J. Phys. Chem. 1995,99, 6509-6518
6509
Laplacians of Fullerenes (c42-c90) K. Balasubramanian Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287-1604 Received: January 4, 1995; In Final Form: February 21, 1995@
Laplacians of fullerenes (C42-C90), which are quite important in the characterization of structures, enumeration of spanning trees, computation of the magnetic properties of fullerenes, computation of the vibrational spectra, and so on, are obtained using high-precision computational algorithms. The spectra and the characteristic polynomials of the Laplacians of fullerenes are computed. From the related Riemann I; functions of fullerenes, the number of spanning trees of fullerenes is computed. Several interesting results pertinent to the spectra of Laplacians of fullerenes are outlined.
Introduction Ever since the discovery of the celebrated buckminsterfullerene (%), fullerenes have become one of the most topical species of the present day.’-25 Synthesis of macroscopic amounts of several fullerenes other than buckminsterfullerene has now made it possible to study the structure, reactivity, and spectroscopic properties of fullerenes in much detail. Fullerenes are cage-structured carbon clusters containing exactly 12 pentagons and a varied number of hexagons. While earlier studies based on the mass spectra of carbon clusters generated by supersonic nozzle expansion of the vaporized graphite revealed magic numbers for 50-, 60-, and 70-carbon clusters, now several other fullerenes such as c78, c 7 6 , C84, and Cm, have become important species. Both Kratshchmer et aL7 and Haufler et a1.6 have made pioneering contributions to the chemistry of fullerenes. Several molecules and radicals arising from fullerene cages have been made, and their properties are being actively investigated. In recent years branches of discrete mathematics such as graph theory, combinatorics, and group theory have been recognized to provide valuable tools in the characterization of structures and spectra of f u l l e r e n e ~ . ~The ~ - ~topics ~ have varied from the enumeration of cages as well as substituted cages to the simulation of the NMR and ESR spectra of molecules and radicals derived from fullerenes. Graph-theoretical methods have been extensively used to study the structural and topological properties of fullerenes. Graph-theoretical polynomials such as the characteristic polynomials, the matching polynomials, sextet polynomials, and Wheland polynomials are known to play important roles in both structural characterizations and computations. For example, the characteristic and matching polynomials of fullerenes have been 0btained.2~9~~ Analysis of such polynomials has provided several fascinating trends in that the coefficients in these polynomials characterize the topological and structural features of the fullerene cages. The matching polynomials and their spectra provide the topological resonance energies of fullerenes. As shown by recent studies from our l a b ~ r a t o r yexact , ~ ~ analytical expressions can be obtained for the first 13 coefficients of the characteristic polynomials of fullerenes. Similar expressions for the coefficients of the matching polynomials have also been derived. The Laplacians of graphs have been the topic of much attention in recent The Laplacians are considered @
Abstract published in Advance ACS Abstracts, April 15, 1995.
0022-3654/95/2099-6509$09.00/0
to be one of the most important structural invariants, although they may not uniquely characterize the structures. Several combinatorial problems pertinent to structures such as walks on the structures, dimer coverings, enumeration of spanning trees, characterization of graphs through the associated Fiedler vector, construction of “Ramanujan graphs”, partitioning of the vertices of the graph using the second eigenvector of Laplacians, and the search for Hamiltonian circuits are related to the Laplacians. As shown by Kir~hhoff:~the topologically independent circuit equations can be obtained as the number of spanning trees of the associated graph. Several recent reviews have appeared on L a p l a ~ i a n s . ~ ~ , ~ ’ In chemical context, Laplacians of structures are useful in many applications. The vibrational spectra, Huckel eigenvalues, number of spanning trees in a structure, enumeration of random walks, computation of the stabilities through the topological resonance energies, etc., are related to the Laplacians of the structures. Chung and S t e m b e ~have t ~ ~ considered the relationship between the Laplacians and vibrational spectra of homogeneous graphs. The Laplacian of a graph is related to both its adjacency matrix and the valency matrix and is thus more complete than the adjacency matrix itself. The spectra of the Laplacians of homogeneous graphs of fullerenes are shown to be related to the bending vibrational modes of fullerenes. Trinajstic et al.36 have considered the Laplacian matrices of graphs in the context of chemistry. The current author38 has studied the spectra and characteristic polynomials of small fullerenes c20-c40. It was shown that some of these fullerenes exhibited integral eigenvalues. Factorization of the Laplacian polynomials of these fullerenes was also considered. The magnetic properties of fullerenes are related to the Laplacians as shown here. In the traditional Mcweeny’s model4 of computing the magnetic properties of conjugated hydrocarbons, the notion of ring current comes into play. This idea is intimately related to the spanning trees of the structure under cons id era ti or^.^^,^^.^^ Brown et al.42 have enumerated the number of spanning trees in buckminsterfullerene as 375 291 866 372 898 816 OOO using a theorem of Gutman et al.4’ and modulo arithmetic in conjunction with the “Chinese remainder theorem”. It is cautioned that counter examples to the theorem of Gutman et al. have been recently found by Kirby et al.,48 although it appears that the result for c 6 0 is unaltered by this. In the current study we verify the same result to all digits but using an entirely different technique based on the Riemann 5 functions of Laplacians. The number of spanning trees of several fullerene cages (c20-C~) are obtained. The Riemann 5 function has a long and rich mathematical history 0 1995 American Chemical Society
Balasubramanian
6510 J. Phys. Chem., Vol. 99, No. 17,1995
involving an Indian mathematical prodigy, R a m a n ~ j a nThe . ~ ~ ~ ~ ~in the Laplacian polynomial differs substantially from the corresponding coefficientsin the characteristicpolynomials. That function which naturally arises in counting the number of prime is, the Laplacian polynomials contain information on both the numbers is likely to have several applications in the characterconnectivity and valencies, while the characteristic polynomials ization of structures. are based purely on the adjacency matrices. Some fundamental In this study, we have computed the spectra and the results emerge from the Laplacians when the Riemann 5 Characteristic polynomials of Laplacians of fullerenes c 4 2 - C ~ . It is shown that the number of spanning trees of fullerenes can functions of the spectra of Laplacians are considered. be obtained using the zeros of the derivative of the Riemann 5 Computation of the Spectra and Laplacian Polynomials function of the computed spectra of Laplacians. A few of Larger Fullerenes interesting results are also obtained concerning the spectra of fullerenes. Factorization of the characteristic polynomials of As noted by the authoS9 before, the computation of Laplacian some of the fullerenes is considered. Some of the fullerenes polynomials is more difficult than the ordinary characteristic considered here have integral Laplacian eigenvalues. polynomial^^^ as a consequence of larger coefficients in the
Laplacian Polynomials The Laplacian matrix of a graph is defined as
I
Vi i f i = j
L.,= -1 if i =kj and vertices i and j are connected 0 otherwise where Vi is the valency (degree) of the ith vertex of the graph under consideration. The related adjacency matrix A of the graph is defined as
I
A,.= 1 if i + j and i and j are connected 0 otherwise It can be easily seen that the Laplacian and the adjaceny matrices are related to each other through the diagonal degree matrix of the graph, defined as
D,,={Vf i f i = j lJ 0 otherwise Consequently, for any graph the Laplacian and the adjacency matrices are related as
L=D-A The Laplacian polynomial of the graph is defined as
PL(G) = IAZ - LI On the other hand, the characteristic polynomial is defined by
Fullerenes are graphs with vertices of the same valency (degree) and are called homogeneous or regular graphs. Since fullerene graphs possess vertices of degree 3, one can relate the Laplacian and the characteristic polynomials readily. That is,
PL(G)= l;lZ - LI = 1(;1- 3)Z
+ AI = IA’Z
A”
= -A
+ AI = lA”Z - AI
+3
As seen from the above relation, the eigenvalues of the adjacency matrix of fullerenes are simply related to the spectra of Laplacians. The spectra of Laplacians can be derived from the eigenvalues of the adjacency matrices by changing the signs and adding 3. The Laplacian polynomials can also be obtained fiom the characteristic polynomials by replacing A“ by -A 3 in the characteristic polynomial. Nevertheless, this is not a convenient method of computing the coefficients of the Laplacian polynomials. Furthermore, the coefficient of e v e j term
+
Laplacian polynomial. Higher arithmetic precision is called for in computing the coefficients of the Laplacian polynomials. The present author’s code46was used in conjunction with quadrupleprecision arithmetic to compute the Laplacian polynomials. The quadruple precision yields approximately 3 1-digit accuracy compared to 15- and 7-digit accuracy obtained in double- and single-precision arithmetic, respectively. First, 13 coefficients are enough to characterize fullerenes, in that structural dependence is manifested only after the 12 coefficient. Generally, quadruple precision is enough for this purpose. However, if all the coefficients are sought especially for a larger fullerene such as CW, then it is evident that even quadruple precision is not enough. The precision is not a mere matter of accuracy of coefficients; the numerical error in computing the subsequent coefficients in the polynomial exponentially propagates compared to the current coefficient. The spectra of the Laplacians were computed using the Givens-Householder method. While the computations of spectra normally do not require beyond double-precision arithmetic, to preserve the accuracy in the computation of the specific properties of the Riemann 5 functions associated with the spectra, quadruple-precision arithmetic is required. Even the quadruple precision was not enough for this purpose for fullerenes larger than c78. Fortunately, existence of either integral eigenvalues or eigenvalues which can be readily brought into a surd form factored the required arithmetic in a natural manner, and thus we could compute such properties for even these larger cages. However, modular arithmetic computation is recommended if such properties are desired for a general large fullerene which may not exhibit such special features. In this case numbers will be divided into two or more parts, each part represented in quadruple-precision arithmetic. Subsequent arithmetic operations will be done in each part in quadrupleprecision arithmetic. In this manner octuple and higher precision could be obtained. As anticipated, the Laplacian spectra and Huckel eigenvalues are related. The Laplacian spectra are obtained by changing the signs of the Huckel eigenvalues and then adding 3 (note that the previous statement3* that the Laplacian spectra are obtained from the Hiickel spectra by adding 3 is incorrect). In the next section we report the first few 13 or more coefficients of the Laplacian polynomials of fullerenes and the complete Laplacian spectra. Some properties pertinent to the Laplacian spectra of fullerenes are also discussed. The Riemann 5 functions of fullerene Laplacians are considered. In the last section we have applied these techniques to the enumeration of the spanning trees of all fullerenes cz0-C~.
