarithmic scale. Buteven with linear scale, one gets correlation coefficients larger than 0 . C into 18 subsets G i = { P ∈ C | P ( P ) = i } according to theirpentagon signature i ∈ { , , , . . . , , , } . Then we performed a linear regression of Newton polynomialsof different degrees vs. relative energies of isomers in G i for every i / ∈ { , , , , , , } as it is requiredin [34]. Our results are listed in Appendix, Table 8. For k ∈ { , } and i ∈ { , } we get Pearson correlationcoefficients and slopes with a negative sign, unlike for all other combinations of k and i . This can be explainedby the fact that G and G do not contain many isomers. More precisely, | G | = 17 , and | G | = 86 holds.So, neglecting these two cases would yield that N ( A , k ) with k = 4 , k = 10 ,
12 one gets positive slopes and Pearson correlation coefficients in all cases, and thereforeNewton polynomials of degree 10 and 12, in particular of degree k ∗ single , entirely fulfil all conditions of a goodstability criterion. Linear regression ρ slope N ( A , ∼ relative energy 0.95 0.45 N ( A , ∼ relative energy 0.95 0.02 log (cid:0) N ( A , (cid:1) ∼ relative energy 0.945 106.6 log (cid:0) N ( A , (cid:1) ∼ relative energy 0.94 81.36 log (cid:0) N ( A , (cid:1) ∼ relative energy 0.94 66.1Table 4: Pearson correlation coefficient ρ and the slope of linear regression between Newton polynomials andthe relative energy of all C –isomers given in [34].To check whether Newton polynomials can distinguish between IPR-isomers, i.e. yield their energeticallycorrect order, we computed Newton polynomials of all 31924 C -isomers. Within the whole set of C theseven IPR-isomers have the smallest Newton polynomials. But ordering the set of IPR-isomers according toNewton polynomials leads to the observation that the most stable IPR-isomers have the greatest Newtonpolynomials. These seven isomers are listed in Table 5. So, it seems that with increasing n Newtonpolynomials N ( A n , k ) for even k ≥ A , ) tr( A , ) tr( A , ) tr( A , ) tr( A , ) tr( A , ) tr( A , )31918 0 1040 12960 19 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × × . × . × . × . × . × × . × . × . × . × . × . × . × . × . × . × . × . × . × Table 5: Seven IPR-isomers of C , their relative energy in kcal/mol − and Newton polynomials. Only theisomers 31918 and 31919 can be chemically separated so far, cf. [23]. C The Fowler asymmetry parameter is claimed to be a good stability criterion [34]. Check whether theasymmetry coefficient θ defined in Section 2 is a good stability criterion as well. The asymmetry coefficientsof isomers shown in Figure 7 are θ = 1 , θ = 1 . , θ = 1 . , θ = 0 . θ = 0. The histogram of θ of all C -isomers is shown in Figure 9.Figure 9: Histogram of 23 different asymmetry coefficients of C -isomersIt turns out that the asymmetry coefficient θ is not a good stability criterion since it does not preservethe energetic order required in [34]. Thus, the asymmetry coefficient of eight isomers is equal 2.4, which is thelargest value. These isomers are C , , C , , C , , C , , C , , C , , C , and C , .Figure 10 shows C , , which has six hexagons with valency two, twelve hexagons with valency four andtwo hexagons with valency six. So, the mean valency is 3.6, the maximal valency is 6 and the resultingasymmetry coefficient equals 2.4. 16igure 10: Schlegel diagram of one of the eight isomers with largest asymmetry coefficient θ = 2 . C , i.e. the ones with smallest and largest θ as well as thoseshown in Figure 7. One can see that the three least stable as well as the two most stable isomers have theasymmetry coefficient less than 2 .
4. In addition, the relative stability of the eight isomers with θ = 2 . θ is not a good stability predictor. i θ i θ and energetic stability of some C –isomers We present an easy to compute functional of spectra of the graphs T n and T n which classifies all C n -isomers.Thereby we focus on the structure of the dual graph of hexagonal facets of C n and its adjacency matrix A n .The spectra of the adjacency matrices are characteristic to combinatorial isomers described above. It becomesapparent that the Newton polynomial of degree 2 , , k ∗ single ) of T n appears to be a good stabilitycriterion. So, Newton polynomials of T can be added to the list presented in [34, Table 3] as indices, which,depending on the degree, fulfil the criteria partly or entirely. We show that Newton polynomials generalizethe Pentagon signature and better describe the fullerene structure. The interpretation of these Newtonpolynomials is very easy for k ≤
3, but gets demanding with increasing k . Acknowledgements
We express our gratitude to Markus Schandar who was involved in programming of spectra of fullerenesat the early stage of this research. We would also like to thank Max von Delius and Konstantin Amsharovfor discussions on the chemistry of fullerenes. We are indebted to Axel Groß for the reference [34] and hislectures on the DFT method. 17 eferences [1] V. Andova, F. Kardoˇs, and R. ˇSkrekovski. Mathematical aspects of fullerenes.
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Matlab - Curve Fitting Toolbox - Linear Regression
Independent variable ρ slope intercept N ( A ,
2) 0.9524 11.6996 -701.3801 N ( A ,
4) 0.9557 0.4501 -96.2619 N ( A ,
6) 0.9514 0.0201 37.6084 N ( A ,
8) 0.9328 9 . · − N ( A ,
10) 0.8974 4 . · − N ( A ,
12) 0.8456 1 . · − log (cid:0) N ( A , (cid:1) − . · log (cid:0) N ( A , (cid:1) log (cid:0) N ( A , (cid:1) Independent variable ∼ relative energy for all C -isomers.20nd. var. i ρ slope intercept N ( A , G -0.4059 -0.2933 217.0715 N ( A , G -0.1603 -0.0701 149.9463 N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G -0.0529 -0.0023 98.0643 N ( A , G -0.0513 -0.001 120.9228 N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G N ( A , G . · − N ( A , G -0.0016 − . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − i ρ slope intercept N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , G . · − N ( A , k ), 4 ≤ k ≤
12 even, vs. relative energy overthe subsets G i of C60