Exhaustive Generation and Analytical Expressions of Matching Polynomials of Fullerenes C20-C50

published in Aduance ACS Abstracts, February 15, 1994. 0095-233819411634-0421$04.50/0 Fueled by the intense activity in fullerene chemistry and intrinsically interesting geometrical and topological features of fullerenes, several studies have focused on the fascinating mathematical properties of fullerenes. These studies included graph theoretical computations of the characteristic polynomials, enumeration of possible structures for a given fullerene formula,3s32 enumeration of isomers of polysubstituted f~ l le renes ,~~ spectral moments of f~llerenes?~ group theoretical studies concerning the nuclear spin statistics and rovibronic levels,35 combinational techniques for the characterization of the NMR and ESR spectral patterns,36 and so on. In spite of significant mathematical studies on fullerenes, the computation of the matching polynomials is restricted to the letter of the author23 in which he considered the matching polynomials of some of the smaller fullerenes. In the present paper we have combined the results of the matching polynomials of some of the fullerenes obtained before23 with the new results on fullerenes C32, C34, c38, c42, Q4,(T), (246, and C4g obtained here for the first time. The matching polynomials of all fullerenes Cn (n = 20-50) for even n with the exception of (222 are thus exhaustively obtained. The coefficients were analyzed which resulted in a systematic exact formula fo the first several coefficients of the matching polynomials of I ullerenes. 2. MATCHING POLYNOMIALS AND THEIR COMPUTATION The matching polynomial M&) of a graph and the related ZG(X) are formally defined as m M&) = & (1 )&P( C , k ) F Z k where n is the number of vertices, the sum terminates with m = [n/2], and P(G,k) is the number of ways of choosing k non-adjacent edges from the graph G. Alternatively P(G,L) is the number of ways of placing k disjoint dimers (dumbbells) on the graph G. Note that P(G,I) is the same as P[21+ 11. For computational convenience we computed results in terms The recurrence relation of Hosoya and co-workers18 was used. This is similar to the recurrence relation for the rook and king polynomials on a chess board, which can be derived of P[21+ 11. Q 1994 American Chemical Society 422 J . Chem. Inf Comput. Sci., Vol. 34, No. 2, 1994 BALASUBRAMANIAN C 4 4 ( D 3 h ) C s o ( D 5 h ) CSO(Ih) Figure 1. Structures of fullerene cages CZO-CM except ‘234, C4, and using the principle of inclusion and exclusion. The recursive relation is as follows. c48. w&3 = wi.&4 %t&) where G-e is the graph obtained by deleting an edge “e” from G while G e e is the graph obtained by deleting this edge, the vertices composing e, and all the edges connected to the vertices that comprise e. This recursive relation facilitates reduction of the graph G to smaller or simpler graphs. The matching polynomial of a tree graph is the same as the characteristic polynomial which is readily computed by the code developed by the authorlIJ2 based on the LeverrierFrame method. The graph G is thus recursively reduced until all the fragments are trees. Then using the characteristic polynomials of the trees computed using the author’s code, the matching polynomials are computed. We used the code developed by Ramaraj and the authorg for recursive pruning of graphs to trees. This is combined with the author’s code1* for computing the characteristic polynomials to generate the matching polynomials. It was found that as the size of the cage increases, the number of pruning steps astronomically grows. 3. RESULTS FOR FULLERENES AND DISCUSSION In this investigation new results for the matching polynomials were obtained for fullerenes C32 (D3 symmetry), C34 ( G U symmetry), C38 ( G U symmetry), c42 (03 symmetry), CU (T symmetry), c46 (C3 symmetry), and c 4 8 (D3 symmetry). The Schlegal diagrams or full three-dimensional structures of all fullerenes C20-C50 are shown in Figures 1 and 2. For the sake of completeness and to provide a complete set of data for further analysis, we also include in tables our previously computed results for other fullerenes from ref 23. We note at the very outset that the computation of matching polynomials of fullerenes is a computationally very intensive problem requiring large amounts of CPU time running into several hours. This is in dramatic contrast with the charc34 (C3V) c38 ( b h ) C4Q (Td) CU ( b h ) C48 (C3) C4e (b) Figure 2. Schlegal diagrams for C34 (Ch), C3s (Dab), two forms of C4, C42 (D3) , two forms of CM, C4 (C3), and CM ( 0 3 ) . acteristic polynomials which are readily obtained in a few seconds of CPU time for such cages using the author’s code based on the Leverrier-Frame method. Table 1 shows the matching polynomials of all fullerenes C2&40 (for C,, n = even number), while Table 2 shows the matching polynomials of C42, CU, CM, C48, and Cso fullerenes. In each case we considered the most attractive structures from the standpoint of strain energy and other considerations. Feyereisen et have considered ab initio studies of some of these fullerenes. The most favorable structures considered by these authors for the fullerenes are included in our study. The constant coefficients in the matching polynomials yield the number of Kekul6 structures. Thus these numbers can be used as a primary measure of stability. Further we note that even if a fullerene cage by itself is not very stable, the introduction of a metal atom can significantly stabilize it. This is the case with the C28 fullerene cage which is stabilized by the introduction of a tetravalent metal atom.2g Figure 3 shows the plot of the constant coefficient (we call this K ) in the matching polynomial divided by n as a function of n for C2&50. As seen from Figure 3, we find a sudden jump in the constant coefficient in the matching polynomial in moving from c28 to C30 (coefficient doubles). The coefficients change very little in moving from 30 to 34, while it increases in moving from c34 to c36. Another jump is seen in going from 38 to 40. Again we see a sharp increase from C48 to C50 which is suggestive of the enhanced stability of C ~ O compared to other smaller fullerenes. This increase in Kln for n = 30 is consistent with the change in the SCF energylcarbon in moving from c 2 8 to The C30 cluster exhibits a local maximum in the SCF energy per carbon. From C30 to C32, we see a drop in the energy per carbon for the fullerene structure. There has to be a large rise in Kln for larger values of n since K is a nonlinear function of n. It is thus interesting that a simple plot of Kln as a function of n correlates reasonably with the ab initio predictions. Let us compare the matching polynomials of two isometric fullerenes. As a first example, consider two C38 fullerene MATCHING POLYNOMIALS OF FULLERENE~ cz&so Table 1. Matching Polynomials of Fullerene Clusters C A M J. Chem. Inf Comput. Sci., Vol. 34, No. 2, 1994 423