Laplacian Polynomials of Larger Fullerenes ( c 4 2 - C ~ ) Table 1 shows the Laplacian polynomials of fullerene cages As is well-known, one could have cage isomers for fullerenes larger than the buckminsterfullerene. A rule of thumb c42-cW.
J. Phys. Chem., Vol. 99, No. 17, 1995 6511
Laplacians of Fullerenes ( c 4 2 - C ~ ) TABLE 1: The Laplacian Polynomials of Fullerenes c 4 2 - C ~ power
coefficient
power
coefficient
power c 4 2
42 40 38 36
1 7686 8 625 897 3369339206
41 39 37 35
-126 -302400 -190 114698 -49 340 658 360
44 42 40 38
1 8448 10 486 410 4561761892
43 41 39 37
-132 -349272 -243 684 120 -70 696 601 568
44 42 40
1 8448 10 486 410
43 41 39
-132 -349272 -243 684 120
48 46 44 42
1 10 080 15 092 712 8014720028
47 45 43 41
-144 -457056 -386 898 312 -137 900 080 704
50 48 46 44
1 10 950 17 895 525 10426638430
49 47 45 43
-150 -518400 -480 196 506 -188 308 587 960
60 58 56 54
1 15 840 38 163 330 33 456 534 020
59 57 55 53
70 68 66 64
1 21 630 72 121 035 88758388770
69 67 65 63
76 74 72 70
1 25 536 101 155 962 148891029924
75 73 71 69
78 76 74 72
1 26 910 112 545 927 175245092090
77 75 73 71
78 76 74 72
1 26 910 112 545 927 175 245 092 090
78 76 74 72
coefficient Fullerene
34 608846305758 32 58621277389212 30 3 264 603 145 133 273 28 110680441758232686 CU Fullerene (D3h) 36 925 085 192 205 34 100812657206352 32 6415371418743782 36 925 085 192 205 CU Fullerene ( r ) 38 4561761892 36 925 085 192 199 34 100 812 657 173 460 c48 Fullerene ( 0 3 ) 40 2 010 268 769 262 38 274919207400144 36 22311461025230750 c50
power
coefficient
(03)
33 31 29 27
-6 424 278 492 216 -466703714911872 -20177058784 179966 -540776298025258076
35 33 31 35
-10372960166084 -856 875 293 066 676 -42553 190624285292 -10372960166084
37 35 33
-70 696 601 568 -10 372 960 165 444 -856 875 291 981 840
39 37 35
-25 203 174 305 196 -2633254771482 192 -168236963 168695368
41 39 37
-38091259914260 -4433689501453080 -317 855 933 618 885 970
51 49 47 45
-235276583396700 -43706364423 180840 -5 139346 156571269680 -401 204 618 556 947 060 096
Fullerene (D5h)
42 40 38
2885719641680 438 181563605608 39756356810226335
61 59 57 61
-1068650167520010 -290 106 403 633 986 960 -50 764 248 608 846 920 680 -1068650167520010
67 65 63
-2 376 308 603 041 788 -785 935 534 970 302 248 -168936582933 293997040
69 67 65
-3055352995959366 -1074879195812784432 -246354604069989666524
77 75 73 71
Buckminsterfullerene(C,) 52 14247 988261 435 -908280 36 3 408 Oii $Si 682 580 -1 252 809 912 48 500 131 066 489 699 160 -747 269 571 360 46 47681197379576857080 c70 Fullerene (D5h) -210 62 53 863 478 347 065 -1 456560 60 18 67 1 804 426 269 458 -2 799 993 018 58 4040 115 394 881 855 670 -2 362 202 763 240 62 53 863478347065 c76 Cage (p=q=O) -228 68 108831024908439 -1 872792 66 45782753 836947732 -4 291 199 832 64 12117987369167052392 -4 344 910 098 528 c78 Fullerene (C2" I) -234 70 135 793 559 628 123 -2027376 68 60693213503586078 -4 909 371 690 66 17 108036655 165529684 -5 263 998 297 384 c 7 8 Fullerene (C2" 11) -234 70 135793559628 123 -2027376 68 60 693 213 503 586 078 -4 909 371 690 66 17 108 036 655 165 529 714 -5 263 998 297 384
69 67 65
-3055352995959366 -1074879 195 812784432 -246354604069989672404
1 26 910 112 545 927 175 245 092 090
77 75 73 71
-234 -2027376 -4 909 371 690 -5 263 998 297 384
135 793 559 628 123 60 693 213 503 586 078 17 108 036 655 165 529 676
69 67 65
-3055352995959366 -1074879195 812784432 -246 354 604 069 989 664 956
78 76 74 72
1 26 910 112 545 927 175245092090
77 75 73 71
-234 -2027376 -4 909 371 690 -5 263 998 297 384
Fullerene (D3h;I Leapfrog) 70 135 793 559 628 123 68 60 693 213 503 586 078 66 17 108 036 655 165 529 652
69 67 65
-3 055 352 995 959 366 -1074879195812784432 -246 354 604 069 989 660 252
78 76 74 72
1 26 910 112 545 927 175245092090
77 75 73 71
-234 -2027376 -4 909 371 690 -5 263 998 297 384
70 68 66
Fullerene (D3h;11) 135793559628 123 60693213503586078 17 108 036 655 165 529 742
69 67 65
-3055352995959366 -1074879 195812784432 -246354604069989677892
80 78 76 74 72
1 28 320 124 871 640 205 376521180 168428716400280
79 77 75 73 71
-240 -2 190240 -5 596 810 824 -6344919460800 -3901685463005 880
70 68 66 64
79839645801202572 23 921 144 113 557 053 290 4 808 295 893 300 235 120 420 676395815 118314978929740
69 67 65 63
-1 457 357 545 117 772 160 -355444955953084343040 -59 502 169 127 886 811 844 664 -7088603414274302402378520
84 82 80 78
1 31 248 152 531 190 278 690 636 660
83 81 79 77
-252 -2541 672 -7 202 448 264 -9 087 141 097 920
Coroninic Cage (p=q=O) 76 254832539259741 74 135 207444340242612 72 45 527 710 463 562 784 116
75 73 71
90 88 86 84
1 35 910 202 328 145 428675 129750
89 87 85 83
-270 -3 136320 -10 281 948 330 -15078 891741240
-180
c78
70 68 66
c78
C 1.n . Fullerene _"" -........
c84
(.,1,,~ ) ~
CW Coroninic Cage (p=q=O) 82 80 78 76
456743012130780 283816613727938772 112526904024004916415 30229816911547751707870
81 79 77 75
-6242341191552156 -2615031372795574536 -718317310315025664184 -12100172245226760 -5952342734957494080 -1930535429884022879950 -434164860337650842173 880
Balasubramanian
6512 J. Phys. Chem., Vol. 99, No. 17,1995
POINT GROUP DQ
P0"T GROUP Ds,,
C7a D n ( l 1 )
Cia D a h ( l )
c7a C l " ( l )
C n Da
c70
CZ"(Il)
Figure 1. Structures of larger fullerenes (c42-C~) considered here. [The structures of CM, (248, and C ~ are O from ref 50.1
is to consider only fullerenes containing isolated pentagons. Even so, there are five fullerenes containing isolated pentagons for c78. The number of structures containing isolated pentagons becomes larger for higher fullerenes. Thus, no efforts were made to consider altemative structures for larger fullerenes; only
for C78 have we considered all five isomers since experimental evidence suggests the existence of more than one isomer for c78. The structures of the fullerenes considered in Table 1 have been shown before, but for the sake of completeness they are shown in Figure 1.
Laplacians of Fullerenes
J. Phys. Chem., Vol. 99, No. 17, 1995 6513
(C42-C90)
TABLE 2: The Laplacian Spectra of c 4 o - C ~ c 4 2 Fullerene with D3 Symmetry o.Oo0 OOO OOO( 1) 0.328 852 559(2) 0.389 519 367(1) 0.881 044 336(2) 3.752 482 153(2) 4.103 720 371(1) 4.383 687 911(2) 4.546 949 476(2) 1.067 037 984(2) 1.0687 323 55(1) 1.635 464 636(1) 1.765 907 349(2) 4.71 1 282 022(2) 4.775 550 298(1) 4.779 403 220(1) 5.169 402 481(1) 1.788 548 061(1) 1.888 728 653(2) 2.000 OOO 000(1) 2.370 228 543(1) 5.248 689 001(2) 5.5228 052 37(1) 5.531 681 217(2) 5.655 981 753(1) 2.433 082 883(2) 2.460 574 454(2) 2.740 643 678( 1) 3.000 OOO OOO(2)
CM Fullerene with D3h Symmetry 0.354 248 689(2) 0.763 932 023(1) 3.688 892 183(2) 4.000 000 OOO(1) 4.414 213 562(2) 4.618 033 989(6) 1.ooO000 000(4) 1.585 786 438(2) 1.697 224 362(2) 1.829 913 513(2) 4.903 211 926(1) 5.236 067 977(1) 5.302 775 638(2) 5.481 194 304(2) 2.000 000 OOo(1) 2.381 966011(6) 2.806063 434(1) 3.000000000(2) 5.645 751 311(2) CM Fullerene with T Symmetry 0.000 OOO 000( 1) 0.326 135 256(3) 0.926 672 695(3) 1.000 000 000(2) 4.618 033 989(3) 4.905 965 607(3) 5.230 844 456(3) 5.302 775 638(1) 1.628 900 925(3) 1.697 224 362(1) 1.846 205 920(3) 2.381 966 Oll(3) 5.639 857 338(3) 2.460 328 165(3) 2.866 195 165(3) 3.859 516 056(3) 4.309 378 417(3)
0.000 000 OW( 1) 0.290 724 641(1)
0.OOO 000 OOO( 1) 0.920 243 331(2) 1.671 104 339(1) 2.302 345 345(2)
0.300 614 762(2) 0.995 481 139(1) 1.758 528 384(2) 2.338 875 663(2)
Fullerene C4.5 with C3 Symmetry 0.850 943 917(2) 2.665 628 781(2) 1.625 166 177(2) 4.262 566 419(1) 2.287 818 983(1) 4.843 852 451(1) 2.656 185 030(1) 5.537 152435(1)
3.382 729 221(2) 4.462 847 044(2) 5.002 375 741(2) 5.568 860 427(2)
4.016 632 750(2) 4.038 320 876(1) 4.558 751 536(1) 4.639 464 380(2) 5.164 744 076(2) 5.310 475 362(1) 5.706 787 060(1)
0.OOO 000 000(1) 0.898 788 390(2) 1.636 146 815(1) 2.264 805 627( 1)
c.48 Fullerene with D3 Symmetry 0.293 443 810(2) 0.315 169 747(1) 0.824 905 036(2) 2.623 605 256(1) 0.916 367 720(1) 1.494 215 624(2) 1.509 575 722(1) 4.178 804 002(2) 1.722 025 267(2) 1.746 928 936(1) 2.203 328 097(2) 4.659 370 324(1) 2.315 338 335(2) 2.343 198 347(1) 2.474 114 783(2) 5.234 101 996(1)
3.064871 455(2) 4.353 262 667(1) 5.024 059 603(1) 5.521 627 556(1)
3.635 162 530(2) 4.487 033 830(2) 5.082 603 764(2) 5.612 115 488(2)
0.000 OOO 000( 1) 0.829 913 513(1) 1.533 268 196(2) 2.262 359 695(2)
0.277 410 862(2) 0.904 706 015(2) 1.728 744 613(2) 2.317 215 813(2)
0.345 605 578(1) 1.530 083 200(1) 1.767 937 195(1) 2.487 878 397(1)
0.324 869 129(1) 1.460 811 127(1) 2.157 711 952(2) 2.585 786 438(1)
c50 Fullerene with D5h Symmetry 0.760 527 039(2) 3.143 980 917(2) 1.495 726 W ( 2 ) 4.355 674 294(2) 2.222 515 749(2) 4.777 484 251(2) 2.688 892 183(1) 5.466 731 804(2)
c52 Fullerene with T Symmetry 0.279 900 316(3) 0.716 002 130(2) 0.869 795 963(3) 3.875 389 122(3) 1.386 493 872(3) 1.585 786 438(4) 2.152 705 993(3) 2.214 738 552(2) 4.957 696 373(3) 2.256 905 977(3) 2.267 949 192(1) 2.763 738 875(3) 3.318 458 897(2) Buckminsterfullerene (C,) o.Oo0 000 000( 1) 0.243 401 746(3) 0.697 224 362(5) 1.179 750 749(3) 3.381 966 Oll(3) 1.438 447 187(4) 2.000 000 000(9) 2.381 966011(5) 3.138 564 265(3) 5.OOO000000(4)
0.OOO 000 OOo(1)
0.000 000 OOO( 1) 0.585 786 438(1) 1.175 152 272(2) 1.780 979 012(2) 2.250 100 969(2) 2.470 683 420( 1)
0.186 393 497(1) 0.648 983 612(2) 1.252 587 390(2) 1.800 594 356(2) 2.267 949 192(1) 3.000 000 OOO( 1)
0.220 239 050(2) 1.000000000(1) 1.585 786 438(1) 1.905 734 256(2) 2.277 709 075(2) 3.094 330 146(2)
c70 Fullerene with D5h Symmetry 0.564 644 208(2) 3.414 213 562(1) 1.071 991 984(2) 4.320 155 827(2) 1.729 466 363(2) 4.618 033 989(2) 2.188 140 895(2) 5.124 381 997(2) 2.381 966 Oll(2) 5.645 135 126(2) 3.405 453 523(2)
0.000 OOO ooO(1) 0.505 729 697( 1) 0.639 351 737(1) 1.066 634 218(1) 1.457 768 347(1) 1.685 646 410(1) 1.879 789 696(1) 2.160 809 124(1) 2.212 062 046(1) 2.476 221 272(1)
0.168 994 061(1) 0.516 534 194(1) 0.936 224 679(1) 1.128 352 170(1) 1.502 878 409(1) 1.707 819 633(1) 2.044 147 240(1) 2.178 599 504(1) 2.256 769 715(1) 2.586 078 759( 1)
0.192417 613(1) 0.553 266 960(1) 0.968 342 412(1) 1.169 386 983(1) 1.591 554 918(1) 1.727 276 528(1) 2.097 756 460(1) 2.201 295 835(1) 2.269 378 679(1) 2.929 649 779(1)
0.216 646 703(1) 0.577 345 980(1) 0.978 150 566(1) 1.175 160 703(1) 1.659 663 237(1) 1.752 102 422(1) 2.102 001 724(1) 2.203 549 849(1) 2.361 504 952(1) 3.083 481 793(1)
0.000 OOO 000( 1) 0.501 762 616(1) 0.610 698 937(1) 0.998 646 018(1) 1.485 759 593(1) 1.606 488 450(1) 2.027 402 174(1) 2.100 422 475(1) 2.222 889 561(1) 2.484 278 411(1)
0.165 242 916(1) 0.504 307 658(1) 0.951 588 371(1) 1.086 046 012(1) 1.523 862 764(1) 1.724 132 542( 1) 2.095 521 288(1) 2.169 369 252(1) 2.251 120418(1) 2.545 368 757(1)
c76
0.000 000 000( 1) 0.172 068 143(1)
0.501 762 616(1) 0.585 411 348(1) 1.053 044 294(1) 1.502 855 394(1) 1.585 786 438(1) 1.770 809 338(1) 2.099 882 855( 1) 2.185 189 442(1) 2.459 704 667( 1)
0.519 922 034(1) 0.951 140 154(1) 1.080 307 384(1) 1.534 778 880(1) 1.619 353 763(1) 2.059 124 080(1) 2.166 525 152(1) 2.185 728 550( 1) 2.484 278 411(1)
4.139 958 517(1) 4.552 625 593(2) 5.160 623 995(2) 5.711 821 169(1)
3.477 259 996(2) 4.000 000 OOO(1) 4.214 319 743(1) 4.481 194 304(1) 4.526 171 169(2) 4.634 574 609(2) 4.848 937 765(2) 5.414 213 562(1) 5.439 543 057(2) 5.669 455 558(2) 4.414 213 562(4) 4.513 662 537(3) 4.750 800 422(2) 5.362 821 787(3) 5.580 889 185(3) 5.732 050 808(1)
4.302 775 638(5) 4.438 283 239(3) 4.618 033 989(5) 5.561 552 813(4) 5.618 033 989(3) 4.000 000 OOO(1) 4.342 923 083(1) 4.689 918 204(2) 5.137 212 074(2) 5.732 050 808(1)
4.054 747 125(2) 4.414 213 562(1) 4.787 363 764(2) 5.557 156 864(2)
4.254 684 551(2) 4.435 664 638(2) 5.000 OOO OOO(1) 5.627 472 719(2)
3.367 773 528( 1) 3.874 524 109(1) 4.241 259 150(1) 4.391 485 138(1) 4.545 303 192(1) 4.806 636 736(1) 5.096 339 895(1) 5.546 160 159(1) 5.684 566 295(1)
3.389 898 600(1) 4.133 701 213(1) 4.252 669 167(1) 4.407 940 136(1) 4.563 609 797( 1) 4.922 959 131(1) 5.117754217(1) 5.580 045 934(1) 5.722 168 879(1)
3.462 626 102(1) 4.147 307 810(1) 4.359 311 810(1) 4.445 372 402(1) 4.679 441 595(1) 4.961 845 620(1) 5.265 272 278( 1) 5.587 918 213(1) 5.754 958 839(1)
3.328 022 495(1) 3.980 914 741(1) 4.238 206 OOl(1) 4.423 103 499(1) 4.668 267 853(1) 4.931 064 561(1) 5.191 269 431(1) 5.563 655 014(1) 5.741 331 152(1)
3.521 296 746(1) 4.109 484 332(1) 4.351 874009(1) 4.440 738 042(1) 4.687 298 276(1) 4.939 140 083(1) 5.210 149 836(1) 5.596 154 676(1) 5.747 414 085(1)
3.283 355 799(1) 3.905 839 123(1) 4.182 644 324(1) 4.366 284 765(1) 4.534 231 198(1) 4.821 798 614(1) 5.019 828 064(1) 5.294 470 685(1) 5.674 865 298(1)
3.489 001 599(1) 4.015 541 052(1) 4.189 024 523(1) 4.414 213 562(1) 4.556 008 660( 1) 4.840 525 505(1) 5.080 677 908(1) 5.537 003 299(1) 5.680 715 884(1)
Cage
3.104 662 155(1) 3.867 102 633(1) 4.182 213 127(1) 4.361 498 754(1) 4.541 577 373(1) 4.763 203 706(1) 4.995 971 386(1) 5.325 958 890(1) 5.628 617 027(1)
c78 Fullerene with C2, Symmetry 0.194 238 472(1) 0.202 046 272(1) 3.206 158 979(1) 3.214 678 328(1) 0.535 686 782(1) 0.565 506 815(1) 3.905 839 123(1) 3.918 144 521(1) 0.958 494 441(1) 0.965 780 211(1) 4.179 594 294(1) 4.182 644 324(1) 1.131 128006(1) 1.131 814592(1) 4.365 942 193(1) 4.414 213 562(2) 1.568 812 974(1) 1.585 786 438(2) 4.530 826 626(1) 4.539 598 446( 1) 1.736 922 590(1) 1.809 227 626(1) 4.839 734 920(1) 4.840 817 495(1) 2.096 094 830( 1) 2.096 104 794( 1) 5.000 000 000( 1) 5.132 036 980(1) 2.183 818 634(1) 2.185 300 288(1) 5.423 319 732(1) 5.496 294 771(1) 2.365 288 481(1) 2.374 945 163(1) 5.677 460 606(1) 5.685 174 246(1) 2.893 504 008(1) 3.142 726 391(1) Second c78 Fullerene with ClvSymmetiJ 0.190 253 510(1) 0.198 171 205(1) 2.942 994 252( 1) 3.167 369 201(1) 0.549 039 217(1) 0.551 795 841(1) 3.582 021 969(1) 3.816 984 537(1) 0.952 687 375(1) 0.977 407 684(1) 4.139 099 691(1) 4.176 115 624(1) 1.090366 199(1) 1.131 128 006(1) 4.306 045 888( 1) 4.364 991 690(1) 1.566 905 650(1) 1.580412 150(1) 4.429 773 504( 1) 4.457 681 501(1) 1.673 941 086(1) 1.736 922 590(1) 4.660 786 147(1) 4.753 919 794(1) 2.096 088 912(1) 2.096 100088(1) 4.936 661 930(1) 5.000 OOO 000(1) 2.175 221 237(1) 2.180036 501(1) 5.210 149 836(1) 5.213 782 556(1) 2.251 120418(1) 2.282 192 898(1) 5.596 154 676(1) 5.644 522 241(1) 2.641 935 331(1) 2.822 176 666(1) 5.709 955 695(1) 5.743 383 902(1)
6514 J. Phys. Chem., Vol. 99,No. 17, 1995
Balasubramanian
TABLE 2 (Continued) Third C78 Fullerene with D3 Symmetry 0.501 823 597(2) 3.349 534 447(2) 0.937 202 996(2) 4.169 038 039(2) 1.427 298 263(2) 4.397 933 417(2) 1.699 545 737(1) 4.787 241 879(1) 2.118 576 795(1) 5.042 787 660(2) 2.347 126 786(2) 5.586 725 656(1) 3.120 478 519(2)
0.000 000 000(1) 0.502 449 614(1) 1.044 173 893(2) 1.534 055 147(1) 1.807 832 008(2) 2.125 596 025(2) 2.389 131 955(2)
0.158 778 992(1) 0.611 661 999(2) 1.174 216 877(1) 1.630 309 233(2) 2.000 000 000(1) 2.177 320 926(2) 2.631 689 765(1)
0.202 616 133(2) 0.930 213 439(1) 1.196 323 096(1) 1.658 735 271(1) 2.078 514 610(2) 2.181 459 191(1) 2.884 840 992(1)
0.000000000(1) 0.504 307 658(2) 0.998 646 018(2) 1.571 475 977(1) 2.000 000 OOO(1) 2.183 818 634(2) 2.484278411(1)
0.158 804432(1) 0.610 698 937(2) 1.131 128 006(1) 1.585 786438(3) 2.095 521 288(2) 2.251 120 418(1) 3.117584723(1)
Fourth C78 Fullerene with D3h Symmetry 0.202 046 272(2) 0.501 762 616(1) 3.328 022 495(2) 3.905 839 123(1) 0.940409 928(1) 0.965 780 211(2) 4.182 644 324(1) 4.351 874009(2) 1.228 785 703(1) 1.485 759 593(2) 4.530 826 626(2) 4.548 043 510(1) 1.736 922 590(1) 1.809 227 626(2) 4.907 701 309(1) 4.939 140 083(2) 2.096 104 794(2) 2.101 277 896(1) 5.423 319 732(2) 5.496 294 771(2) 2.365 288 481(2) 2.374 945 163(2) 5.741 331 152(2) 5.755 980 122(1) 3.142 726 391(2) 3.267 345 929(1)
0.000 000 000(1) 0.549 039 217(2) 1.071 791 352(1) 1.566 905 650(2) 1.736 922 590(1) 2.180 036 501(2) 2.484 278 41 l(1)
0.179 347 914(1) 0.551 795 841(2) 1.090 366 199(2) 1.585 786 438(1) 2.096 088 912(2) 2.185 189 442(2) 2.641 935 331(2)
0.190 253 510(2) 0.951 140 154(2) 1.131 128 006(1) 1.628 197 254(1) 2.099 507 861(1) 2.197 101 070(1) 2.714 914 924(1)
Fifth C78fullerene with D3h Symmetry 0.501 762 616(1) 3.489 001 599(2) 0.967476 675(1) 4.176 115 624(2) 1.534 778 880(2) 4.366 284 765(2) 1.673 941 086(2) 4.753 919 794(2) 2.166 525 152(2) 5.080 677 908(2) 2.251 120418(1) 5.596 154 676(1) 2.942 994 252(2)
3.490 665 688(1) 4.267 798 518(1) 4.448 724 955(2) 4.791 930464(2) 5.221 083 785(1) 5.593 007 594(1)
3.711 118 315(1) 4.182 644 324(1) 4.414 213 562(1) 4.821 798 614(2) 5.210 149 836(1) 5.672 790 793(1)
3.777 506 071(1) 4.296 474 323(2) 4.523 370 618(2) 4.905 478 413(1) 5.312 939 901(2) 5.726 993 967(2)
3.964 105 704(2) 4.326 082 063(1) 4.633 267 023(1) 4.984 665 886(1) 5.575 079 562(2) 5.755 592 506(1)
3.980 914 741(2) 4.414 213 562(3) 4.819 049 756(1) 5.000000 000(1) 5.583 540 717(1)
4.109 484 332(2) 4.423 103 499(2) 4.840 817 495(2) 5.210 149 836(1) 5.596 154 676(1)
3.816 984 537(2) 4.249 076 744(1) 4.457 681 501(2) 5.000 000 000(1) 5.296 233 004(1) 5.680 715 884(2)
3.905 839 123(1) 4.306 045 888(2) 4.647 194 880(1) 5.019 828 064(2) 5.565 249 213(1) 5.709 955 695(2)
Cgo Fullerene with f h symmetry 0.000 000 000(1) 0.181 885 072(3) 0.527 166 091(5) 0.918 718 793(3) 4.200 080 940(3) 4.618 033 989(8) 4.935 432 332(5) 5.198 691 244(4) 1.087 770 822(4) 1.537 401 577(5) 1.622 797 146(4) 2.000 OOO OOO(6) 5.651 093 409(4) 5.699 315 196(3) 2.381 966 Oll(8) 2.726 109 445(4) 3.713 537 935(4) 4.000 OOO 000(6)
0.000 000 000(1) 0.442 291 912(1) 0.599 354 921(1) 0.964 639 804( 1) 1.242 108 599(1) 1.501 716 442(1) 1.759 544 000(1) 2.000 000 000( 1) 2.155 944 342(1) 2.298 975 941(1) 2.43 1 952 574(1)
0.142 759 767(1) 0.443 351 099(1) 0.830 346 512(1) 0.969 362 375(1) 1.420 015 699(1) 1.552 053 149(1) 1.844 965 901(1) 2.055 841 313(1) 2.158 135 414(1) 2.337 432 660( 1) 2.502 529 918(1)
0.172 751 645(1) 0.493 653 536(1) 0.834 676 614(1) 1.111 920764(1) 1.453 411 105(1) 1.626 011 935(1) 1.846 822 803(1) 2.082 820 070(1) 2.180411 505(1) 2.344472472(1) 3.116 782 978(1)
C84 Coroninic Cage (p=q=O) 0.212 225 756(1) 3.154428 958(1) 0.579 767 336(1) 3.836 988 570(1) 0.938 488 743(1) 4.073 964 633(1) 1.124567 306(1) 4.313 415 416(1) 1.493 282 487(1) 4.439 676 354(1) 1.723 996 320(1) 4.536 2OO773(1) 1.935 057 381(1) 4.840 463 064(1) 2.088 463 461(1) 5.058 492 249(1) 2.257 111 165(1) 5.458 695 353(1) 2.370467 530(1) 5.653 648 338(1) 3.137 383 644(1)
0.000 OOO OOO( 1) 0.413 094016(1) 0.867 818 738(2) 1.338 137 479(2) 1.819 022 916(2) 2.152240935(1) 2.359 470 242(1)
0.117 246 097( 1) 0.596 146 151(2) 1.136 932 946(1) 1.497 753 351(2) 1.819 313 462(2) 2.168 540221(2) 2.405 832 129(2)
0.191 842 606(2) 0.758 554 155(1) 1.157 509 635(2) 1.667 429 053(1) 1.939 979 333(2) 2.228 316 965(2) 2.585 786 438(1)
C90 Coroninic Cage (p=q=O) 0.410 760 289(2) 3.160 601 106(2) 0.815 239 483(2) 3.936 651 472(2) 1.319 549 023(2) 4.192 718 642(2) 1.801 714 417(2) 4.503 268 754(2) 2.151 962 401(2) 4.790 008 495(2) 2.278 588 053(2) 5.377 194 031(2) 3.084 640 562(1) 5.736 200 305(2)
a
3.248 267 407(1) 3.847 335 007(1) 4.116901 851(1) 4.388 189 761(1) 4.455 604 109(1) 4.712 035 321(1) 4.846 069 138(1) 5.205 209 321(1) 5.520906057(1) 5.657 311 211(1)
3.280 983 825( 1) 4.036 973 191(1) 4.180 828 501(1) 4.388 521 662(1) 4.496 919 669(1) 4.777 754 696(1) 4.895 593 066( 1) 5.336 278 041(1) 5.565 603 124(1) 5.786 279 311(1)
3.341 629 866(1) 4.057 418 465(1) 4.222 782 190(1) 4.431 230 169(1) 4.532 039 159(1) 4.836 582 944(1) 4.954 510 014(1) 5.345 803 265(1) 5.590 788 454(1) 5.799 898 599(1)
3.234 633 135(1) 4.000 000 000(1) 4.372 130 134(1) 4.507 757 021(2) 4.892 560 842(2) 5.414 213 562(1) 5.847 759 065(1)
3.299 461 831(2) 4.052 217 258(2) 4.392 756 247( 1) 4.699 258 288(2) 4.967 579 794(2) 5.583 160 628(2)
3.697 746 549(1) 4.17 1372 438(2) 4.43 1 425 450(2) 4.765 366 865(1) 5.277 198 930(2) 5.613 338 061(2)
Numbers in parentheses are the frequencies of the corresponding eigenvalues.
The coefficient of the constant term of the Laplacian polynomial is always 0 since there exists a zero eigenvalue in the Laplacian spectrum. This can also be seen from the fact that all fullerenes possess 3.0 (not -3.0 as inadvertently stated before38)as a Huckel eigenvalue. In contrast with the characteristic polynomial which possesses a zero coefficient for An-1 (n = number of vertices), the coefficient of AnP1 in PLis always -3n, for fullerenes; this is a consequence of the fact that fullerenes are 3-regular homogeneous graphs. Table 2 shows the Laplacian spectra of the Laplacians of all fullerenes considered here. Some efforts were made to factorize the Laplacian polynomials of high-symmetric fullerenes such as buckminsterfullerene and c80 shown in Table 1 using the computed eigenvalues reported in Table 2. This was discussed for smaller fullerenes such as (22.0, c24 , C ~ Xand , C36. The factorization involves identifying those eigenvalues which can be reduced into surd forms and then solving symmetric equations for the remaining eigenvalues which cannot be expressed in surd forms. For example, as shown by the author38before, the four eigenvalues of c24(&) fullerene denoted as AI = 5.050 959,;12 = 2.698 857,;13 = 1.673 094, and A4 = 0.569 092 1
satisfy the following four symmetric equations:
+ A, + A3 + A, = 10 412,+ AlA3 + AlA4 + A2A3 + AJ, + A3A4 = 32 + + + = 38 Al
AlA2A3
AIA2A4
AIA3A4
AZA3A4
A 1A2A3A4 = 13 It was thus shown that the Laplacian polynomial of the fullerene cage is factored into
C24
pL(c24) = A{(A - 3)2 - 5}{(A - 3)2 - 2I2(A - 2){(A - 4)2 - 2}2 x (A - 3)2(A - 5)2(A4 - 10A3 32A2 - 381 13)’
+
+
The factored form is also useful in that it provides the prime factors in the coefficients, and thus it can be verified that the
J. Phys. Chem., Vol. 99,No. 17, 1995 6515
Laplacians of Fullerenes (c42-c90) coefficient of A of C24 is
132 x 4 x 72 x 2 x 142 x 32 x 52 which is the same as 2 921 536 800. The factored form of the characteristic polynomial of the buckminsterfullerene is available. Chung and Stemberg35have considered a more generalized version of the buckminsterfullerene for which different weights are given for single and double bonds. If 1 is the weight given to the single bond and t is the weight for the double bond, then these authors have obtained an elegant representation for the characteristic polynomial of the buckminsterfullerene. The factored form for such a weighted buckminsterfullerene is (x2
+
+ t - 1 1 7 ~ 3- tx2 - - t2x + 2tx - 3x + t 3 - + t - 2)5(x2+ - t2 - 1)4 + - ( t + 1)2>4{x2+ (2t + 1lx + t2 + t - 1 j 3 (x4 - 3x3 + (-2t2 + t - 1)x2 + (3t2 - 4t + 8)x + t4 t3 + t2 + 4t - 4}3(x - t - 2) - t2
x2
t2
{x2
The Laplacian polynomial of the buckyball can be readily factored. The occurrence of a 9-fold degenerate 2.0 eigenvalue and 4-fold degenerate 5.0 eigenvalues suggests that (A - 2)9(x - 5)4 should be a factor. The 5-fold degenerate pairs (4.6180339887, 2.381 9660) can be expressed as 712 f f i .This yields the factor (A2 - 7A ll)5. The %fold degenerate pairs (5.618 033 988 7, 3.381 966 011 3) can be expressed as 912 f &, yielding (A2 - 9A 19)3. Likewise, the eigenvalues 1.438 447 187 2 and 5.561 552 812 8 can be written as 712 It yielding (A2 - 72 8)4. The 5-fold generate eigenpairs 4.302 775 637 and 0.697 224 362 3 can be written as 512 f thus contributing (A2 - 5 1 3)5. This leaves us with a 3-fold quartet of eigenvalues A, = 0.243 401 746 1, A2 = 1.179 750 749, A3 = 4.438 283 239, and A4 = 3.138 564 265 1. These four eigenvalues satisfy the following symmetric equations.
+
+ +
m, m,
+
+ A2 + A, + A, = 9 + + A1Ad + + + = 25 A,A2A3 + AiA2A4 + A,A3A4 + A2A3A4 = 22 A2A3
iZIA3
4 1 4
Thus, the contribution to the Laplacian polynomials by these eigenvalues is {A4 - 9A3 25A2 - 22A 4}3. Consequently, the factorized form for the Laplacian polynomial of buckminsterfullerene is
+
A(A - 219@- 514(n2- 7~ + 1 1 ) 5 ( ~-2 9~ + 1 9 ) ~ (A2 - 7A + 8)4(A2 - 5A + 3)5 x (A4 - 9A3 25A2 - 221
+
35
54
1i5
A2
A3
A4
A1A2A3A4 = 4 Thus, they contribute the (A4 - llA3 + 35A2 - 28A + 4)3 term to the Laplacian polynomial. The 4-fold degenerate triplets (1.622797 146, 2.726 10944, 5.6510934) yield (A3 - 10A2 291 - 25)4. Likewise, the 4-fold degenerate triplets (1.087 770 82, 3.713 537 9,5.198 691 2) yield (A3 - 10A2 29A - 21)4. Finally, the 5-fold degenerate triplets (0.527 166 091, 1.537 401 577, 4.935 432 332) generated (A3 - 7A 11A2 - 4)5.Combining all of these terms we obtain the Laplacian polynomial of the CSO(Z~) fullerene as
+
+
+
A(A - 2y(A - 4y(A - 7A + 11y x
+
+
- 1 1 ~ 3 3 9 2 - 28A 4)3 (A3 - 10A2 29A - 25)4(A3- 10A2+ 29A - 21)4 x (A3 - 7iZ 11A - 4)5 The coefficient of A is thus factored into primes as ( ~ 4
+
+
34
234
58
74
118
Properties of the Computed Laplacian Eigenvalues The sum of all Laplacian eigenvalues should be the total degrees of all vertices. We thus have n
where n is the number of vertices and vi is the valency (degree) of the ith vertex, which is always 3 for fullerenes. The Laplacian spectra of fullerenes can always be arranged as 0 = A1 A 2 I1 3 ...IA,,. The eigenvector corresponding to A2 is called the Friedel vector and is a characteristic of the structure. The following results were found to be valid for all Laplacian eigenvalues of all fullerenes. n
= 12n i= 1 n
+ 4)3
Thus, the coefficient of the 1 term of the buckyball is
227
+ + + = 11 AlA2 + AlA3 + iZlA4 + A2A3 + A2A4 + A3A4 = 35 A,A2A3 + AlA2A4 + iZ1A3iZ4 + A2A3A4 = 28 A1
A3iZ4
A,A2A3A4 = 4
+
+
n
A1
1112
occurrence of integers 2 and 4 six times in the Laplacian spectra suggests that (A - 2)6(A - 4)6 should be a factor. The 8-fold degenerate (4.618 033 987, 2.381 966 011 3) pairs of eigenvalues can be expressed as 712 f 4514 surd form. This yields the (A2 - 71 1l)s factor in the Laplacian polynomial. The 3-fold degenerate eigenvalues denoted by A1 = 0.181 885 07, A2 = 0.918 718 793, A3 = 4.200 089 396, and A4 = 5.699 315 195 8 satisfy the following symmetric equations.
ig3
= 54n i= 1 n
ZA;= 258n i= 1
or 22517511982373928960000 The high symmetry of the CEO fullerene cage considered here
(Zh) facilitated factoring of the Laplacian polynomial. First, the
where a is a real number between 0 and 5. The first three results are rigorous and independent of the structure of fullerenes in
Balasubramanian
6516 J. Phys. Chem., Vol. 99, No. 17, 1995 question. These results depend only on n, the number of vertices in the fullerene. However, the last result includes a real factor a which is structure-dependent. This suggests that one needs at least the fifth and higher powers of the eigenvalue to discriminate the structures. We were able to find a straightforward proof for the first result. The eigenvalues of the adjacency matrix of regular graphs, which we shall denote by A;, satisfy
(zA:)Td = 6 823 152 i
= 6 823 120 1
Consequently, the power should be at least 8 to discriminate closely related isomers of fullerenes. Note that although we have expressed the sum of the sixth and seventh power as a function of n, the result is not valid for all fullerenes.
The Riemann 5 Functions and the Spanning Trees The Riemann 5 function, which plays an important role in the distribution of prime numbers,39is defined as
Since
il;= -Ai
+3
C(x) = 1
+ 1-" + 2-" + 3- + ...
The distribution of the primes is well-known and is given by
we have n
ZAF = C(4,+ 3)2 i= 1
I
i
i
where the product is over all primes. The corresponding Riemann 5 function for the Laplacian eigenvalues3' can be defined as n
Using
C(2x) = 1 i
we get
CA:= 12n i
i=2 where n is any complex or real variable. The motivation for this definition is the Laplace differential operation on the 1-sphere whose nth eigenvalue is A,, = n2, n = 0, 1, 2, ... Consequently, m
((2.x) = 1
The result for the sum of the cubic powers of Laplacian eigenvalues follows from the fact that the sum of the cubes of eigenvalues of regular graphs must be 0. The result for the fourth power of the Laplacian suggests that the sum of the fourth powers of Hiickel eigenvalues of all fullerene graphs must satisfy
CAY = 15n i
It is evident that the coefficient of the ilterm in all Laplacian polynomials must be
+ EA,-"
+ EAn-" n=l
The Riemann 5 functions defined for graphs are likely to be important functions of structures. It is believed that many enumerative combinational numbers pertinent to the structure are contained in the Riemann 5 function. While this is continuing to be an open topic of research, an important wellknown result can be immediately exploited. The number of spanning trees of a graph containing n vertices, K, is given by the zero of the derivative of the Riemann 5 function. That is,
n
This result can be simplified as This result was used in obtaining the prime factoring of the coefficient of il as well as in verifying the accuracies of the computed results. In the case of the fifth-power sum, the coefficient a is exactly 0 for the dodecahedral c20fdlerene, which exhibits symmetry. For the C80 fullerene the a factor was found to be 4.5. For both C40 fullerenes with D5 and T symmetry we found that the a factor is the same and equal to 3. It is thus evident that the sum of the fifth powers is not enough to discriminate the isomers of fullerenes. For c 4 0 ( T d ) and c40(D5) structures we also find that
XA: = 257 400 = 643% ZA:= 1 318 080 32952n i
i
r
n
where it may be recalled that the Laplacian eigenvalues are arranged in the ascending order 0 = I I < AZ I I3 ... II,. Consequently, the eigenvalues of the Laplacians enumerate the number of spanning trees. The enumeration of spanning trees of conjugated systems such as polycyclic aromatics has been the topic of considerable discussion in the chemical literature.41,42045The McWeeny modelu of estimating the magnetic properties of these systems requires computation of the contributions from the n-electron ring currents. This in tum reduces to finding the spanning trees. Hence, Gutman, Mallion, and others41,42,45 have investigated this problem recently. These authors have related the number of spanning trees of benzenoid structures to the characteristic polynomial of the dualist graph. That is, if D is the dualist
Laplacians of Fullerenes
J. Phys. Chem., Vol. 99, No. 17, 1995 6517
(C42-cW)
graph of the benzenoid obtained by joining the centers of the rings then
-
K(G)= PD(A" r)
TABLE 3: Number of Spanning Trees of Fullerenes
czo-cw
5 184 000 121 730 700 594 350 400 2921811200 13959271220 1699400908 800 7882437888000 41 161926159360 41289587750400 204531036514713 1008092658367 872 999762353 129232 4981371080647 104 24631728710615040 123383 185685984000 610769348 140210000 375291866372898 816000 1 136544737068261950000000 39843301563635858064622592 696 993 394 614 782 941 957 757 000 699172755 274636812638280000 695297921694344493944525028 694225492484225056067590500 700782 175001531756548800000 3497 104100935210106880000000 [85377 185 1685577883052366848001 [10434029908911740510051 176035700]
where A f f ,the dummy variable in the characteristic polynomial, should be replaced by the parameter r, which was found to be 6 for benzenoids. Several interesting results are obtained by Gutman and co-workers related to the spanning trees of benzenoids. From the previous section it is clear that the Riemann 5 functions for all fullerenes for s = -1, -2, -3, ... -7 are given by
5(0>= 1
+ 3n 5(-4) = 1 + 12n 5(-6) = 1 + 54n 5(-8) = 1 + 258n = 1 + (1272 + ~ ) n ,0 Iu I5 5(-2) = 1
I;(-lO)
The Reimann 5 functions for positive integral values come out to be real numbers. For the C 2 0 fullerene we found the Reimann 5 at -2 to be 10.133 333 33, which can be expressed as the rational number 152/15. However, for c 2 4 the value came out to be 12.769 963 369 963 37. For c 2 g this came out to be 15.448 051 948 051 948; the recurrence of the 051 948 sequence is interesting. This number can thus be expressed as 154480504042/(99999999 x I 03). Of course, the number can be simplified further. The 5(-2) value for Cm was found to be 5(C6,;-2)
39.236 204 146 730 625
From this discussion it is clear that the Reimann 5 functions of fullerenes contain many interesting structural and number theoretical results. There is considerable scope for further investigation of the Reimann 5 functions of fullerenes.
Enumeration of Spanning Trees of Fullerenes As shown in the previous section, the number of spanning trees of fullerenes is simply the exponential of the negative derivative of the Riemann 5 function at zero. This was shown to be related to the product of the Laplacian eigenvalues omitting the zero eigenvalue. It may be recalled that this is also the coefficient of the A term in the Laplacian polynomial. Thus
= coefficient of A/n This also the product of the Laplacian eigenvalues divided by n. This was used to evaluate the number of spanning trees of fullerenes C 2 0 - c ~ . The results were computed with quadruple precision arithmetic and are shown in Table 3. The number of spanning trees present in a structure also measures the complexity of the structure. It is evident from Table 3 that the complexity of fullerenes increases exponentially. In fact a plot of log K versus n is a straight line, indicating exponential complexity as a function of n. The number of spanning trees of several fullerenes can also be expressed in prime-factored form. For example, the number of spanning trees of the dodecahedral (3.0 is 5 184 000 or 29 x 34 x 53. The number of spanning trees of c 2 4 is 121 730 700 or 22 x 32 x 52 x 74 x 13*. The prime factoring for the c 2 8 cage is 28 x 52 x 73 x 113. Likewise the prime factoring for the number of spanning trees of C 3 6 is 220 x 33 x 52 x 74. K
no. of spanning trees
fullerene
The number of spanning trees of the buckminsterfullerene has received considerable attention. Brown et al.42have used a theorem of Gutman et aL4' discussed before. This calls for the construction of the dual graph of the carbon cage. In fact Brown et have devoted an entire paper for this topic of enumerating the spanning trees of buckminsterfullerene. They obtained the complete geometric dual of buckminsterfullerene first. Then they used the modulo arithmetic and the Chinese remainder theorem to compute the cofactor of the equicofactorial geometric dual matrix. The final number they compute for the buckminsterfullerene using this procedure is 375291866372898816000 As seen from Table 3, our computed result for buckminsterfullerene using an entirely different technique derived from the zero of the derivative of the Riemann 5 function exactly agrees with this number to every digit. The prime factors reported by these authors can also be directly verified from the A coefficient of the Laplacian polynomial of the buckminsterfullerene, which was shown to be in a previous section prime-factorable as 237 54 1 i 5 ig3 Since the number of spanning trees is given by this divided by 60, the prime factoring we obtain for the number of spanning trees is 235 34 53 1 i 5 1g3 which agrees exactly with the result of Brown et al.42 The number of spanning trees of c 8 0 shown in Table 3 can also be factored using the factored form of the Laplacian polynomial of C g o that we derived before. Simply dividing the coefficient of the 2 term of Cgo by 80, we obtain the prime factors in the number of spanning trees of Cgo as 230
34
s7
74
ii8
Conclusion We considered the Laplacians of fullerene cages C 4 2 - c W . Both the Laplacian polynomials and the spectra of Laplacians
Balasubramanian
6518 J. Phys. Chem., Vol. 99, No. 17, 1995 were obtained for all these fullerene cages. The spectra of Laplacians were found to generate interesting number-theoretic results for fullerenes. It was shown that at least eighth powers of the spectra are required to discriminate isomers of fullerene cages. The Laplacian polynomials of several fullerenes were brought into factored forms. The Riemann 5 functions which play a key role in number theory (that is, in determining the distribution of prime numbers) were introduced for the Laplacian spectra of fullerene cages. An exponential function of the zero of the derivativeof the Riemann 5 function was used to compute the number of spanning trees of fullerenes c20-C~. Our computed result for the number of spanning trees of buckminsterfullerene agreed exactly with the result of Brown et al.$* who used the complete geometric dual of the cage together with the modulo arithmetic and Chinese remainder theorem to determine the same number. We believe that there is considerable scope for further investigation of the properties of the Riemann 5 functions of fullerenes. The authoflg has recently obtained analytical expressions for the coefficients of Laplacian polynomials. He has also enumerated the coefficients of the A* term as well as spanning trees of all fullerenes C2o-CW, some of which were not considered here.
Acknowledgment. This research was supported by the National Science Foundation Grant CHE 9417459. References and Notes (1) Haddon, R. C. Acc. Chem. Res. 1989, 21, 243. (2) Kroto, H. W. Nature 1981, 329, 529. Kroto, H. W. Comput. Math. Appl. 1989, 17, 417. Kroto, H. W.; Allaf, A. W.; Balm, S. P. Chem. Rev. 1991, 91, 1213. (3) Duncan, M. A.; Rouvray, D. H. Sci. Am. 1990, 260, 110. (4) Chibante. L. P. F.: Smallev. < . R. E. ComDlete Buckminsterfullerene Bibli’ography; 1992. ( 5 ) Kroto, H.: Heath. W. J. R.: O’Brian. S. C.; Curl. R. F.: Smallev, R. E. Nature 1985, 318, 162. (6) Haufler, R. E.; Conceicao, J.; Chibante, L. P. F.; Chai, Y.; Byme, N. E.; Flangan; Haley, M. M.; O’Brian, S. C.; Pan, C.; Billups, W. E.; Ciufolini, M. A.; Hauge, R. H.; Margrave, J. L.; Wilson, L. J.; Curl, R. F.; Smalley, R. E. J. Phys. Chem. 1990, 94, 8634. (7) Katschmer, W.; Fositropoulos, K.; Huffman, D. R. Chem. Phys. Lett. 1990, 170, 167. Kratschmer, W.; Lamb, L. D.; Fositropoulos, K.; Huffman, D. R. Nature 1990, 347, 354. Frum, C. I.; Ergleman, R., Jr.; Hedderich, H. G.; Bemath, P. F.; Lamb, L. D.; Huffman, D. R. C(um. Phys. Lett. 1991, 176, 504. (8) Haufler, R. E.; Chai, Y.; Chibante, L. P. F.; Fraelich, M. R.; Weisman, R. B.; Curl, R. F.; Smalley, R. E. J. Chem. Phys. 1991,95,2197. (9) Leach, S.; Vervloet, M.; Despres, A.; Breheret, E.; Hare, J. P.; Dennis, T. J.; Kroto, H. W.; Taylor, R.; Walter, D. R. M. Chem. Phys. 1992, 160, 431. (10) Terazima, M.; Hirota, N.; Shinohara, H.; Saito, Y. Chem. Phys. Lett. 1992, 195, 333. (11) Williams, R. M.; Verhoeven, J. M. Chem. Phys. Lett. 1992, 94, 446. (12) Lane, P. A.; Swanson, L. W.; Ni, Q.X.; Shinar, J.; Engel, J. P.; Barton, T. J.; Jones, L. Phys. Rev. Lett. 1992, 68, 887. (13) Bethune, D. S.; Meijer, G.; Tang, W. C.; Rosen, H. J. Chem. Phys. Lett. 1990, 174, 219.
(14) Gasyna, 2.; Schatz, P. N.; Hare, J. P.; Dennis, T. J.; Kroto, H. W.; Taylor, R.; Walton, D. R. W. Chem. Phys. Lett. 1991, 183, 283. (15) Hoinkis, M.; Yannoni, C. S.; Bethune, D. S.; Salem, J. R.; Johsnon, R. D.; Crowder, M. S.; de Vries, M. S. Chem. Phys. Lett. 1992, 198, 461. (16) Yabana, K.; Bertsch, G. F. Chem. Phys. Lett. 1992, 197, 32. (17) Hino, S.; Matsumoto, K.; Hasegawa, S.; Inokuchi, H.; Morikawa, T.; Takahashi, T.; Seki, K.; Kikuchi, K.; Suzuki, S.; Ikemoto, I.; Achiba, Y.Chem. Phys. k t t . 1992, 197, 38. (18) Diederich, F.; Whetten, R. L.; Thilgen, C.; Ettl, R.; Chao, I.; Alvarez, M. M. Science 1991,254, 1768. (19) Saito, S.; Sawada, S.; Hamada, N. Phys. Rev. B 1992, 45, 845. (20) Fowler, P. W. J. Chem. Soc., Faraday Trans. 1991, 87, 1965. (21) Saito, S.; Sawada, S. Chem. Phys. Lett. 1992, 198, 466. (22) Schmalz, T. G.; Klein, D. J.; Liu, X. Mater. Res. SOC.Symp. Proc. 1992,270, 129. (23) Ettl, R.; Chao, I.; Diederich, F.; Whetten, R. L. Nature 1991, 353, 149. (24) Wakabayashi, T.; Shivomaru, H.; Kikuchi, K. K.; Achiba, Y. Chem. Phys. Lett. 1993, 201, 470. (25) Klein, D. J.; Schmalz, R. G.; Hite, T. G.; Seitz, W. A. J. Am. Chem. SOC. 1986, 108, 1301. Klein, D. J.; Seitz, W. A.: Schmalz, T. G. Nature 1986, 323, 703. (26) Colpa, J. P.; Temme, F. P. Chem. Phys. 1990, 154,97. Z. Phys. D 1991, 23, 187. Temme, F. P.; Mitchell, C. J.; Krishnan, M. Mol. Phy., in press, Temme, F. P. Chem. Phys. Lett. 1992, 200, 534. (27) Liu, X. Y.; Schmaltz, T. G.; Klein, D. J. Chem Phys. Lett. 1992, 188, 550. On, 0.; D’Mello, M. Chem. Phys. Lett. 1992, 197, 49. Fowler, P. W.; Balten, R. C.; Manopoulos, D. E. J. Chem. Soc., Faraday. Trans. 1991,87,3103. Manopoulos, D. E.; Woodall, D. R.; Fowler, P. W. J. Chem. Soc., Faraday Trans. 1992, 88, 2427. (28) Dias, J. R. Molecular Orbitlas Using Chemical Graph Theory; Springer-Verlag: Berlin, 1993; p 113; Chem Phys. Lett. 1991, 185, 10; J. Chem. Educ. 1989, 66, 1012. (29) Balasubramanian, K. Chem. Phys. Lett. 1993,210, 153; Polycyclic Aromat. Comp. 1993, 3, 247; Chem. Phys. Lett. 1992, 198, 577; Chem. Phys. Lett. 1991, 183, 292. (30) Zhang, H. X.; Balasubramanian, K. J. Phys. Chem. 1993.97, 10341. (31) Balasubramanian, K.; Liu, X. J. Comput. Chem. 1988, 9, 406. (32) Balasubramanian, K. Chem. Phys. Lett. 1990, 175, 273. (33) Mems, R. A Survey of Graph Laplacians, ICMS Workshop of Algebraic Graph Theory at the University of Edinburgh, July 1993. (34) Mems, R. Linear Multilinear Algebra 1987, 22, 115. (35) Chung, F. R. K.; Stemberg, S. J. Graph Theory 1992, 16, 605. (36) TrinajstiC , N.; BabiC , D.; NikoliC , S.; PlaGsiC , D.; AmiC, D.; MihaliC, Z. J. Chem. In$ Comput. Sci., 1994, 34, 368. (37) Mohar, B. Discrete Math. 1992, 109, 171. (38) Balasubramanian, K. Chem. Phys. Lett. 1994, 224, 325. (39) Hardy, G. H. Ramanujan: Twelve lectures on subjects suggested by his life and work; Chelsea Publishers: New York, 1959. (40) Kanigel, R. The Man Who Knew Infiniry; Maxwell International: New York, 1991. (41) Gutman, I.; Mallion. R. B. Z. Natur. A 1993, 489, 1026. (42) Brown, T. J. N.; Mallion, R. B.; Pollak, P.; de Castro, B. R. M.; Gomes, J. A. N. F. J. Comput. Chem. 1991, 12, 1118. (43) Kirchhoff, G. R. Annul. Phys. Chem. 1847, 72, 497. (44) McWeeny, R. Mol. Phys. 1958, I , 311. (45) Gutman, I.; Mallion, R. G.; Essam, J. W. Mol. Phys. 1983, 50, 859. (46) Balasubramanian, K. J. Comput. Chem. 1984,3,357; 1988,6,656; 1990, 11, 829. (47) Balasubramanin, K. Chem Rev. 1985, 85, 599. (48) Kirby, E. C.; Mallion, R.; Pollak, P. Mol. Phys. 1994, 83, 599. (49) Balasubramanin, K. Mol. Phys., in press. (50) Boo, W. 0. J. J. Chem. Educ. 1992, 69, 605.
JF’9500246
